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Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 08

2014-09-30T21:14:38Z

ChristinaKoch:

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Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 08/Statement

2014-09-30T21:13:47Z

ChristinaKoch: Created page with "Given the differential equation and initial condition : <math>\frac{dy}{dt} = y^2(y - a), y(0) = 2a</math> where <math>a > 0</math> is a constant, which of the following is ..."

Given the differential equation and initial condition

: <math>\frac{dy}{dt} = y^2(y - a), y(0) = 2a</math>

where <math>a > 0</math> is a constant, which of the following is true?

(a) <math>\lim_{t \to \infty} y(t) = 0</math>

(b) <math>\lim_{t \to \infty} y(t) = \infty</math>

(c) <math>\lim_{t \to \infty} y(t) = a</math>

(d) <math>\lim_{t \to \infty} y(t) = 2a</math>

(e) None of the above.

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 07

2014-09-30T21:11:56Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 07/Statement

2014-09-30T21:10:40Z

ChristinaKoch: Created page with "The number of sunspots (solar storms on the sun) varies with a period of roughly 11 years reaching a high of 120 and a low of 0 sunspots detected. A peak of 120 sunspots was ..."

The number of sunspots (solar storms on the sun) varies with a period of roughly 11 years reaching a high of 120 and a low of 0 sunspots detected. A peak of 120 sunspots was detected in the year 2000. Which of the following trigonometric functions could be used to approximate this cycle?

(a) <math>N = 60 + 120\sin\left(\frac{2\pi}{11}(t - 2000 ) + \frac{\pi}{2}\right)</math>

(b) <math>N = 60 + 60\sin\left(\frac{11}{2\pi}(t + 2000 ) \right)</math>

(c) <math>N = 60 + 60\cos\left(\frac{11}{2\pi}(t + 2000) \right)</math>

(d) <math>N = 60 + 60\sin\left(\frac{2\pi}{11}(t - 2000) \right)</math>

(e) <math>N = 60 + 60\cos\left(\frac{2\pi}{11}(t - 2000 ) \right)</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 06/Statement

2014-09-30T21:06:51Z

ChristinaKoch: Created page with "The slope field shown below corresponds to which differential equation? (img) (a) <math>\frac{dy}{dt} = y(y+1)</math> (b) <math>\frac{dy}{dt} = (y - 1)(y + 1)</math> (c)..."

The slope field shown below corresponds to which differential equation?

(img)

(a) <math>\frac{dy}{dt} = y(y+1)</math>

(b) <math>\frac{dy}{dt} = (y - 1)(y + 1)</math>

(c) <math>\frac{dy}{dt} = -(y - 1)(y + 1)</math>

(d) <math>\frac{dy}{dt} = y(y - 1)</math>

(e) <math>\frac{dy}{dt} = -y(y + 1)</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 05

2014-09-30T21:05:07Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 05/Statement

2014-09-30T21:04:45Z

ChristinaKoch: Created page with "Consider the function <math>y = \cos(x)</math> and the tangent line to this function at the point <math>x = \pi/2</math>. Using that tangent line as a linear approximation of..."

Consider the function <math>y = \cos(x)</math> and the tangent line to this function at the point <math>x = \pi/2</math>. Using that tangent line as a linear approximation of the function would lead to

(a) overestimating the value of the actual function for any nearby ''x''.

(b) underestimating the value of the actual function for any nearby ''x''.

(c) overestimating the function when <math>x > \pi/2</math> and underestimating the function when <math>x < \pi/2 </math>.

(d) overestimating the function when <math>x < \pi/2</math> and underestimating the function when <math>x > \pi/2 </math>.

(e) overestimating the function when <math>x < 0 </math> and underestimating the function when <math>x > 0</math>.

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 04/Statement

2014-09-30T21:02:16Z

ChristinaKoch:

Shown in the figure below is a function and its tangent line at <math>x = x_0</math>.

