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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07

2020-03-19T03:53:22Z

MingfengQiu:

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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07/Solution 1

2020-03-19T03:52:44Z

MingfengQiu:

By definition,

<math>
\begin{align}
f'(2) &= \lim_{x\to 2}\frac{f(x) - f(2)}{x - 2} \\
&= \lim_{x\to 2}\frac{\frac{1}{1-2x} - \frac{1}{1-4}}{x - 2} \\
&= \lim_{x\to 2}\frac{\frac{1}{1-2x} + \frac{1}{3}}{x - 2} \\
&= \lim_{x\to 2}\left( \frac{1}{x-2}\frac{3 + (1 - 2x)}{3(1-2x)} \right) \\
&= \lim_{x\to 2}\left( \frac{1}{x-2}\frac{-2(x - 2)}{3(1-2x)} \right) \\
&= \lim_{x\to 2} \frac{-2}{3(1 - 2x)} \\
&= \frac{-2}{3\cdot(-3)} \\
&= \frac{2}{9}.
\end{align}
</math>

Alternatively,

<math>
\begin{align}
f'(2) &= \lim_{h\to 0}\frac{f(2+h) - f(2)}{h} \\
&= \lim_{h\to 0}\frac{\frac{1}{1-4-2h} - \frac{1}{1-4}}{h} \\
&= \lim_{h\to 0}\frac{\frac{1}{-3-2h} + \frac{1}{3}}{h} \\
&= \lim_{h\to 0}\left( \frac{1}{h}\frac{3 - 3 - 2h}{3(-3-2h)} \right) \\
&= \lim_{h\to 0}\left( \frac{1}{h}\frac{- 2h}{3(-3-2h)} \right) \\
&= \lim_{h\to 0}\frac{2}{3(3 + 2h)} \\
&= \frac{2}{9}.
\end{align}
</math>

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07/Solution 1

2020-03-19T03:33:45Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07/Solution 1

2020-03-19T03:24:57Z

MingfengQiu: Created page with "By definition <math>f'(2) = \lim_{x\to 2}\frac{f(x) - f(2)}{x - 2} = 3</math>"

By definition

<math>f'(2) = \lim_{x\to 2}\frac{f(x) - f(2)}{x - 2}
= 3</math>

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07/Hint 1

2020-03-19T03:23:04Z

MingfengQiu: Created page with "The derivative is a limit."

The derivative is a limit.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07/Statement

2020-03-19T03:22:44Z

MingfengQiu: Created page with "Use the '''definition of the derivative''' to calculate <math>f'(2)</math> for <math>f(x) = \frac{1}{1 - 2x}</math>."

Use the '''definition of the derivative''' to calculate <math>f'(2)</math> for <math>f(x) = \frac{1}{1 - 2x}</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 07

2020-03-19T03:20:32Z

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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06

2020-03-19T03:19:22Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06/Hint 1

2020-03-19T03:18:55Z

MingfengQiu:

The definition of continuity relates the value of the function at a point to a limit.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06/Hint 1

2020-03-19T03:18:27Z

MingfengQiu:

The definition relates the value of the function at a point to a limit.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06/Solution 1

2020-03-19T03:17:49Z

MingfengQiu: Created page with "<math>\lim_{x\to a} f(x) = f(a)</math>."

<math>\lim_{x\to a} f(x) = f(a)</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06/Hint 1

2020-03-19T03:17:08Z

MingfengQiu: Created page with "Just refer to the definition. No tricky business here."

Just refer to the definition. No tricky business here.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06/Statement

2020-03-19T03:16:31Z

MingfengQiu: Created page with "Write the definition of a function <math>f(x)</math> being continuous at a point <math>x = a</math> in its domain."

Write the definition of a function <math>f(x)</math> being continuous at a point <math>x = a</math> in its domain.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 06

2020-03-19T03:15:15Z

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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-18T04:41:11Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Solution 1

2020-03-18T04:37:33Z

MingfengQiu: Created page with "The slope of the tangent line to the curve <math>y = f(x)</math> is equal to <math>f'(x)</math>. <math>f'(10) = \frac{1}{(1+10)^2} = \frac{1}{121}</math>. Using the slope-poi..."

The slope of the tangent line to the curve <math>y = f(x)</math> is equal to <math>f'(x)</math>.

