A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Chapter 5 True Or False and Multiple Choice Problems

Answer the following questions.

1. For each of the following ten statements answer TRUE or FALSE as appropriate:

   1. If \(f\) is differentiable on \([-1,1]\) then \(f\) is continuous at \(x=0\text{.}\)

   2. If \(f'(x)\lt 0\) and \(f"(x)>0\) for all \(x\) then \(f\) is concave down.

   3. The general antiderivative of \(f(x)=3x^2\) is \(F(x)=x^3\text{.}\)

   4. \(\ln x\) exists for any \(x>1\text{.}\)

   5. \(\ln x=\pi\) has a unique solution.

   6. \(e^{-x}\) is negative for some values of \(x\text{.}\)

   7. \(\ln e^{x^2}=x^2\) for all \(x\text{.}\)

   8. \(f(x)=|x|\) is differentiable for all \(x\text{.}\)

   9. \(\tan x\) is defined for all \(x\text{.}\)

   10. All critical points of \(f(x)\) satisfy \(f'(x)=0\text{.}\)

2. Answer each of the following either TRUE or FALSE.

   1. The function \(f(x)=\left\{ \begin{array}{lll} 3+\frac{\sin (x-2)}{x-2}\amp \mbox{if} \amp x\not=2 \\ 3\amp \mbox{if} \amp x=2 \end{array} \right.\) is continuous at all real numbers \(x\text{.}\)

   2. If \(f'(x)=g'(x)\) for \(0\lt x\lt 1\text{,}\) then \(f(x)=g(x)\) for \(0\lt x\lt 1\text{.}\)

   3. If \(f\) is increasing and \(f(x)>0\) on \(I\text{,}\) then \(\ds g(x)=\frac{1}{f(x)}\) is decreasing on \(I\text{.}\)

   4. There exists a function \(f\) such that \(f(1)=-2\text{,}\) \(f(3)=0\text{,}\) and \(f'(x)>1\) for all \(x\text{.}\)

   5. If \(f\) is differentiable, then \(\ds \frac{d}{dx}f(\sqrt{x})=\frac{f'(x)}{2\sqrt{x}}\text{.}\)

   6. \(\displaystyle \ds \frac{d}{dx}10^x=x10^{x-1}\)

   7. Let \(e=\exp (1)\) as usual. If \(y=e^2\) then \(y'=2e\text{.}\)

   8. If \(f(x)\) and \(g(x)\) are differentiable for all \(x\text{,}\) then \(\ds \frac{d}{dx}f(g(x))=f'(g(x))g'(x)\text{.}\)

   9. If \(g(x)=x^5\text{,}\) then \(\ds \lim _{x\to 2}\frac{g(x)-g(2)}{x-2}=80\text{.}\)

   10. An equation of the tangent line to the parabola \(y=x^2\) at \((-2,4)\) is \(y-4=2x(x+2)\text{.}\)

   11. \(\displaystyle \ds \frac{d}{dx}\tan ^2x=\frac{d}{dx}\sec ^2x\)

   12. For all real values of \(x\) we have that \(\ds \frac{d}{dx}|x^2+x|=|2x+1|\text{.}\)

   13. If \(f\) is one-to-one then \(\ds f^{-1}(x)=\frac{1}{f(x)}\text{.}\)

   14. If \(x>0\text{,}\) then \((\ln x)^6=6\ln x\text{.}\)

   15. If \(\ds \lim _{x\to 5}f(x)=0\) and \(\ds \lim _{x\to 5}g(x)=0\text{,}\) then \(\ds \lim _{x\to 5}\frac{f(x)}{g(x)}\) does not exist.

   16. If the line \(x=1\) is a vertical asymptote of \(y=f(x)\text{,}\) then \(f\) is not defined at 1.

   17. If \(f'(c)\) does not exist and \(f'(x)\) changes from positive to negative as \(x\) increases through \(c\text{,}\) then \(f(x)\) has a local minimum at \(x=c\text{.}\)

   18. \(\sqrt{a^2}=a\) for all \(a>0\text{.}\)

   19. If \(f(c)\) exists but \(f'(c)\) does not exist, then \(x=c\) is a critical point of \(f(x)\text{.}\)

   20. If \(f"(c)\) exists and \(f'''(c)>0\text{,}\) then \(f(x)\) has a local minimum at \(x=c\text{.}\)

3. Are the following statements TRUE or FALSE.

   1. \(\ds \lim _{x\to 3}\sqrt{x-3}=\sqrt{\lim _{x\to 3}(x-3)}\text{.}\)

   2. \(\displaystyle \ds \frac{d}{dx}\left( \frac{\ln 2^{\sqrt{x}}}{\sqrt{x}}\right) =0\)