(img)

The tangent line intersects the ''x'' axis at the point <math>x = x_1</math>. The coordinate of the point <math>x_1</math> is

(a) <math>x_1 = x_0 + \frac{f(x_0)}{f'(x_0)}</math>

(b) <math>\displaystyle x_1 = x_0 - f'(x_0)(x-x_0)</math>

(c) <math>x_1 = x_0 - \frac{f'(x_1)}{f(x_1)}</math>

(d) <math>x_1 = x_0 + \frac{f'(x_1)}{f(x_1)}</math>

(e) <math>x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 04/Statement

2014-09-30T21:01:50Z

ChristinaKoch: Created page with "Shown in the figure below is a function and its tangent line at <math>x = x_0</math>. (img) The tangent line intersects the ''x'' axis at the point <math>x = x_1</math>. ..."

Shown in the figure below is a function and its tangent line at <math>x = x_0</math>.

(img)

The tangent line intersects the ''x'' axis at the point <math>x = x_1</math>. The coordinate of the point <math>x_1</math> is

(a) <math>x_1 = x_0 + \frac{f(x_0)}{f'(x_0)}</math>

(b) <math>x_1 = x_0 - f'(x_0)(x-x_0)</math>

(c) <math>x_1 = x_0 - \frac{f'(x_1)}{f(x_1)}</math>

(d) <math>x_1 = x_0 + \frac{f'(x_1)}{f(x_1)}</math>

(e) <math>x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 04

2014-09-30T20:58:14Z

ChristinaKoch:

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{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 04

2014-09-30T20:57:23Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 04/Statement

2014-09-30T20:57:06Z

ChristinaKoch: Created page with "Shown in the figure below is the view from above of the path taken by a penguin from point A to a feeding area on the shore at point C. The penguin must choose the point B to..."

Shown in the figure below is the view from above of the path taken by a penguin from point A to a feeding area on the shore at point C. The penguin must choose the point B toward which it starts walking. It takes twice as much energy per unit distance for the penguin to walk over land (AB) as to swim through water (BC). The distance AD is 300 m and the distance DC is 400 m. Calculate the value of the distance ''x'' (and hence the location of the point B - see figure) that minimizes the energy spent on the entire trip.

(img)

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 03

2014-09-30T20:55:07Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 03/Statement

2014-09-30T20:54:35Z

ChristinaKoch: Created page with "Consider the function : <math>f(x) = x^2e^{-x}</math> defined on the whole real line. Find all zeros of <math>f, f'</math> and <math>f''</math> and determine all local mi..."

Consider the function

: <math>f(x) = x^2e^{-x}</math>

defined on the whole real line.

Find all zeros of <math>f, f'</math> and <math>f''</math> and determine all local minima, maxima and inflection points of ''f''. Sketch the graph of ''f''. You may use the fact that <math>\lim_{x \to \infty} x^2 e^{-x} = 0</math>.

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (d)

2014-09-30T20:52:43Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (d)/Statement

2014-09-30T20:52:18Z

ChristinaKoch: Created page with "The population of fish in a particular lake is given by the function ''F(t)'' where ''F'' is measured in number of fish and ''t'' is measured in days. A company that manages ..."

The population of fish in a particular lake is given by the function ''F(t)'' where ''F'' is measured in number of fish and ''t'' is measured in days. A company that manages fish stocks is hired to restock the lake, adding fish at a constant rate. Only ''N'' fishers are allowed to fish in the lake at a time. A simple model for this scenario is given by the equation:

<math>\frac{dF}{dt} = I - \alpha N F </math>

Where ''I'' and <math>\alpha</math> are constant and two cases for ''N'' are considered.

Case 2: The agency that administers fishing permits decides that the number of fishers should be proportional to the number of fish currently in the lake, <math>N = F/3</math>. What is the population size <math>F_*</math> to which ''F(t)'' approaches as <math>t \to \infty</math> in this scenario?

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (c)

2014-09-30T20:50:38Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (c)/Statement

2014-09-30T20:50:15Z

The population of fish in a particular lake is given by the function ''F(t)'' where ''F'' is measured in number of fish and ''t'' is measured in days. A company that manages fish stocks is hired to restock the lake, adding fish at a constant rate. Only ''N'' fishers are allowed to fish in the lake at a time. A simple model for this scenario is given by the equation:

<math>\frac{dF}{dt} = I - \alpha N F </math>

Where ''I'' and <math>\alpha</math> are constant and two cases for ''N'' are considered.

