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04:55, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (c)/Hint 1 (Created page with "First, solve for the steady state. Then determine the sign of <math>\frac{dS}{dt}</math> for <math>S</math> on either side of the steady state by considering very small and very large values of <math>S</math>.")
04:53, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (c)/Solution 1 (Created page with "Let <math>0<R<1</math>. Solving for the steady state <math>0=R-\frac{S}{2+S}</math>, we obtain <math>S = \frac{2R}{1-R}</math>. For <math>S</math> very close to <math>0</math>, we would have <math>R-\frac{S}{2+S}</math> close to <math>R</math>, which is positive, so <math>\frac{dS}{dt}>0</math> for <math>S<\frac{2R}{1-R}</math>. On the other hand, for <math>S</math> very large, we would have <math>\frac{S}{2+S}</math> close to <math>1</math>, so <math>R-\frac{S}{2+S}<...")
04:38, 15 April 2026 AlanGetz talk contribs created page File:Q12(c) diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard)
04:38, 15 April 2026 AlanGetz talk contribs uploaded File:Q12(c) diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard) Tag: Upload Wizard
04:34, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (b)/Solution 1 (Created page with "For all <math>S\geq 0</math>, <math>\frac{S}{2+S}<1</math>. Since <math>\frac{3}{2}>1</math>, we have that <math display="block"> \frac{3}{2}-\frac{S}{2+S}> \frac{3}{2} - 1 > 0, </math> so <math>\frac{dS}{dt}>0</math> for all nonnegative <math>S</math>. That is, over time the substrate builds up without bound.")
04:09, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (a)/Solution 1 (Created page with "The phase diagram is File:Q12(a) diagram, UBC math 100 December 2024 exam.png")
04:08, 15 April 2026 AlanGetz talk contribs created page File:Q12(a) diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard)
04:08, 15 April 2026 AlanGetz talk contribs uploaded File:Q12(a) diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard) Tag: Upload Wizard
04:06, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (a)/Hint 1 (Created page with "Set <math>R=0</math> and consider the differential equation <math>\frac{dS}{dt}=-\frac{S}{2+S}</math>. Find the equilibrium points by solving <math>\frac{dS}{dt}=0</math>, then determine the sign of <math>\frac{dS}{dt}</math> for the values of <math>S</math> on either side.")
04:00, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (c)/Statement (Created page with "Suppose a chemical is being converted to a product in tank. The amount of the chemical is <math>S</math>, which is nonnegative and changes with respect to time. If the chemical is being added to the tank at a constant nonnegative rate <math>R</math>, then the amount of the chemical in the tank satisfies the differential equation <math display="block"> \frac{dS}{dt}=R-\frac{S}{2+S}. </math> Describe in words what happens to the amount of the chemical in the tank over a l...")
03:59, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (b)/Statement (Created page with "Suppose a chemical is being converted to a product in tank. The amount of the chemical is <math>S</math>, which is nonnegative and changes with respect to time. If the chemical is being added to the tank at a constant nonnegative rate <math>R</math>, then the amount of the chemical in the tank satisfies the differential equation <math display="block"> \frac{dS}{dt}=R-\frac{S}{2+S}. </math> Describe in words what happens to the amount of the chemical in the tank over a l...")
03:55, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (a)/Statement (Created page with "Suppose a chemical is being converted to a product in tank. The amount of the chemical is <math>S</math>, which is nonnegative and changes with respect to time. If the chemical is being added to the tank at a constant nonnegative rate <math>R</math>, then the amount of the chemical in the tank satisfies the differential equation <math display="block"> \frac{dS}{dt}=R-\frac{S}{2+S}. </math> Draw a phase diagram for the differential equation, for <math>S\geq 0</math>, whe...")
03:40, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (d)/Solution 1 (Created page with "From the sketch, we see <math>y(x)</math> is concave down. That means its tangent lines lie above the curve, so linear approximations produce over approximations. Hence, the first step of Euler's method produces an over approximation. Additionally, <math>y(x)</math> is increasing, so an initial over approximation plus a linear approximation will produce an over approximation in the second step as well.")
03:36, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (d)/Hint 1 (Created page with "From your sketch in part (a), does the curve look concave up or concave down? What does this imply about whether Euler's method will yield an over approximation or an under approximation?")
03:31, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (d)/Statement (Created page with "In this question, we’ll investigate solutions to the differential equation <math display="block"> \frac{dy}{dt}= (y-1)(3-y). </math> Is your numerical approximation from (c) using Euler’s method larger or smaller than <math>y(1)</math>? You may use your previous work, but you must explain your answer.")