<math>f'(10) = \frac{1}{(1+10)^2} = \frac{1}{121}</math>. Using the slope-point form, the tangent line equation is

<math>y - \frac{10}{11} = \frac{1}{121}\left(x - 10\right)</math>,

i.e.

<math>y = \frac{1}{121}x + \frac{100}{11}</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Hint 1

2020-03-18T04:29:14Z

MingfengQiu: Created page with "For a curve described by a function, at any point, the slope of the tangent line is equal to the derivative of the function."

For a curve described by a function, at any point, the slope of the tangent line is equal to the derivative of the function.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Statement

2020-03-18T04:20:49Z

MingfengQiu: Created page with "For this question, let <math>f(x) = 1 - (1+x)^{-1}</math> You may use the following facts without proving them: * <math>f(x) = \frac{x}{1+x}</math> * <math>f(x) = \frac{1}{(..."

For this question, let

<math>f(x) = 1 - (1+x)^{-1}</math>

You may use the following facts without proving them:
* <math>f(x) = \frac{x}{1+x}</math>
* <math>f(x) = \frac{1}{(1+x)^2}</math>
* <math>f(x) = \frac{-2}{(1+x)^3}</math>

(a) Give an equation of the tangent line to <math>y = f(x)</math> at the point <math>\left( 10, \frac{10}{11} \right)</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-18T04:18:40Z

MingfengQiu: Replaced content with "  <!-- second letter is..."

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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (d)

2020-03-18T04:18:01Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (d)/Solution 1

2020-03-18T04:15:20Z

MingfengQiu: Created page with "By definition, the average rate of change from <math>x = 0</math> to <math>x = 10</math> is <math>\frac{f(10) - f(0)}{10 - 0} = \frac{1}{10}\left( \frac{10}{11} - 0 \right) =..."

By definition, the average rate of change from <math>x = 0</math> to <math>x = 10</math> is

<math>\frac{f(10) - f(0)}{10 - 0} = \frac{1}{10}\left( \frac{10}{11} - 0 \right) = \frac{1}{11}</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (d)/Hint 1

2020-03-18T04:11:35Z

MingfengQiu: Created page with "Use the definition of the average rate of change of a function. Remember to simplify your answer."

Use the definition of the average rate of change of a function. Remember to simplify your answer.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (d)/Statement

2020-03-18T04:10:42Z

For this question, let

<math>f(x) = 1 - (1+x)^{-1}</math>

You may use the following facts without proving them:
* <math>f(x) = \frac{x}{1+x}</math>
* <math>f(x) = \frac{1}{(1+x)^2}</math>
* <math>f(x) = \frac{-2}{(1+x)^3}</math>

(d) Find the average rate of change of <math>f(x)</math> from <math>x = 0</math> to <math>x = 10</math>. Simplify your answer.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (d)

2020-03-18T04:09:17Z

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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (b)/Solution 1

2020-03-18T04:08:23Z

MingfengQiu:

<math>f''(10) = \frac{-2}{(1+10)^3} = \frac{-2}{11^3} < 0.</math>

Therefore, the tangent line as a linear approximation is an <i>overestimate</i> of <math>f(x)</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (c)

2020-03-18T04:05:20Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (c)/Solution 1

2020-03-18T04:04:09Z

MingfengQiu:

Recall that the template for a Hill function is

<math>\frac{Ax^n}{a^n + x^n}</math>.

The half-maximal activation level is obtained when <math>x = a</math>. In our case here, the half-max is reached at <math>x = 1</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (c)/Solution 1

2020-03-18T04:03:43Z

MingfengQiu: Created page with "Recall that the template for a Hill function is <math>\frac{Ax^n}{a^n + x^n}<\math>. The half-maximal activation level is obtained when <math>x = a</math>. In our case here,..."

Recall that the template for a Hill function is

<math>\frac{Ax^n}{a^n + x^n}<\math>.

The half-maximal activation level is obtained when <math>x = a</math>. In our case here, the half-max is reached at <math>x = 1</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (c)/Hint 1

2020-03-18T03:58:40Z

MingfengQiu: Created page with "Use the definition of half-max of a Hill function. Note that the current problem asks for the value of <math>x</math> that gives the half-max."