Case 1: Suppose ''N'' is a constant. What is the steady state number of fish <math>(F_*)</math> in the lake? If the lake has no fish in it initially, at what time <math>(t_*)</math> does the population size reach half its steady state value <math>F_*/2</math>? (You do not have to show a derivation of the solution ''F(t)'' of the equation for full points - simply stating it is sufficient - but you must show the rest of the calculation.)

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (b)

2014-09-30T20:48:11Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (b)/Statement

2014-09-30T20:47:52Z

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (a)

2014-09-30T20:47:32Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (a)/Statement

2014-09-30T20:47:14Z

Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 01/Statement

2014-09-30T20:44:13Z

ChristinaKoch: Created page with "A mollusc grows inside a shell and produces new shell material at its opening (indicated in the image below in bold) (img) The volume of the shell can be approximated as tha..."

A mollusc grows inside a shell and produces new shell material at its opening (indicated in the image below in bold)

(img)

The volume of the shell can be approximated as that of a cone of base radius ''r'' and height ''h''.

(img)

Given that the height of the cone grows at a constant rate of 0.1 cm/year, at what rate will the volume of the cone be changing when ''h = 10'' and ''r = 1'' cm? Leave your answer in terms of <math>/pi</math>

Note that:

<math>V_{cone} = \frac13 \pi r^2 h </math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 08

2014-09-29T23:19:29Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 08/Statement

2014-09-29T23:19:09Z

ChristinaKoch: Created page with "The velocity of a crawling cell has been modelled as <math>v(x) = \frac{1}{ax + e^{1/x}}</math> where <math> x > 0 </math> is the number of actin filaments driving the motio..."

The velocity of a crawling cell has been modelled as

<math>v(x) = \frac{1}{ax + e^{1/x}}</math>

where <math> x > 0 </math> is the number of actin filaments driving the motion and <math>a > 0</math> is a constant. Determine the following:

(i) <math>\lim_{x \to 0^+} e^{1/x}</math>

(ii) <math>\lim_{x \to \infty} e^{1/x} </math>

(iii) <math>\lim_{x \to 0^+} v(x)</math>

(iv) <math>\lim_{x \to \infty} v(x) </math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 07

2014-09-29T23:16:42Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 07/Statement

2014-09-29T23:16:13Z

ChristinaKoch: Created page with "Initially, a patient has <math>1000</math> copies of the HIV virus. If the smallest detectable viral load is <math>350,000</math> virus particles, how long (in days) will it ..."

Initially, a patient has <math>1000</math> copies of the HIV virus. If the smallest detectable viral load is <math>350,000</math> virus particles, how long (in days) will it take until the HIV infection is detectable? Assume that the number of virus particles ''y'' grows according to the equation

<math>\frac{dy}{dt} = 0.05y</math>

where ''t'' is time in days. Leave your answer in terms of logarithms.

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 06

2014-09-29T23:14:21Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 06/Statement

2014-09-29T23:14:04Z

ChristinaKoch: Created page with "Let <math>f(x) = \begin{cases} x^2 & x < 1 \\ a-x & x \geq 1 \end{cases}</math> where ''a'' is a constant. What value of ''a'' ensures that ''f'' is continuous for all ''x''?"

Let

<math>f(x) = \begin{cases} x^2 & x < 1 \\ a-x & x \geq 1 \end{cases}</math>

where ''a'' is a constant. What value of ''a'' ensures that ''f'' is continuous for all ''x''?

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 05

2014-09-29T23:12:08Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 05/Statement

2014-09-29T23:11:51Z

ChristinaKoch: Created page with "Compute the limit <math>L = \lim_{h \to 0} \frac{\sqrt{3+h} - \sqrt{3}}{h} </math>"

Compute the limit

<math>L = \lim_{h \to 0} \frac{\sqrt{3+h} - \sqrt{3}}{h} </math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 04

2014-09-29T23:10:50Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 04/Statement

2014-09-29T23:10:22Z

ChristinaKoch: Created page with "What function satisfies <math>y'' + 4y = 0</math> with initial condition <math>y(0) = 1</math>?"