   3. If \(f(x)=(1+x)(1+x^2)(1+x^3)(1+x^4)\text{,}\) then \(f'(0)=1\text{.}\)

   4. If \(y=f(x)=2^{|x|}\text{,}\) then the range of \(f\) is the set of all non-negative real numbers.

   5. \(\ds \frac{d}{dx}\left( \frac{\log x^2}{\log x}\right) =0\text{.}\)

   6. If \(f'(x)=-x^3\) and \(f(4)=3\text{,}\) then \(f(3)=2\text{.}\)

   7. If \(f"(c)\) exists and if \(f"(c)>0\text{,}\) then \(f(x)\) has a local minimum at \(x=c\text{.}\)

   8. \(\ds \frac{d}{du}\left( \frac{1}{\csc u}\right) =\frac{1}{\sec u}\text{.}\)

   9. \(\ds \frac{d}{dx}(\sin ^{-1}(\cos x)=-1\) for \(0\lt x\lt \pi\text{.}\)

   10. \(\sinh ^2x-\cosh ^2x=1\text{.}\)

   11. \(\ds \int \frac{dx}{x^2+1}=\ln (x^2+1)+C\text{.}\)

   12. \(\ds \int \frac{dx}{3-2x}=\frac{1}{2}\ln |3-2x|+C\text{.}\)

4. Answer each of the following either TRUE or FALSE.

   1. For all functions \(f\text{,}\) if \(f\) is continuous at a certain point \(x_0\text{,}\) then \(f\) is differentiable at \(x_0\text{.}\)

   2. For all functions \(f\text{,}\) if \(\ds \lim _{x\to a^-}f(x)\) exist, and \(\ds \lim _{x\to a^+}f(x)\) exist, then \(f\) is continuous at \(a\text{.}\)

   3. For all functions \(f\text{,}\) if \(a\lt b\text{,}\) \(f(a)\lt 0\text{,}\) \(f(b)>0\text{,}\) then there must be a number \(c\text{,}\) with \(a\lt c\lt b\) and \(f(c)=0\text{.}\)

   4. For all functions \(f\text{,}\) if \(f'(x)\) exists for all \(x\text{,}\) then \(f"(x)\) exists for all \(x\text{.}\)

   5. It is impossible for a function to be discontinues at every number \(x\text{.}\)

   6. If \(f\text{,}\) \(g\text{,}\) are any two functions which are continuous for all \(x\text{,}\) then \(\ds \frac{f}{g}\) is continuous for all \(x\text{.}\)

   7. It is possible that functions \(f\) and \(g\) are not continuous at a point \(x_0\text{,}\) but \(f+g\) is continuous at \(x_0\text{.}\)

   8. If \(\ds \lim _{x\to \infty }(f(x)+g(x))\) exists, then \(\ds \lim _{x\to \infty }f(x)\) exists and \(\ds \lim _{x\to \infty }g(x)\) exists.

   9. \(\displaystyle \ds \lim _{x\to \infty}\frac{(1.00001)^x}{x^{100000}}=0\)

   10. Every continuous function on the interval \((0,1)\) has a maximum value and a minimum value on \((0,1)\text{.}\)

5. Answer each of the following either TRUE or FALSE.

   1. Let \(f\) and \(g\) be any two functions which are continuous on \([0,1]\text{,}\) with \(f(0)=g(0)=0\) and \(f(1)=g(1)=10\text{.}\) Then there must exist \(c,d\in [0,1]\) such that \(f'(c)=g'(d)\text{.}\)

   2. Let \(f\) and \(g\) be any two functions which are continuous on \([0,1]\) and differentiable on \((0,1)\text{,}\) with \(f(0)=g(0)=0\) and \(f(1)=g(1)=10\text{.}\) Then there must exist \(c\in [0,1]\) such that \(f'(c)=g'(c)\text{.}\)

   3. For all \(x\) in the domain of \(\sec ^{-1}x\text{,}\)

      \begin{equation*} \sec (\sec ^{-1}(x))=x\text{.} \end{equation*}

6. Answer each of the following either TRUE or FALSE.

   1. The slope of the tangent line of \(f(x)\) at the point \((a,f(a))\) is given by \(\ds \frac{f(a+h)-f(a)}{h}\text{.}\)

   2. Using the Intermediate Value Theorem it can be shown that \(\ds \lim _{x\to 0}x\sin \frac{1}{x}=0\text{.}\)

   3. The graph on Figure 5.0.1 exhibits three types of discontinuities.

      

   4. If \(w=f(x)\text{,}\) \(x=g(y)\text{,}\) \(y=h(z)\text{,}\) then \(\ds \frac{dw}{dz}=\frac{dw}{dx}\cdot \frac{dx}{dy}\cdot \frac{dy}{dz}\text{.}\)