Use the definition of half-max of a Hill function. Note that the current problem asks for the value of <math>x</math> that gives the half-max.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (c)/Statement

2020-03-18T03:56:06Z

MingfengQiu: Created page with "For this question, let <math>f(x) = 1 - (1 + x)^{-1}</math> You may use the following facts without proving them: * <math>f(x) = \frac{x}{1+x}</math> * <math>f(x) = \frac{1}..."

For this question, let

<math>f(x) = 1 - (1 + x)^{-1}</math>

You may use the following facts without proving them:
* <math>f(x) = \frac{x}{1+x}</math>
* <math>f(x) = \frac{1}{(1+x)^2}</math>
* <math>f(x) = \frac{-2}{(1+x)^3}</math>

(c) Note <math>f(x)</math> is a Hill function. What value of <math>x</math> gives the half-maximal activation level?

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (c)

2020-03-18T03:54:07Z

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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (b)

2020-03-18T00:17:40Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (b)/Solution 1

2020-03-18T00:16:38Z

MingfengQiu: Created page with "<math>f''(10) = \frac{-2}{(1+10)^3} = \frac{-2}{11^3} < 0.</math> Therefore, the tangent line as a linear approximation is an overestimate of <math>f(x)</math>."

<math>f''(10) = \frac{-2}{(1+10)^3} = \frac{-2}{11^3} < 0.</math>

Therefore, the tangent line as a linear approximation is an overestimate of <math>f(x)</math>.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (b)/Hint 1

2020-03-18T00:13:08Z

MingfengQiu: Created page with "Think about the concavity of <math>f(x)</math>. For example, if it is concave up, its graph will lie on top of its tangent line."

Think about the concavity of <math>f(x)</math>. For example, if it is concave up, its graph will lie on top of its tangent line.

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (b)/Statement

2020-03-18T00:10:24Z

For this question, let

<math>f(x) = 1 - (1+x)^{-1}</math>

You may use the following facts without proving them:
* <math>f(x) = \frac{x}{1+x}</math>
* <math>f(x) = \frac{1}{(1+x)^2}</math>
* <math>f(x) = \frac{-2}{(1+x)^3}</math>
(b) Does the function from (a) give an overestimate or an underestimate of <math>f(x)</math> for values of <math>x</math> close to (but different from) 10? Why?

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (b)

2020-03-18T00:00:27Z

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User:MingfengQiu

2020-03-17T23:50:41Z

MingfengQiu: Created page with "Test test test"

Test test test

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:42:37Z

MingfengQiu:

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
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  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
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=== Question 05 (a) <font size="1"></font>===
|-
| style="padding:1em" |
For this question, let<span></span><math>f(x) = 1-(1+x)^{-1}.</math><span></span>You may use the following facts without proving them:<span></span>
* <span></span><math>f(x)=\frac{x}{1+x}</math><span></span>
* <span></span><math>f'(x)=\frac{1}{(1+x)^2}</math><span></span>
* <span></span><math>f''(x)=\frac{-2}{(1+x)^3}</math><span></span>
(a) Give an equation of the tangent line to <math>y = f(x)</math> at the point <math>\left( 10,\frac{10}{11} \right)</math>.
|}<span></span><span></span><span></span><span></span><span></span><span></span><span></span>
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=== Hint <font size="1"></font>===
|-
| style="padding:1em" |
For a curve described by a function, at any point, the slope of its tangent line is equal to the derivative of that function, if the derivative exists.
|}
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=== Solution <font size="1"></font>===
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The slope of the tangent line at the point <math>\left(10,\frac{10}{11}\right)</math>is <math>f'(10) = \frac{1}{(1+10)^2} = \frac{1}{121}</math>. Using the point-slope form, the equation of the tangent line is

<math>y-\frac{10}{11} = \frac{1}{121}(x - 10),</math>

that is

<math>y = \frac{1}{121}(x - 10) + \frac{10}{11} = \frac{1}{121}x + \frac{100}{121}</math>.
|}
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Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:36:23Z

MingfengQiu: /* Solution */

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
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Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.</div>