03:28, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (c)/Solution 1 (Created page with "Let us denote <math>f(y) = (y-1)(3-y)</math>. At <math>t=0</math>, we have <math>y_0=y(0)=2</math> and <math>f(y_0)=f(2)=(2-1)(3-2)=1</math>. Hence, at our next time step, we have <math>y_1=1+(1/2)=5/2</math> and <math>f(y_1)f(5/2)=(3/2)(1/2)=3/4</math>. At our final time step, we have <math>y_2=5/2+(1/2)(3/4)=23/8</math>, which is our approximation.")
03:17, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (c)/Hint 1 (Created page with "Since we are using Euler's method with two steps, beginning at <math>t=0</math> and ending at <math>t=1</math>, our time step will need to be <math>\Delta t = 1/2</math>.")
03:14, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (c)/Statement (Created page with "In this question, we’ll investigate solutions to the differential equation <math display="block"> \frac{dy}{dt}= (y-1)(3-y). </math> Use Euler’s method with two steps to approximate y(1).")
03:11, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (b)/Hint 1 (Created page with "Where does it look like the curve sketched out in part (a) intersects with the line <math>x=1</math>.")
03:05, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (b)/Solution 1 (Created page with "Based on the sketch in part (a), <math>y(1)</math> is approximately <math>2.75</math>, though any answer in the interval <math>(2.5,3)</math> is acceptable.")
03:01, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (b)/Statement (Created page with "In this question, we’ll investigate solutions to the differential equation <math display="block"> \frac{dy}{dt}= (y-1)(3-y). </math> Using your sketch from (a), give an approximate value for <math>y(1)</math>.")
02:58, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (a)/Solution 1 (Created page with "The solution is given by File:Q11(a) diagram, UBC math 100 December 2024 exam.png")
02:57, 15 April 2026 AlanGetz talk contribs created page File:Q11(a) diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard)
02:57, 15 April 2026 AlanGetz talk contribs uploaded File:Q11(a) diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard) Tag: Upload Wizard
02:53, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (a)/Hint 1 (Created page with "Start at the point <math>(0,2)</math> and trace out a curve that agrees with the slope lines shown in the diagram.")
02:44, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (a)/Statement (Created page with "a File:Q11 diagram, UBC math 100 December 2024 exam.png a")
02:44, 15 April 2026 AlanGetz talk contribs created page File:Q11 diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard)
02:44, 15 April 2026 AlanGetz talk contribs uploaded File:Q11 diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard) Tag: Upload Wizard
01:54, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 10/Hint 1 (Created page with "Since <math>A(x) = k(x^2+ 10(1+\sqrt{5})x</math> and <math>C(x)=5x^2</math>, the leading terms agree when <math>k=5</math>. Consider the behaviour of the cell in the three separate cases <math>k<5</math>, <math>k=5</math>, and <math>k>5</math>.")
01:49, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 10/Solution 1 (Created page with "We are given that the surface area of the prism is <math>x^2+ 10(1+\sqrt{5})x</math> and and the volume is <math>5x^2</math>, so <math display="block"> A(x) = k(x^2+ 10(1+\sqrt{5})x \text{ and } C(x)=5x^2. </math> The leading terms in the above expressions are <math>kx^2</math> and <math>5x^2</math>, so we consider the cases <math>k<5</math>, <math>k=5</math>, and <math>k>5</math>. First suppose <math>k<5</math>. Since <math>x</math> starts out small, linear terms domi...")
01:13, 15 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 10/Statement (Created page with "Consider a cell that is shaped like a triangular prism. Its triangular sides have base length <math>x</math> and height <math>x</math> (which may change over time, and are both measured in micrometers, μm) and its depth is fixed at <math>10 \mu m</math> (not changing). Its volume is <math>5x^2 \ \mu m^3</math>, and its surface area is <math>x^2 + 10(1+\sqrt{5})x \ \mu m^2</math>. File:Q10 diagram, UBC math 100 December 2024 exam.png")
01:04, 15 April 2026 AlanGetz talk contribs created page File:Q10 diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard)
01:04, 15 April 2026 AlanGetz talk contribs uploaded File:Q10 diagram, UBC math 100 December 2024 exam.png (Uploaded a work by {{Unknown|author}} from UBC December 2024 math 100 B exam with UploadWizard) Tag: Upload Wizard
23:34, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 14/Solution 1 (Created page with "For any nonzero <math>x</math> that is not of the form <math>\frac{1}{n}</math>, the function is differentiable. Indeed, one can find a small interval around such an <math>x</math> that does not contain any points of the form <math>\frac{1}{n}</math>, so the function is identically zero in that interval. Hence, <math display="block"> f'(x) = 0. </math> For any <math>a = \frac{1}{n}</math>, the function is not continuous because <math display="block"> \lim_{x \to a} f(x)...")
23:33, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 14/Hint 1 (Created page with "Consider three cases separately: <math>x \ne \frac{1}{n},\, x = \frac{1}{n}</math>, and <math>x = 0</math>.")