   5. Suppose that on the open interval \(I\text{,}\) \(f\) is a differentiable function that has an inverse function \(f^{-1}\) and \(f'(x)\not= 0\text{.}\) Then \(f^{-1}\) is differentiable and \(\ds \left( f^{-1}(x)\right) '=\frac{1}{f'(f^{-1}(x))}\) for all \(x\) in the domain of \(f^{-1}\text{.}\)

   6. If the graph of \(f\) is on Figure 5.0.2 to the left, the graph to the right must be that of \(f^\prime\text{.}\)

      

      

   7. The conclusion of the Mean Value Theorem says that the graph of \(f\) has at least one tangent line in \((a,b)\text{,}\) whose slope is equal to the average slope on \([a,b]\text{.}\)

   8. The linear approximation \(L(x)\) of a function \(f(x)\) near the point \(x=a\) is given by \(L(x)=f'(a)+f(a)(x-a)\text{.}\)

   9. The graphs in Figure 5.0.3 are labeled correctly with possible eccentricities for the given conic sections:

      

      

      

      

   10. Given \(h(x)=g(f(x))\) and the graphs of \(f\) and \(g\) on Figure 5.0.4 then a good estimate for \(h'(3)\) is \(-\frac{1}{4}\text{.}\)

       

7. Answer TRUE or FALSE to the following questions.

   1. If \(f(x)=7x+8\) then \(f'(2)=f'(17.38)\text{.}\)

   2. If \(f(x)\) is any function such that \(\ds \lim _{x\to 2}f(x)=6\) the \(\ds \lim _{x\to 2^+}f(x)=6\text{.}\)

   3. If \(f(x)=x^2\) and \(g(x)=x+1\) then \(f(g(x))=x^2+1\text{.}\)

   4. The average rate of change of \(f(x)\) from \(x=3\) to \(x=3.5\) is \(2(f(3.5)-f(3))\text{.}\)

   5. An equivalent precise definition of \(\ds \lim _{x\to a}f(x)=L\) is: For any \(0\lt \epsilon \lt 0.13\) there is \(\delta >0\) such that

      \begin{equation*} \mbox{if } |x-a|\lt \delta \mbox{ then } |f(x)-L|\lt \epsilon\text{.} \end{equation*}

      The last four True/False questions ALL pertain to the following function. Let

      \begin{equation*} f(x)\left\{ \begin{array}{lll} x-4\amp \mbox{if} \amp x\lt 2\\ 23\amp \mbox{if} \amp x=2\\ x^2+7\amp \mbox{if} \amp x>2 \end{array} \right. \end{equation*}

   6. \(\displaystyle f(3)=-1\)

   7. \(\displaystyle f(2)=11\)

   8. \(f\) is continuous at \(x=3\text{.}\)

   9. \(f\) is continuous at \(x=2\text{.}\)

8. Answer TRUE or FALSE to the following questions.

   1. If a particle has a constant acceleration, then its position function is a cubic polynomial.

   2. If \(f(x)\) is differentiable on the open interval \((a,b)\) then by the Mean Value Theorem there is a number \(c\) in \((a,b)\) such that \((b-a)f'(c)=f(b)-f(a)\text{.}\)

   3. If \(\ds \lim _{x\to \infty }\left( \frac{k}{f(x)}\right) =0\) for every number \(k\text{,}\) then \(\ds \lim _{x\to \infty }f(x)=\infty\text{.}\)

   4. If \(f(x)\) has an absolute minimum at \(x=c\text{,}\) then \(f'(c)=0\text{.}\)

9. True or False. Give a brief justification for each answer.

   1. There is a differentiable function \(f(x)\) with the property that \(f(1)=-2\) and \(f(5)=14\) and \(f^\prime (x)\lt 3\) for every real number \(x\text{.}\)

   2. If \(f"(5)=0\) then \((5,f(5))\) is an inflection point of the curve \(y=f(x)\text{.}\)

   3. If \(f^\prime (c)=0\) then \(f(x)\) has a local maximum or a local minimum at \(x=c\text{.}\)

   4. If \(f(x)\) is a differentiable function and the equation \(f^\prime (x)=0\) has 2 solutions, then the equation \(f(x)=0\) has no more than 3 solutions.