<span></span>
  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
noresultsheader=\n
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</dpl>
<span></span>
{| class="collapsible wikitable " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Question 05 (a) <font size="1"></font>===
|-
| style="padding:1em" |
For this question, let<span></span><math>f(x) = 1-(1+x)^{-1}.</math><span></span>You may use the following facts without proving them:<span></span>
* <span></span><math>f(x)=\frac{x}{1+x}</math><span></span>
* <span></span><math>f'(x)=\frac{1}{(1+x)^2}</math><span></span>
* <span></span><math>f''(x)=\frac{-2}{(1+x)^3}</math><span></span>
(a) Give an equation of the tangent line to <math>y = f(x)</math> at the point <math>\left( 10,\frac{10}{11} \right)</math>.
|}<span></span><span></span><span></span><span></span><span></span><span></span><span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''Make sure you understand the problem fully:''' What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
|-
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''If you are stuck''', check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
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</dpl>
<span></span>
{| class="collapsible wikitable collapsed " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |

=== Hint <font size="1"></font>===
|-
| style="padding:1em" |
For a curve described by a function, at any point, the slope of its tangent line is equal to the derivative of that function, if the derivative exists.
|}
<span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
* '''If you are stuck on a problem:''' Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
* '''If you want to check your work:''' Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br>
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed solutionMER" width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Solution <font size="1"></font>===
|-
| style="padding:1em" |<div style="color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;"><span class="plainlinks">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? [https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578 Let us know here].<br>
Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>
try try
|}
<span></span><div class="mw-customtoggle-11" style="cursor:pointer;background-color:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:center"> Click here for similar questions </div><span></span><div class="mw-collapsible mw-collapsed" style=""><div class="mw-collapsible-content" style="">
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</div></div></div><span></span><div class="noprint" style="float:none; border:5px solid #7491A3;width:280px; padding:10px;">
{| cellspacing="0"
| [[File:Bulbgraph.png|none|31x31px|lightbulb image]]
| style="padding-left:20px;padding-right: 20px;background-color : #FFFFCC;" | '''<span class="plainlinks">[http://www.math.ubc.ca/~MLC/ Math Learning Centre]</span>'''
* A space to study math together.
* Free math graduate and undergraduate TA support.
* Mon - Fri: 11 am - 5 pm, <span class="plainlinks">[http://www.maps.ubc.ca/PROD/index_detail.php?show=y,n,n,n,n,y&bldg2Search=n&locat1=308 LSK 301&302].</span>
'''Private tutor'''
* We can help to <span class="plainlinks">[https://docs.google.com/forms/d/1nV32JI4R3Mp96lYtLIpNEE-LB_T-fEHz-8lfAZXd430/viewform find a private tutor].</span>
|}
</div><span></span>

{{DISPLAYTITLE:Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)}}

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:33:00Z

MingfengQiu: /* Question 05 (a) */

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
[[File:Men_at_work_sign.gif|left|120x120px]]
<span> </span><div style="width:50%; color:red;"><br>
<br>
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.</div>

<span></span>
  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Question 05 (a) <font size="1"></font>===
|-
| style="padding:1em" |
For this question, let<span></span><math>f(x) = 1-(1+x)^{-1}.</math><span></span>You may use the following facts without proving them:<span></span>
* <span></span><math>f(x)=\frac{x}{1+x}</math><span></span>
* <span></span><math>f'(x)=\frac{1}{(1+x)^2}</math><span></span>
* <span></span><math>f''(x)=\frac{-2}{(1+x)^3}</math><span></span>
(a) Give an equation of the tangent line to <math>y = f(x)</math> at the point <math>\left( 10,\frac{10}{11} \right)</math>.
|}<span></span><span></span><span></span><span></span><span></span><span></span><span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''Make sure you understand the problem fully:''' What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
|-
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''If you are stuck''', check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |

