Differential Calculus for Life Sciences, Term Test 2, November 8, 2019
Aids Permitted: None
Time Allotted: 110 Minutes
Instructions
- Please have your student card ready for inspection, turn off all cellular phones, and read all the instructions carefully.
- This test contains two parts: Part A (36 marks) contains nine short questions, and Part B (64 marks) contains four questions. All answers are to be given in this booklet.
- Please do not write anything on the QR codes that appear at the top of each page.
- Check that this test has 12 pages, including this cover page
- There is a formula sheet on page 11, and space for rough work. You can also use page 10 and 12 for rough work. You may tear off the formula sheet, but you must submit it together with the rest of the test.
- Make sure to provide exact answers, using symbols such as \(\sqrt{ }, e\) and \(\pi\), if needed.
PART A (36 marks)
In this part, clearly indicate your final short answer in the appropriate box.
You must show your work (if any), even though only the final answer will be graded.
Simplify your answers as much as possible. Each question is worth 4 marks.
- Let \(f(x)=\frac{3+x}{9-x^2}\). Find \(\lim_{x\to-3^-}f(x)\quad,\quad\lim_{x\to3^+}f(x)\) and \(\lim_{x\to+\infty}f(x)\). Your answer should be a number, \(+\infty \), \(-\infty \) or DNE if the limit does not exist.
- Suppose that \(g(x)\) is a differentiable function, satisfying \(g(3)=g'(3)=4\). Find the equation of the tangent line to the graph of \(g(x)\) at \(x=3\). Write your answer in the form \(y=mx+b\).
- Which of the following limits represents the derivative of \(y=\ln x\) at \(x=1\)? Circle the correct limit from the options below.
- \(\lim_{z\to1}\frac{\ln z}{z}\)
- \(\lim_{h\to0}\frac{\ln(1+h)}{h}\)
- \(\lim\limits_{h\to0}\frac{\ln(1+h)}{h}\)
- \(\lim_{h\to0}\frac{\ln h}{h-1}\)
- \(\lim_{z\to1}\frac{\ln(1+z)}{z-1}\)
- \(\lim_{h\to0}\frac{\ln(h-1)}{1+h}\)
- If \(f(x)=\frac{e^x}{\sqrt{x}}\), what is \(f'(1)\)? Simplify your answer, and express it using \(e\), if needed.
- Calculate the second derivative of \(g(x)=\sin(x^{2})\) at \(x=\sqrt{\pi}\). Use \(\pi\) in your answer, if needed.
- Consider the function \(y=\sqrt{x^{1-c}}+\ln x\) (where \(c\) is a constant). If \(\frac{dy}{dx}=\frac{7}{6}\) when \(x=1\), what is the value of the constant \(c\)?
- A ball is dropped from a tall building. The height of the ball (in meters) at time \(t\) (in seconds) is given by \(h(t)=320-5t^2\quad(\mathrm{for}\quad t\geq0)\). How fast is the ball travelling when it hits the ground?
- Find the slope of the tangent line to the curve \(e^x-2y{=}\cos y{-}2x\quad\text{at}\quad(0 ,0)\).
- Suppose that a function \(f(x)\) has an inverse. If the graph of the inverse function \(f^{-1}(x)\) passes through the point \((-2,3)\), and has a slope 5 at that point, what are \(f(3)\) and \(f'(3)\)?
PART B (64 marks)
In this part you are required to provide full solutions and to show all your work.
A correct answer obtained with false reasoning or with no reasoning will not receive any marks.
Each question is worth 16 marks.
Find the equations of all the horizontal and vertical asymptotes of the function \(g(x)=\frac{\sqrt{8x}}{\sqrt{2x}-3}\), if there are any. Show and explain your work.
Here is the graph of a function \(f(x)\) for \(-2<x<4\). Answer the questions below.
(a) There are two x-values where \(f(x)\) exists but has no derivative. What are they? Explain why there is no derivative at each of those values.
(b) If we drew the graph of the derivative \(f'(x)\), what would we see at \(x=1\)? A corner? A jump? A hole? Something else? Explain.
Let \(h(x)=x^{\tan^{-1}x}\).
(a) Calculate \(h'(x)\). Make sure your final answer is given in terms of \(x\) only.
(b) Find the x-intercept of the line that is tangent to the graph of \(h(x)\) at \(x=1\). Express your answer using \(\pi\), if needed.
- Use the definition of the derivative as a limit (i.e., first principles) to calculate the derivative of \(g(x)=\sqrt{x^2+7}\) at \(x=3\). Show your work, and do not use any differentiation rules.
FORMULAS
\[\sin(A+B)=\sin A\cos B+\cos A\sin B\]
\[\cos^2\theta=\frac{1+\cos(2\theta)}{2}\]
\[\tan\theta=\frac{\sin\theta}{\cos\theta}\]
\[\sin\left(\frac\pi4\right)=\cos\left(\frac\pi4\right)=\frac{\sqrt2}2\]
\[\sin\left(\frac\pi6\right)=\cos\left(\frac\pi3\right)=\frac12\]
\[[\log_a(x)]'=\frac{1}{x\cdot\ln a}\quad\text{(for} 0<a\neq1)\]
\[\cos(A+B)=\cos A\cos B-\sin A\sin B\]
\[\sin^2\theta=\frac{1-\cos(2\theta)}{2}\]
\[\sin^2\theta+\cos^2\theta=1\]
\[\sin\left(\frac\pi3\right)=\cos\left(\frac\pi6\right)=\frac{\sqrt3}2\]
\[\tan\left(\frac{\pi}{4}\right)=1\]
\[(a^x)'=a^x\cdot\ln a\quad\text{(for} \quad a>0)\]
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