What function satisfies <math>y'' + 4y = 0</math> with initial condition <math>y(0) = 1</math>?

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 03

2014-09-29T23:08:57Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 03/Statement

2014-09-29T23:08:36Z

ChristinaKoch: Created page with "List the ''x'' coordinates of all local minima and maxima of <math>f(x) = x^5 - \frac54 x^4</math>."

List the ''x'' coordinates of all local minima and maxima of <math>f(x) = x^5 - \frac54 x^4</math>.

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 02

2014-09-29T23:07:46Z

ChristinaKoch:

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{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 02/Hint 1

2014-09-29T23:06:49Z

ChristinaKoch:

You made need the following formula (cosine law)

<math>a^2 = b^2 + c^2 - 2bc * \cos(\theta)</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 02/Hint 1

2014-09-29T23:06:18Z

ChristinaKoch: Created page with "You made need the following formula (cosine law) <math>a^2 = b^2 + c^2 - 2bc cos(�\theta)</math>"

You made need the following formula (cosine law)

<math>a^2 = b^2 + c^2 - 2bc cos(�\theta)</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 02/Statement

2014-09-29T23:04:42Z

ChristinaKoch: Created page with "A triangle has two sides with fixed lengths 3 cm and 4 cm respectively. The angle between them changes at a constant rate of 1 radian/sec. How quickly is the length of the t..."

A triangle has two sides with fixed lengths 3 cm and 4 cm respectively. The angle between them changes at a constant rate of 1 radian/sec. How quickly is the length of the third side (S) changing when the angle between the sides of fixed length is <math>\pi/2</math> ?

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 01

2014-09-29T23:04:20Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 03

2014-09-29T23:03:10Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question B 01/Statement

2014-09-29T23:02:32Z

ChristinaKoch: Created page with "Use implicit differentiation to find <math>dy / dx </math> where <math>y(x)</math> is defined implicitly by <math>\tan(y) = x</math>. Your answer should not include any trig ..."

Use implicit differentiation to find <math>dy / dx </math> where <math>y(x)</math> is defined implicitly by <math>\tan(y) = x</math>. Your answer should not include any trig functions. Show your work.

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 03/Statement

2014-09-29T23:01:05Z

ChristinaKoch:

Consider the differential equation and initial condition

<math>\frac{dy}{dt} = 2 - y^2, y(0) = 1</math>

Use Euler's method with one step of size <math>\Delta t = 0/1</math> to approximate the value of the solution at time <math>t = 0.1</math>

(a) <math> y(0.1) = 2 </math>

(b) <math> y(0.1) = 2.1 </math>

(c) <math> y(0.1) = 2.2</math>

(d) <math> y(0.1) = 1.2</math>

(e) <math> y(0.1) = 1.1</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 03/Statement

2014-09-29T23:00:38Z

ChristinaKoch: Created page with "Consider the differential equation and initial condition <math>\frac{dy}{dt} = 2 - y^2, y(0) = 1</math>. Use Euler's method with one step of size <math>\Delta t = 0/1</mat..."

Consider the differential equation and initial condition

<math>\frac{dy}{dt} = 2 - y^2, y(0) = 1</math>.

Use Euler's method with one step of size <math>\Delta t = 0/1</math> to approximate the value of the solution at time <math>t = 0.1</math>

(a) <math> y(0.1) = 2 </math>

(b) <math> y(0.1) = 2.1 </math>

(c) <math> y(0.1) = 2.2</math>

(d) <math> y(0.1) = 1.2</math>

(e) <math> y(0.1) = 1.1</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 02

2014-09-29T22:59:56Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 02/Statement

2014-09-29T22:57:43Z

ChristinaKoch: Created page with "Which of the following gives a value of <math>\cos \left( \arcsin \left( \frac{\sqrt{9-x^2}}{3} \right) \right) </math> ? (a) <math>\frac{3}{x}</math> (b) <math>\frac{x}{3}<..."