=== Hint <font size="1"></font>===
|-
| style="padding:1em" |
For a curve described by a function, at any point, the slope of its tangent line is equal to the derivative of that function, if the derivative exists.
|}
<span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
* '''If you are stuck on a problem:''' Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
* '''If you want to check your work:''' Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br>
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed solutionMER" width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Solution <font size="1"></font>===
|-
| style="padding:1em" |<div style="color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;"><span class="plainlinks">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? [https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578 Let us know here].<br>
Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Solution 1]]
|}
<span></span><div class="mw-customtoggle-11" style="cursor:pointer;background-color:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:center"> Click here for similar questions </div><span></span><div class="mw-collapsible mw-collapsed" style=""><div class="mw-collapsible-content" style="">
__NOTOC__
<div id="MER-tag">
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</div></div></div><span></span><div class="noprint" style="float:none; border:5px solid #7491A3;width:280px; padding:10px;">
{| cellspacing="0"
| [[File:Bulbgraph.png|none|31x31px|lightbulb image]]
| style="padding-left:20px;padding-right: 20px;background-color : #FFFFCC;" | '''<span class="plainlinks">[http://www.math.ubc.ca/~MLC/ Math Learning Centre]</span>'''
* A space to study math together.
* Free math graduate and undergraduate TA support.
* Mon - Fri: 11 am - 5 pm, <span class="plainlinks">[http://www.maps.ubc.ca/PROD/index_detail.php?show=y,n,n,n,n,y&bldg2Search=n&locat1=308 LSK 301&302].</span>
'''Private tutor'''
* We can help to <span class="plainlinks">[https://docs.google.com/forms/d/1nV32JI4R3Mp96lYtLIpNEE-LB_T-fEHz-8lfAZXd430/viewform find a private tutor].</span>
|}
</div><span></span>

{{DISPLAYTITLE:Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)}}

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:31:01Z

MingfengQiu: /* Hint */

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
[[File:Men_at_work_sign.gif|left|120x120px]]
<span> </span><div style="width:50%; color:red;"><br>
<br>
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.</div>

<span></span>
  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Question 05 (a) <font size="1"></font>===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Statement]]
|}
<span></span>For this question, let

<span></span><math>f(x) = 1-(1+x)^{-1}.</math>

<span></span>You may use the following facts without proving them:
* <span></span><math>f(x)=\frac{x}{1+x}</math>
* <span></span><math>f'(x)=\frac{1}{(1+x)^2}</math>
* <span></span><math>f''(x)=\frac{-2}{(1+x)^3}</math>
<span></span>(a) Give an equation of the tangent line to <math>y = f(x)</math> at the point <math>\left( 10,\frac{10}{11} \right)</math>.
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''Make sure you understand the problem fully:''' What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
|-
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''If you are stuck''', check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |

=== Hint <font size="1"></font>===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Hint 1]]
|}
<span></span>For a curve described by a function, at any point, the slope of its tangent line is equal to the derivative of that function, if the derivative exists.
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
* '''If you are stuck on a problem:''' Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
* '''If you want to check your work:''' Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br>
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed solutionMER" width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Solution <font size="1"></font>===
|-
| style="padding:1em" |<div style="color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;"><span class="plainlinks">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? [https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578 Let us know here].<br>
Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Solution 1]]
|}
<span></span><div class="mw-customtoggle-11" style="cursor:pointer;background-color:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:center"> Click here for similar questions </div><span></span><div class="mw-collapsible mw-collapsed" style=""><div class="mw-collapsible-content" style="">
__NOTOC__
<div id="MER-tag">
Extension:DynamicPageList (DPL), version 2.3.0 : Warning: No results.
</div></div></div><span></span><div class="noprint" style="float:none; border:5px solid #7491A3;width:280px; padding:10px;">
{| cellspacing="0"
| [[File:Bulbgraph.png|none|31x31px|lightbulb image]]
| style="padding-left:20px;padding-right: 20px;background-color : #FFFFCC;" | '''<span class="plainlinks">[http://www.math.ubc.ca/~MLC/ Math Learning Centre]</span>'''
* A space to study math together.
* Free math graduate and undergraduate TA support.
* Mon - Fri: 11 am - 5 pm, <span class="plainlinks">[http://www.maps.ubc.ca/PROD/index_detail.php?show=y,n,n,n,n,y&bldg2Search=n&locat1=308 LSK 301&302].</span>
'''Private tutor'''
* We can help to <span class="plainlinks">[https://docs.google.com/forms/d/1nV32JI4R3Mp96lYtLIpNEE-LB_T-fEHz-8lfAZXd430/viewform find a private tutor].</span>
|}
</div><span></span>

{{DISPLAYTITLE:Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)}}

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:28:49Z

MingfengQiu: /* Question 05 (a) */

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
[[File:Men_at_work_sign.gif|left|120x120px]]
<span> </span><div style="width:50%; color:red;"><br>
<br>
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.</div>

<span></span>
  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
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<span></span>
{| class="collapsible wikitable " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Question 05 (a) <font size="1"></font>===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Statement]]
|}
<span></span>For this question, let