   5. If \(f(x)\) is increasing on \([0,1]\) then \([f(x)]^2\) is increasing on \([0,1]\text{.}\)

10. Answer the following questions TRUE or False.

    1. If \(f\) has a vertical asymptote at \(x=1\) then \(\ds \lim _{x\to 1}f(x)=L\text{,}\) where \(L\) is a finite value.

    2. If has domain \([0,\infty )\) and has no horizontal asymptotes, then \(\lim _{x\to \infty }f(x)=\pm \infty\text{.}\)

    3. If \(g(x)=x^2\) then \(\ds \lim _{x\to 2}\frac{g(x)-g(2)}{x-2}=0\text{.}\)

    4. If \(f"(2)=0\) then \((2,f(2))\) is an inflection point of \(f(x)\text{.}\)

    5. If \(f^\prime(c)=0\) then \(f\) has a local extremum at \(c\text{.}\)

    6. If \(f\) has an absolute minimum at \(c\) then \(f^\prime (c)=0\text{.}\)

    7. If \(f^\prime (c)\) exists, then \(\ds \lim _{x\to c}f(x)=f(c)\text{.}\)

    8. If \(f(1)\lt 0\) and \(f(3)>0\text{,}\) then there exists a number \(c\in (1,3)\) such that \(f(c)=0\text{.}\)

    9. If \(\ds f^\prime (g)=\frac{1}{(3-g)^2}\text{,}\) then \(f(g)\) is differentiable on \((-\infty ,3)\cup (3,\infty )\text{.}\)

    10. If \(\ds f^\prime (g)=\frac{1}{(3-g)^2}\text{,}\) the equation of the tangent line to \(f(g)\) at \((0,1/3)\) is \(y=\frac{1}{9}g+\frac{1}{3}\text{.}\)

11. Are the following statements true or false?

    1. The points described by the polar coordinates \((2,\pi /4)\) and \((-2,5\pi /4)\) are the same.

    2. If the limit \(\displaystyle \lim _{x\to \infty }\frac{f^\prime (x)}{g^\prime (x)}\) does not exist, then the limit \(\displaystyle \lim _{x\to \infty }\frac{f(x)}{g(x)}\) does not exist.

    3. If \(f\) is a function for which \(f"(x)=0\text{,}\) then \(f\) has an inflection point at \(x\text{.}\)

    4. If \(f\) is continuous at the number \(x\text{,}\) then it is differentiable at \(x\text{.}\)

    5. Let \(f\) be a function and \(c\) a number in its domain. The graph of the linear approximation of \(f\) at \(c\) is the tangent line to the curve \(y=f(x)\) at the point \((c,f(c))\text{.}\)

    6. Every function is either an odd function or an even function.

    7. A function that is continuous on a closed interval attains an absolute maximum value and an absolute minimum value at numbers in that interval.

    8. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points is a constant.

12. For each statement indicate whether is True or False.

    1. There exists a function \(g\) such that \(g(1)=-2\text{,}\) \(g(3)=6\) and \(g^\prime(x)>4\) for all \(x\text{.}\)

    2. If \(f(x)\) is continuous and \(f^\prime(2)=0\) then \(f\) has either a local maximum to minimum at \(x=2\text{.}\)

    3. If \(f(x)\) does not have an absolute maximum on the interval \([a,b]\) then \(f\) is not continuous on \([a,b]\text{.}\)

    4. If a function \(f(x)\) has a zero at \(x=r\text{,}\) then Newton's method will find \(r\) given an initial guess \(x_0\not= r\) when \(x_0\) is close enough to \(r\text{.}\)

    5. If \(f(3)=g(3)\) and \(f^\prime(x)=g^\prime(x)\) for all \(x\text{,}\) then \(f(x)=g(x)\text{.}\)

    6. The function \(\ds g(x)=\frac{7x^4-x^3+5x^2+3}{x^2+1}\) has a slant asymptote.

13. For each statement indicate whether is True or False.

    1. If \(\ds \lim_{x\to a}f(x)\) exists then \(\ds \lim_{x\to a}\sqrt{f(x)}\) exists.

    2. If \(\ds \lim_{x\to 1}f(x)=0\) and \(\ds \lim_{x\to 1}g(x)=0\) then \(\ds \lim_{x\to 1}\frac{f(x)}{g(x)}\) does not exist.

    3. \(\ds \sin^{-1}\left(\sin \left(\frac{7\pi}{3}\right)\right)=\frac{7\pi}{3}\text{.}\)

    4. If \(h(3)=2\) then \(\ds \lim_{x\to 3}h(x)=2\text{.}\)

    5. The equation \(\ds e^{-x^2}=x\) has a solution on the interval \((0,1)\text{.}\)

    6. If \((4,1)\) is a point on the graph of \(h\) then \((4,0)\) is a point on the graph \(f\circ h\) where \(f(x)=3^x+x-4\text{.}\)

    7. If \(-x^3+3x^2+1\leq g(x)\leq (x-2)^2+5\) for \(x\geq 0\) then \(\ds \lim _{x\to 2}g(x)=5\text{.}\)

    8. If \(g(x)=\ln x\text{,}\) then \(g(g^{-1}(0))=0\text{.}\)

14. For each of the following, circle only one answer.

    1. If \(h(x)=\ln (1-x^2)\) where \(-1\lt x\lt 1\text{,}\) then \(h^\prime(x)=\)

       A. \(\ds \frac{1}{1-x^2}\text{,}\) B. \(\ds \frac{1}{1+x}-\frac{1}{1-x}\text{,}\) C. \(\ds \frac{2}{1-x^2}\text{,}\) D. None of these.