23:32, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 14/Statement (Created page with "Let <math> f(x) </math> be a function that is defined for all real numbers by <math display="block"> f(x)=\begin{cases} x^2 & \text{if } x=\frac{1}{n}, \text{where} \ n \ \text{is a positive integer}\\ 0 & \text{otherwise.} \end{cases} </math> A portion of the function is sketched below, to help you visualize it. For what values of x is this function differentiable? For those values, compute <math> f'(x) </math>. Remember to fully justify your answer. File: Sketch...")
23:30, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 13/Solution 1 (Created page with "A = 2 \pi r^2 + 2\pi r h = 2 \pi r(r+h) </math> and the volume as <math display="block"> V = \pi r^2 h. </math> Our goal is to find the maximum and minimum volume of the cylinder, so we must find a formula for the volume that depends on only one variable. To do this, we solve for <math>h</math> in the expression for the area: <math display="block"> h = \frac{A}{2\pi r} - r = \frac{A- 2\pi r^2}{2\pi r}. </math> Substituting this expression into the formula for the volume...")
23:30, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 13/Hint 1 (Created page with "Write formulas for the surface area and volume of a cylinder in terms of its radius <math>r</math> and height <math>h</math>. Use the surface area constraint to eliminate one variable, so that the volume is expressed as a function of a single variable.")
23:29, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 13/Statement (Created page with "Consider a cylinder. If its total surface area, including both its top and bottom, is A, what is the largest possible volume it can enclose? What is the smallest possible volume? Your answers may depend on the constant A.")
23:24, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 09 (b)/Solution 1 (Created page with "Plugging in <math>x=0.1</math> to the answer from (a) we get <math display="block"> e^{0.3}\cos(0.1) \approx 1 + 0.3 + 4 (0.1)^2. </math>")
23:23, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 09 (b)/Hint 1 (Created page with "Match the expression <math>e^{0.3}\cos(0.1)</math> to the form <math>e^{3x} \cos (x)</math>. What value of <math>x</math> makes these expressions the same?")
23:23, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 09 (b)/Statement (Created page with "Use your answer from (a) to approximate <math>e^{0.3}\cos(0.1)</math> by a rational number. You may leave your answer in calculator-ready form.")
23:22, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 09 (a)/Solution 1 (Created page with "The second order Taylor polynomial for <math>\cos(x)</math> is given by <math display="block"> \cos (x) \approx 1- \frac{x^2}{2}. </math> Similarly, the second order Taylor polynomial for <math>e^y</math> is given by <math display="block"> e^y \approx 1 + y + \frac{y^2}{2}, </math> and setting <math>y=3x</math> gives <math display="block"> e^{5x} \approx 1 + 3x + \frac{9x^2}{2}. </math> Multiplying these polynomials together and truncating at the second order gives <math...")
23:16, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 09 (a)/Hint 1 (Created page with "Rather than deriving the Taylor polynomial from scratch you can find it by multiplying the Taylor polynomials for <math>e^{3x}</math> and <math>\cos (x)</math>.")
23:15, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 09 (a)/Statement (Created page with "Find the second-order Taylor polynomial for <math>e^{3x} \cos (x)</math> about <math>x=0</math>.")
23:14, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 08/Solution 1 (Created page with "This question is testing implicit differentiation, since <math> y </math> is a function of <math> x </math>. We start by taking logarithm of both sides of equation to get <math display="block">\begin{align} \log(y)=\log(x^{x^4})=x^4\log x \end{align}</math> Then differentiating with respect to <math> x </math>, we apply chain rule to <math> \log(y) </math>, and product rule to <math> =x^4\log x </math>: <math display="block">\begin{align} \frac{y'}{y}&=4x^3\log x + \fra...")
23:10, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 08/Hint 1 (Created page with "The variable appears both in the base and the exponent. Try taking the natural logarithm of both sides first, then differentiate implicitly.")
23:10, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 08/Statement (Created page with "Differentiate the function <math display="block"> y=x^{x^4} </math> with respect to <math> x </math>. Give your answer as a function of <math> x </math> only (not <math> y </math>).")
23:07, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 07/Solution 1 (Created page with "We begin by implicitly differentiating the equation, which gives <math display="block"> 2x +2y y' = 2(x^2 + y^2 -2x) (2x + 2y y' -2). </math> Hence, when <math>x=0 \ \text{and} \ y=1</math> this becomes <math display="block"> 2y' = 2(1)(2y'-2) = 4y' - 4, </math> so <math>y'=5</math>. The tangent line is therefore the line with slope <math>5</math> passing through the point <math>(0,1)</math>, and hence is given by <math display="block"> y=5x+1. </math>")
23:06, 14 April 2026 AlanGetz talk contribs created page Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 07/Hint 1 (Created page with "Don't try to solve for <math>y</math> before differentiating. Begin by differentiating implicitly, and then solve for <math>y'</math>.")