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

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.

5.

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. \(y=e^x\)

6.

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. \(\ds \frac{3\pi }{4}\)

7.

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. \((0,\infty )\)

8.

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. \(\ln 2\)

9.

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 \((a,b)\)

10.

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

11.

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. \(\ds \frac{5}{3}\)

12.

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. \(2.4\)

13.

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. \(1\)

14.

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 \(x=0\)

15.

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. \(2x\cos (2x)\)

16.

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

17.

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. \(\sin e\)

18.

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 \(\sqrt{5}\)

19.

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

20.

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. \(-\cot \theta\)

21.

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. \(4\)

22.

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.

23.

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.

24.

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

25.

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

26.

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

27.

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

28.

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

29.

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

30.

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

31.

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

32.

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

33.

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 \(\ds \lim_{x\to a} f(x)=f(a)\)

34.

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

35.

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 \(F"(x)=f(x)\)

36.

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 \(\ds (c,f(c))\)

37.

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

38.

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

39.

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

40.

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

41.

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

42.

Does differentiability imply continuity?

43.

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

44.

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

45.

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.

46.

Function that is continuous but not differentiable at a point.

47.

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

48.

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

49.

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

50.

State the definition of a critical number of a function.

51.

State the Extreme Value Theorem.