    2. The derivative of \(f(x)=x^2\tan x\) is

       A. \(2x\sec^2x\text{,}\) B. \(2x\tan x+x^2\cot x\text{,}\) C. \(2x\tan x+(x\sec x)^2\text{,}\) D. None of these.

    3. If \(\cosh y=x+x^3y\text{,}\) then at the point \((1,0)\) we have \(y^\prime\)

       A. \(0\text{,}\) B. \(3\text{,}\) C. \(-1\text{,}\) D. Does not exist.

    4. The derivative of \(\ds g(x)=e^{\sqrt{x}}\) is

       A. \(e^{\sqrt{x}}\text{,}\) B. \(\sqrt{x}e^{\sqrt{x}-1}\text{,}\) C. \(\frac{0.5e^{\sqrt{x}}}{\sqrt{x}}\text{,}\) D. None of these.

15. For each of the following, circle only one answer.

    1. Suppose \(y^{\prime\prime}+y=0\text{.}\) Which of the following is a possibility for \(y=f(x)\text{.}\)

       A. \(y=\tan x\text{,}\) B. \(y=\sin x\text{,}\) C. \(y=\sec x\text{,}\) D. \(y=1/x\text{,}\) E. \(y=e^x\)

    2. Which of the following is \(\ds \arcsin \left( \sin \left( \frac{3\pi }{4}\right) \right)\text{?}\)

       A. \(0\text{,}\) B. \(\ds \frac{\pi }{4}\text{,}\) C. \(\ds -\frac{\pi }{4}\text{,}\) D. \(\ds \frac{5\pi }{4}\text{,}\) E. \(\ds \frac{3\pi }{4}\)

    3. Let \(f(x)\) be a continuous function on \([a,b]\) and differentiable on \((a,b)\) such that \(f(b)=10\text{,}\) \(f(a)=2\text{.}\) On which of the following intervals \([a,b]\) would the Mean Value Theorem guarantee a \(c\in (a,b)\) such that \(f'(c)=4\text{.}\)

       A. \([0,4]\text{,}\) B. \([0,3]\text{,}\) C. \([2,4]\text{,}\) D. \([1,10]\text{,}\) E. \((0,\infty )\)

    4. Let \(P(t)\) be the function which gives the population as a function of time. Assuming that \(P(t)\) satisfies the natural growth equation, and that at some point in time \(t_0\text{,}\) \(P(t_0)=500\text{,}\) \(P'(t_0)=1000\text{,}\) find the growth rate constant \(k\text{.}\)

       A. \(\ds -\frac{1}{2}\text{,}\) B. \(\ds \ln \left( \frac{1}{2}\right)\text{,}\) C. \(\ds \frac{1}{2}\text{,}\) D. \(2\text{,}\) E. \(\ln 2\)

    5. Suppose that \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) If \(f^\prime(x)>0\) on \((a,b)\text{.}\) Which of the following is necessarily true?

       A. \(f\) is decreasing on \([a,b]\text{,}\)

       B. \(f\) has no local extrema on \((a,b)\text{,}\)

       C. \(f\) is a constant function on \((a,b)\text{,}\)

       D. \(f\) is concave up on \((a,b)\text{,}\)

       E. \(f\) has no zero on \((a,b)\)

16. For each of the following, circle only one answer.

    1. The equation \(x^5+10x+3=0\) has

       A. no real roots, B. exactly one real root, C. exactly two real roots, D. exactly three real roots, E. exactly five real roots

    2. The value of \(\cosh (\ln 3)\) is

       A. \(\ds \frac{1}{3}\text{,}\) B. \(\ds \frac{1}{2}\text{,}\) C. \(\ds \frac{2}{3}\text{,}\) D. \(\ds \frac{4}{3}\text{,}\) E. \(\ds \frac{5}{3}\)

    3. The function \(f\) has the property that \(f(3)=2\) and \(f'(3)=4\text{.}\) Using a linear approximation to \(f\) near \(x=3\text{,}\) an approximation to \(f(2.9)\) is