<span></span><math>f(x) = 1-(1+x)^{-1}.</math>

<span></span>You may use the following facts without proving them:
* <span></span><math>f(x)=\frac{x}{1+x}</math>
* <span></span><math>f'(x)=\frac{1}{(1+x)^2}</math>
* <span></span><math>f''(x)=\frac{-2}{(1+x)^3}</math>
<span></span>(a) Give an equation of the tangent line to <math>y = f(x)</math> at the point <math>\left( 10,\frac{10}{11} \right)</math>.
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''Make sure you understand the problem fully:''' What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
|-
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''If you are stuck''', check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|}
<span></span>
<dpl>
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<span></span>
{| class="collapsible wikitable collapsed " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |

=== Hint <font size="1"></font>===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Hint 1]]
|}
<span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
* '''If you are stuck on a problem:''' Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
* '''If you want to check your work:''' Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br>
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
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{| class="collapsible wikitable collapsed solutionMER" width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Solution <font size="1"></font>===
|-
| style="padding:1em" |<div style="color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;"><span class="plainlinks">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? [https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578 Let us know here].<br>
Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Solution 1]]
|}
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{| cellspacing="0"
| [[File:Bulbgraph.png|none|31x31px|lightbulb image]]
| style="padding-left:20px;padding-right: 20px;background-color : #FFFFCC;" | '''<span class="plainlinks">[http://www.math.ubc.ca/~MLC/ Math Learning Centre]</span>'''
* A space to study math together.
* Free math graduate and undergraduate TA support.
* Mon - Fri: 11 am - 5 pm, <span class="plainlinks">[http://www.maps.ubc.ca/PROD/index_detail.php?show=y,n,n,n,n,y&bldg2Search=n&locat1=308 LSK 301&302].</span>
'''Private tutor'''
* We can help to <span class="plainlinks">[https://docs.google.com/forms/d/1nV32JI4R3Mp96lYtLIpNEE-LB_T-fEHz-8lfAZXd430/viewform find a private tutor].</span>
|}
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{{DISPLAYTITLE:Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)}}

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:26:36Z

MingfengQiu: /* Question 05 (a) */

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
[[File:Men_at_work_sign.gif|left|120x120px]]
<span> </span><div style="width:50%; color:red;"><br>
<br>
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.</div>

<span></span>
  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
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</dpl>
<span></span>
{| class="collapsible wikitable " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Question 05 (a) <font size="1"></font>===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Statement]]
|}
<span></span>For this question, let

<span></span><math>f(x) = 1-(1+x)^{-1}.</math>

<span></span>You may use the following facts without proving them:
* <span></span><math>f(x)=\frac{x}{1+x}</math>
* <span></span><math>f'(x)=\frac{1}{(1+x)^2}</math>
* <span></span><math>f''(x)=\frac{-2}{(1+x)^3}</math>
<span></span>(a) Given an equation of the tangent line to
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''Make sure you understand the problem fully:''' What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
|-
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''If you are stuck''', check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Hint <font size="1"></font>===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Hint 1]]
|}
<span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
* '''If you are stuck on a problem:''' Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
* '''If you want to check your work:''' Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br>
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed solutionMER" width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Solution <font size="1"></font>===
|-
| style="padding:1em" |<div style="color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;"><span class="plainlinks">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? [https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578 Let us know here].<br>
Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Solution 1]]
|}
<span></span><div class="mw-customtoggle-11" style="cursor:pointer;background-color:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:center"> Click here for similar questions </div><span></span><div class="mw-collapsible mw-collapsed" style=""><div class="mw-collapsible-content" style="">
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{| cellspacing="0"
| [[File:Bulbgraph.png|none|31x31px|lightbulb image]]
| style="padding-left:20px;padding-right: 20px;background-color : #FFFFCC;" | '''<span class="plainlinks">[http://www.math.ubc.ca/~MLC/ Math Learning Centre]</span>'''
* A space to study math together.
* Free math graduate and undergraduate TA support.
* Mon - Fri: 11 am - 5 pm, <span class="plainlinks">[http://www.maps.ubc.ca/PROD/index_detail.php?show=y,n,n,n,n,y&bldg2Search=n&locat1=308 LSK 301&302].</span>
'''Private tutor'''
* We can help to <span class="plainlinks">[https://docs.google.com/forms/d/1nV32JI4R3Mp96lYtLIpNEE-LB_T-fEHz-8lfAZXd430/viewform find a private tutor].</span>
|}
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{{DISPLAYTITLE:Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)}}

Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)

2020-03-17T23:09:51Z

MingfengQiu: Created page with "<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div> File:..."