Which of the following gives a value of <math>\cos \left( \arcsin \left( \frac{\sqrt{9-x^2}}{3} \right) \right) </math> ?

(a) <math>\frac{3}{x}</math>

(b) <math>\frac{x}{3}</math>

(c) <math>\frac{\sqrt{9-x^2}}{x} </math>

(d) <math>\frac{x}{\sqrt{9-x^2}}</math>

(e) <math>\frac{3}{\sqrt{9-x^2}}</math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 01

2014-09-29T22:54:26Z

ChristinaKoch:

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 01/Statement

2014-09-29T22:54:03Z

ChristinaKoch:

Let <math>f(x) = \sin(x) + ax^2</math>. Which of the following conditions describes all values of ''a'' for which ''f'' HAS inflection points?

(a) <math>|a| > 1/2 </math>

(b) <math>a > 1/2 </math>

(c) <math>a \geq 1/2 </math>

(d) <math>|a| < 1/2 </math>

(e) <math>|a| \leq 1/2 </math>

Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 01/Statement

2014-09-29T22:52:36Z

ChristinaKoch: Created page with "Let <math>f(x) = sin(x) + ax^2</math>. Which of the following conditions describes all values of ''a'' for which ''f'' HAS inflection points? (a) <math>|a| > 1/2 </math> (b)..."

Let <math>f(x) = sin(x) + ax^2</math>. Which of the following conditions describes all values of ''a'' for which ''f'' HAS inflection points?

(a) <math>|a| > 1/2 </math>

(b) <math>a > 1/2 </math>

(c) <math>a �\geq 1/2 </math>

(d) <math>|a| < 1/2 </math>

(e) <math>|a| \leq� 1/2 </math>

Sandbox:MER Main Page

2014-09-22T04:32:42Z

ChristinaKoch:

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| style="background:{{#var:secondary_Colour2}}; vertical-align:middle;" | <p style="margin:0; font-size:20pt; font-weight:bold; text-align:right; color:white; padding:0.5em;">Usage Data</p> || style="text-align:center;" | </br> [[File:MER usage data spikes.png|350px|link=Science:Math_Exam_Resources/Usage]]
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| style="width:45%; text-align:center; font-size:14pt;" | </br> [[File:MER infographic.png|400px|link=Science:Math_Exam_Resources/Guidelines]] </br> </br> [[Science:Math_Exam_Resources/About|About]] • [[Science:Math_Exam_Resources/Guidelines|How To Use This Resource]] • [[Science:Math_Exam_Resources/FAQ|FAQ]] || style="background:{{#var:secondary_Colour2}}; vertical-align:middle;" | <p style="margin:0; font-size:20pt; font-weight:bold; text-align:left; color:white; padding:0.5em;">Why this Resource?</p>
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| style="background:{{#var:secondary_Colour2}}; vertical-align:middle;" | <p style="margin:0; font-size:20pt; font-weight:bold; text-align:right; color:white; padding:0.5em;">Get Involved</p> || style="text-align:center;" | <span class="plainlinks">[http://wiki.ubc.ca/Science:Math_Exam_Resources/Contribute http://webprofessionals.org/wp-content/uploads/2009/08/join-the-wow-organization.jpg] </span>
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! <h2 style="margin:0; background:{{#var:main_Colour}}; font-size:20pt; font-weight:bold; border:1px solid #cccccc; text-align:left; color:white; padding:0.5em;">Resources by Topic</h2>
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* Caution is needed when working through problems on the same topic from a different course. While problems are often applicable cross-courses, sometimes questions from a particular course require a technique that was not covered in another course, even though both courses covered the same general topic.
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! <h2 style="margin:0; background:{{#var:main_Colour}}; font-size:20pt; font-weight:bold; border:1px solid #cccccc; text-align:left; color:white; padding:0.5em;">Resources by Course</h2>
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{{MER Course table}}
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<div style="font-size:8pt">Original photograph by [http://www.flickr.com/photos/tywak tywak] deposited [http://www.flickr.com/photos/tywak/6317919860 on Flickr].</div>
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