       A. \(1.4\text{,}\) B. \(1.6\text{,}\) C. \(1.8\text{,}\) D. \(1.9\text{,}\) E. \(2.4\)

    4. Suppose \(F\) is an antiderivative of \(f(x)=\sqrt[3]{x}\text{.}\) If \(\ds F(0)=\frac{1}{4}\text{,}\) then \(F(1)\) is

       A. \(-1\text{,}\) B. \(\ds -\frac{3}{4}\text{,}\) C. \(0\text{,}\) D. \(\ds \frac{3}{4}\text{,}\) E. \(1\)

    5. Suppose \(f\) is a function such that \(f'(x)=4x^3\) and \(f"(x)=12x^2\text{.}\) Which of the following is true?

       A. \(f\) has a local maximum at \(x=0\) by the first derivative test

       B. \(f\) has a local minimum at \(x=0\) by the first derivative test.

       C. \(f\) has a local maximum at \(x=0\) by the second derivative test.

       D. \(f\) has a local minimum at \(x=0\) by the second derivative test.

       E. \(f\) has an inflection point at \(x=0\)

17. Circle clearly your answer to the following 10 multiple choice question.

    1. Evaluate \(\ds \frac{d}{dx}\sin (x^2)\)

       A. \(2x\cos (x^2)\text{,}\) B. \(2x\sin (x^2)\text{,}\) C. \(2x\cos (x)\text{,}\) D. \(2x\cos (2x)\text{,}\) E. \(2x\cos (2x)\)

    2. Evaluate \(\ds \lim _{x\to 0^+}\frac{\ln x}{x}\)

       A. \(0\text{,}\) B. \(\infty\text{,}\) C. \(1\text{,}\) D. \(-1\text{,}\) E. none of the above

    3. Evaluate \(\ds \lim _{x\to 0^+}\frac{1-e^x}{\sin x}\)

       A. \(1\text{,}\) B. \(-1\text{,}\) C. \(0\text{,}\) D. \(\infty\text{,}\) E. \(\sin e\)

    4. The circle described by the equation \(x^2+y^2-2x-4=0\) has center \((h,k)\) and radius \(r\text{.}\) The values of \(h\text{,}\) \(k\text{,}\) and \(r\) are

       A. \(0\text{,}\) \(1\text{,}\) and \(\sqrt{5}\text{,}\) B. \(1\text{,}\) \(0\text{,}\) and \(5\text{,}\) C. \(1\text{,}\) \(0\text{,}\) and \(\sqrt{5}\text{,}\) D. \(-1\text{,}\) \(0\text{,}\) and \(5\text{,}\) E. \(-1\text{,}\) \(0\text{,}\) and \(\sqrt{5}\)

    5. The edge of the cube is increasing at a rate of 2 cm/hr. How fast is the cube's volume changing when its edge is \(\sqrt{x}\) cm in length?

       A. 6 cm\(^3\)/hr, B. 12 cm\(^3\)/hr, C. \(3\sqrt{2}\) cm\(^3\)/hr, D. \(6\sqrt{2}\) cm\(^3\)/hr, E. none of the above

    6. Given the polar equation \(r=1\text{,}\) find \(\ds \frac{dy}{dx}\)

       A. \(\cot \theta\text{,}\) B. \(-\tan \theta\text{,}\) C. \(0\text{,}\) D. \(1\text{,}\) E. \(-\cot \theta\)

    7. Let \(A(t)\) denote the amount of a certain radioactive material left after time \(t\text{.}\) Assume that \(A(0)=16\) and \(A(1)=12\text{.}\) How much time is left after time \(t=3\text{?}\)

       A. \(\ds \frac{16}{9}\text{,}\) B. \(8\text{,}\) C. \(\ds \frac{9}{4}\text{,}\) D. \(\ds \frac{27}{4}\text{,}\) E. \(4\)

    8. Which of the following statements is always true for a function \(f(x)\text{?}\)

       1. If \(f(x)\) is concave up on the interval \((a,b)\text{,}\) then \(f(x)\) has a local minimum \((a,b)\text{.}\)

       2. It is possible for \(y=f(x)\) to have an inflection point at \((a,f(a))\) even if \(f'(x)\) does not exist at\(x=a\text{.}\)

       3. It is possible for \((a,f(a))\) to be both a critical point and an inflection point of \(f(x)\text{.}\)

       A. i. and ii.

       B. only iii.

       C. i., ii., and iii.

       D. ii. and iii.

       E. only i.

    9. Which of the following statements is always true for a function \(f(x)\text{?}\)

       1. If \(f(x)\) and \(g(x)\) are continuous at \(x=a\text{,}\) then \(\ds \frac{f(x)}{g(x)}\) is continuous at \(x=a\text{.}\)