<span></span><div style="font-size:18pt; line-height:100%; color:#ffffff; background:#2F4F4F; position: relative; padding:1ex;">'''MATH102''' '''December 2019'''</div>
[[File:Men_at_work_sign.gif|left|120x120px]]
<span> </span><div style="width:50%; color:red;"><br>
<br>
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.</div>

<span></span>
  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (a)|Q4 (a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (b)|Q4 (b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)|Q4 (c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (d)|Q4 (d)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(a)|Q8(a)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(b)|Q8(b)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 08(c)|Q8(c)]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11|Q11]]  •  <div class="mw-customtoggle-2" style="cursor:pointer;background:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:left;">   Other [[Science:Math Exam Resources/Courses/MATH102|<span style="color: white">MATH102</span>]] Exams</div><span></span><div class="mw-collapsible " style=""><div class="mw-collapsible-content" style="">  •  [[Science:Math Exam Resources/Courses/MATH102/December 2016|December 2016]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2015|December 2015]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2014|December 2014]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2013|December 2013]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2012|December 2012]]  •  [[Science:Math Exam Resources/Courses/MATH102/December 2011|December 2011]]  •  </div></div><dpl>
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</dpl>
<span></span>
{| class="collapsible wikitable " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Question 05 (a) <font size="1"></font> ===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Statement]]
|}
<span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''Make sure you understand the problem fully:''' What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
|-
| style="text-align:justify; font-size:8pt; color:#666666;" |
'''If you are stuck''', check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed " width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Hint <font size="1"></font> ===
|-
| style="padding:1em" |
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Hint 1]]
|}
<span></span>
{| width="100%" style="background:#ffffff;"
| style="text-align:justify; font-size:8pt; color:#666666;" |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
* '''If you are stuck on a problem:''' Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
* '''If you want to check your work:''' Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.<br>
|}
<span></span>
<dpl>
noresultsheader=\n
titlematch={{PAGENAME}}
skipthispage=no
replaceintitle=$ $,+
format=,{{#vardefine:class|{{#titleparts:%TITLE%|1|-3}}}}{{#vardefine:exam|{{#titleparts:%TITLE%|1|-2}}}}{{#vardefine:question|{{#titleparts:%TITLE%|1|-1}}}},,
</dpl>
<span></span>
{| class="collapsible wikitable collapsed solutionMER" width="100%" style="background: #FFFFFF;"
! style="padding-left:1em; text-align: left; background:#FFFFFF; color:#666666;" |
=== Solution <font size="1"></font> ===
|-
| style="padding:1em" |<div style="color:#2F4F4F; background: #DDDDDD; border: 3px solid #2F4F4F; margin-bottom:1em; padding:10px;"><span class="plainlinks">Found a <strong>typo</strong>? Is this solution <strong>unclear</strong>? [https://docs.google.com/forms/d/1n6Pf-fegzLB4n1QYTHmxvIcIWr4swidLaTm1kekZk38/viewform?entry.86513335=&entry.1104878578&entry.554346562=&entry.488130431&entry.2101116491=&entry.572580842&entry.336854578 Let us know here].<br>
Please <strong>rate my easiness</strong>! It's quick and helps everyone guide their studies.</span></div>
[[Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)/Solution 1]]
|}
<span></span><div class="mw-customtoggle-11" style="cursor:pointer;background-color:#7491A3; font-weight:normal; color: white; font-size:10pt; width:100%; text-align:center"> Click here for similar questions </div><span></span><div class="mw-collapsible mw-collapsed" style=""><div class="mw-collapsible-content" style="">
__NOTOC__
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Extension:DynamicPageList (DPL), version 2.3.0 : Warning: No results.
</div></div></div><span></span><div class="noprint" style="float:none; border:5px solid #7491A3;width:280px; padding:10px;">
{| cellspacing="0"
| [[File:Bulbgraph.png|none|31x31px|lightbulb image]]
| style="padding-left:20px;padding-right: 20px;background-color : #FFFFCC;" | '''<span class="plainlinks">[http://www.math.ubc.ca/~MLC/ Math Learning Centre]</span>'''
* A space to study math together.
* Free math graduate and undergraduate TA support.
* Mon - Fri: 11 am - 5 pm, <span class="plainlinks">[http://www.maps.ubc.ca/PROD/index_detail.php?show=y,n,n,n,n,y&bldg2Search=n&locat1=308 LSK 301&302].</span>
'''Private tutor'''
* We can help to <span class="plainlinks">[https://docs.google.com/forms/d/1nV32JI4R3Mp96lYtLIpNEE-LB_T-fEHz-8lfAZXd430/viewform find a private tutor].</span>
|}
</div><span></span>