       2. If \(f(x)+g(x)\) is continuous at \(x=a\) and \(f'(a)=0\text{,}\) then \(g(x)\) is continuous ta \(x=a\text{.}\)

       3. If \(f(x)+g(x)\) is differentiable at \(x=a\text{,}\) then \(f(x)\) and \(g(x)\) are both differentiable at \(x=a\text{.}\)

       A. only i.

       B. only ii.

       C. only iii.

       D. i. and ii.

       E. ii. and iii.

    10. The slant asymptote of the function \(\ds f(x)=\frac{x^2+3x-1}{x-1}\) is

        A. \(y=x+4\text{,}\) B. \(y=x+2\text{,}\) C. \(y=x-2\text{,}\) D. \(y=x-4\text{,}\) E. none of the above

18. This is a multiple choice question. No explanation is required

    1. The derivative of \(\ds g(x)=e^{\sqrt{x}}\) is

       A. \(\sqrt{x}e^{\sqrt{x}-1}\text{,}\)

       B. \(2e^{\sqrt{x}}x^{-0.5}\text{,}\)

       C. \(\ds \frac{0.5e^{\sqrt{x}}}{\sqrt{x}}\text{,}\)

       D. \(e^{\sqrt{x}}\text{,}\)

       E. None of these

    2. If \(\cosh y=x+x^3y\text{,}\) then at the point \((1,0)\) \(y^\prime =\)

       A. \(0\text{,}\)

       B. \(-1\text{,}\)

       C. \(1\text{,}\)

       D. \(3\text{,}\)

       E. Does not exist

    3. An antiderivative of \(f(x)=x-\sin x+e^x\) is

       A. \(1-\cos x +e^x\text{,}\)

       B. \(x^2+\ln x-\cos x\text{,}\)

       C. \(\ds 0.5x^2+e^x-\cos x\text{,}\)

       D. \(\cos x +e^x+0.5x^2\text{,}\)

       E. None of these

    4. If \(h(x)=\ln (1-x^2)\) where \(-1\lt x\lt 1\text{,}\) then \(h^\prime (x)=\)

       A. \(\ds \frac{1}{1-x^2}\text{,}\)

       B. \(\ds \frac{1}{1+x}+\frac{1}{1-x}\text{,}\)

       C. \(\ds \frac{2}{1-x^2}\text{,}\)

       D. \(\ds \frac{1}{1+x}-\frac{1}{1-x}\text{,}\)

       E. None of these

    5. The linear approximation to \(f(x)=\sqrt[3]{x}\) at \(x=8\) is given by

       A. \(2\text{,}\)

       B. \(\ds \frac{x+16}{12}\text{,}\)

       C. \(\ds \frac{1}{3x^{2/3}}\text{,}\)

       D. \(\ds \frac{x-2}{3}\text{,}\)

       E. \(\ds \sqrt[3]{x}-2\)

19. This is a multiple choice question. No explanation is required

    1. If a function \(f\) is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\text{,}\) then there exists \(c\in (a,b)\) such that \(\ds f(b)-f(a)=f^\prime(c)(b-a)\) is:

       A. The Extreme Value Theorem,

       B. The Intermediate Value Theorem,

       C. The Mean Value Theorem,

       D. Rolle's Theorem,

       E. None of these

    2. If \(f\) is continuous function on the closed interval \([a,b]\text{,}\) and \(N\) is a number between \(f(a)\) and \(f(b)\text{,}\) then there is \(c\in [a,b]\) such that \(f(c)=N\) is:

       A. Fermat's Theorem

       B. The Intermediate Value Theorem

       C. The Mean Value Theorem

       D. Rolle's Theorem

       E. The Extreme Value Theorem

    3. If \(f\) is continuous function on the open interval \((a,b)\) then \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c,d\in (a,b)\) is:

       A. The Extreme Value Theorem,

       B. The Intermediate Value Theorem,

       C. The Mean Value Theorem,

       D. Rolle's Theorem,

       E. None of these

    4. A function \(f\) is continuous at a number \(a\) …

       A. … if \(f\) is defined at \(a\text{,}\)

       B. … if \(\ds \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\) exists,

       C. … if \(\ds \lim_{x\to a} f(x)\) exists,

       D. … if \(f\) is anti-differentiable at \(a\text{,}\)

       E. … if \(\ds \lim_{x\to a} f(x)=f(a)\)

    5. A function \(f\) is differentiable at a number \(a\) …

       A. … if \(\ds \lim_{x\to a} f(x)=f(a)\text{,}\)

       B. … if \(\ds \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\) exists,

       C. … if \(f\) is defined at \(a\text{,}\)

       D. … if \(f\) is continuous at \(a\text{,}\)

       E. … if we can apply the Intermediate Value Theorem

    6. An antiderivative of a function \(f\) …

       A. … is a function \(\ds F\) such that \(F^\prime(x)=f(x)\text{,}\)

       B. … is a function \(\ds F\) such that \(F(x)=f^\prime(x)\text{,}\)

       C.… is a function \(\ds F\) such that \(F^\prime(x)=f^\prime(x)\text{,}\)

       D. … is a function \(\ds F\) such that \(F(x)=f(x)\text{,}\)

       E. … is a function \(\ds F\) such that \(F"(x)=f(x)\)

    7. A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that …

       A. … \(\ds f^\prime(c)=0\text{,}\)

       B. … \(\ds f(c)\) is a local extremum,

       C. … either \(\ds f^\prime(c)=0\) or \(f^\prime(x)\) is not defined,

       D.… \(\ds (c,f(c))\) is an inflection point,

       E. … we can apply the Extreme Value Theorem in the neighbourhood of the point \(\ds (c,f(c))\)

20. Answer the following questions. You need not show work for this section.

    1. What is the period of \(f(x)=\tan x\text{?}\)

    2. What is the derivative of \(f(x)=x\ln |x| -x\text{?}\)

    3. If \(y=\sin ^2 x\) and \(\ds \frac{dx}{dt}=4\text{,}\) find \(\ds \frac{dy}{dt}\) when \(x=\pi\text{.}\)

    4. What is the most general antiderivative of \(f(x)=2xe^{x^2}\text{?}\)

    5. Evaluate \(\ds \lim _{t\to \infty }(\ln (t+1)-\ln t)\text{?}\)

    6. Does differentiability imply continuity?

    7. Convert the Cartesian equation \(x^2+y^2=25\) into a polar equation.

    8. Simplify \(\cosh ^2x-\sinh ^2x\text{.}\)

21. Give an example for the each of the following:

    1. Function \(F=f\cdot g\) so that the limits of \(F\) and \(f\) at \(a\) exist and the limit of \(g\) at \(a\) does not exist.

    2. Function that is continuous but not differentiable at a point.

    3. Function with a critical number but no local maximum or minimum.

    4. Function with a local minimum at which its second derivative equals 0.

22. 1. State the definition of the derivative of function \(f\) at a number \(a\text{.}\)

    2. State the definition of a critical number of a function.

    3. State the Extreme Value Theorem.

23. Match the start of each definition/theorem with its conclusion.

    1. The Mean Value Theorem states that …

    2. The chain rule states that …

    3. A critical number is a number that …

    4. The Extreme Value Theorem states that …

    5. Fermat's Theorem states that

    6. An antiderivative of a function \(f\) is …

    7. The natural number \(e\) is …

    8. An inflection point is a point …

    9. The derivative of a function \(f\) at a number \(a\) is …

    10. L'Hospital's Rule states that …

    11. The Intermediate Value Theorem states that …

    12. A function \(f\) is continuous at a number \(a\) …

    13. The Squeeze Theorem states that …

        1. … if \(f\) is continuous on the closed interval \([a,b]\) and let \(N\) be any number between \(f(a)\) and \(f(b)\text{,}\) \(f(a)\not= f(b)\text{.}\) Then there exists a number \(c\) in \((a,b)\) such that \(f(c)=N\text{.}\)

        2. … if \(f\) is a function that satisfies the following hypotheses:

           1. \(f\) is continuous on the closed interval \([a,b]\text{.}\)

           2. \(f\) is differentiable on the open interval \((a,b)\text{.}\)

           Then there is a number \(c\) in \((a,b)\) such that \(\displaystyle f^\prime (c)=\frac{f(b)-f(a)}{b-a}\text{.}\)

        3. … \(\displaystyle f^\prime (a)=\lim _{h\to 0}\frac{f(a+h)-f(a)}{h}\) if this limit exists.

        4. … If \(f\) is continuous on a closed interval \([a,b]\text{,}\) then \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c,d\in [a,b]\text{.}\)

        5. … is in the domain of \(f\) such that either \(f^\prime (c)=0\) or \(f^\prime (c)\) does not exist.

        6. … on a continuous curve where the curve changes from concave upward to concave downward or from concave downward to concave upward.

        7. … the base of the exponential function which has a tangent line of slope \(1\) at \((0,1)\text{.}\)

        8. … If \(f\) and \(g\) are both differentiable then \(\displaystyle \frac{d}{dx}\left[ f(g(x))\right] =f^\prime (g(x))\cdot g^\prime (x)\text{.}\)

        9. … If \(f(x)\leq g(x)\leq h(x)\) and \(\ds \lim_{x\to a}f(x)=\lim_{x\to a}h(x)=L\) then \(\lim_{x\to a}g(x)=L\text{.}\)