{{DISPLAYTITLE:Science:Math Exam Resources/Courses/MATH102/December 2019/Question 05 (a)}}

Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04

2020-02-15T23:49:53Z

MingfengQiu:

Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04/Solution 1

2020-02-15T23:45:39Z

MingfengQiu: Created page with "Let <math> (x_0, y_0) </math> be a point on the curve. Then <math> x_0^2 + 2y_0^2 = 8.~~~~(1) </math> To find the slope of the tangent line passing through <math> (x_0, y_0)..."

Let <math> (x_0, y_0) </math> be a point on the curve. Then

<math> x_0^2 + 2y_0^2 = 8.~~~~(1) </math>

To find the slope of the tangent line passing through <math> (x_0, y_0) </math>, use implicit differentiation. Differentiate both sides of the equation of the curve with respect to <math>x</math>. By the chain rule,

<math> 2x + 4yy' = 0. </math>

Let <math>y_0'</math> be the slope of the tangent line at point <math>(x_0, y_0)</math>. Assume <math>y_0 \neq 0</math>. Then

<math>y'_0 = -\frac{x_0}{2y_0}.</math>

Using the point-slope formula, the tangent line equation to the curve at the point <math>(x_0, y_0)</math> can be written as

<math> y - y_0 = -\frac{x_0}{2y_0}(x - x_0). </math>

Let this tangent line pass through <math>(0, -6)</math>. Putting <math>x = 0</math> and <math>y = -6</math>,

<math> -6 - y_0 = \frac{x_0^2}{2y_0}. </math>

Together with Equation (1) we can solve for the point. <math>(x_0, y_0) = \left(\frac{8}{3}, -\frac{2}{3}\right)</math> or <math>\left(-\frac{8}{3}, -\frac{2}{3}\right).</math>

Now consider if <math>y_0 = 0</math>, then <math>x_0 = \pm2\sqrt{2}.</math> The slope of the tangent line DNE. It follows that the tangent line is a vertical line, either <math>x = 2\sqrt{2}</math> or <math>x = -2\sqrt{2}</math>. Neither of them passes through the point <math>(0, -6).</math>

Therefore, the points on the curve that satisfy the requirements are <math>\left(\frac{8}{3}, -\frac{2}{3}\right),~\left(-\frac{8}{3}, -\frac{2}{3}\right)</math>.

Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04/Hint 2

2020-02-15T23:08:47Z

MingfengQiu:

Write the equation of the tangent line at an arbitrary point <math>(x_0, y_0)</math> on this curve, using the point-slope formula. Let this line go through (0, -6).

Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04/Hint 2

2020-02-15T23:08:25Z

MingfengQiu: Created page with "Write the equation of the tangent line at an arbitrary point <math>(x_0, y_0)</math>on this curve, using the point-slope formula. Let this line go through (0,-6)."

Write the equation of the tangent line at an arbitrary point <math>(x_0, y_0)</math>on this curve, using the point-slope formula. Let this line go through (0,-6).

Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04/Hint 1

2020-02-15T23:03:32Z

MingfengQiu: Created page with "To find the slope of the tangent line at a point of a curve, use implicit differentiation."

To find the slope of the tangent line at a point of a curve, use implicit differentiation.

Science:Math Exam Resources/Courses/MATH100/December 2018/Question 04/Statement

2020-02-15T22:59:45Z

MingfengQiu: Created page with "Find the <math>x</math> and <math>y</math> coordinates of all points on the curve given by the equation: <math> x^2 + 2y^2 = 8 </math> for which the tangent line at each of..."

Find the <math>x</math> and <math>y</math> coordinates of all points on the curve given by the equation:

<math> x^2 + 2y^2 = 8 </math>

for which the tangent line at each of those points passes through the point (0, −6).