## Section 5.2 Multiple Choice

### Exercises 5.2.1 Exercises

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.

For each of the following, circle only one answer.

###### 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 \((a,b)\)

For each of the following, circle only one answer.

###### 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 \(x=0\)

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

###### 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 \(f(x)\text{?}\)

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

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

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 \(f(x)\text{?}\)

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

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

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

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

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

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

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

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

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

Give an example for the each of the following:

###### 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.

Answer the following.

###### 49.

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

The derivative of function \(f\) at a number \(a\text{,}\) denoted by \(f^\prime(a)\text{,}\) is \(\ds f'(a)= \lim _{h \to 0} \frac{f (a+h)- f (a)}{h}\) if this limit exists.

###### 50.

State the definition of a critical number of a function.

A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that \(f'(c)=0\) or \(f'(c)\) does not exist.

###### 51.

State the Extreme Value Theorem.

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

###### 52.

Match the start of each definition/theorem \(a - m\) with its conclusion \(i - ix\text{.}\) If there is no match, please state “no match”.

The

*Mean Value Theorem states*that …The

*chain rule*states that …A

*critical number*is a number that …The

*Extreme Value Theorem*states that …*Fermat's Theorem*states thatAn

*antiderivative*of a function \(f\) is …The

*natural number*\(e\) is …An

*inflection point*is a point …The

*derivative*of a function \(f\) at a number \(a\) is …*L'Hospital's Rule*states that …The

*Intermediate Value Theorem*states that …A function \(f\) is continuous at a number \(a\) …

The Squeeze Theorem states that …

… 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 \(f(c)=N\text{.}\)

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

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

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

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

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

… 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 \(c,d\in [a,b]\text{.}\)

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

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

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

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

… 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 \(\lim_{x\to a}g(x)=L\text{.}\)

\(\displaystyle ii. \)

\(\displaystyle viii. \)

\(\displaystyle v. \)

\(\displaystyle iv. \)

*no match.*no match.

\(\displaystyle vii. \)

\(\displaystyle vi. \)

\(\displaystyle iii.\)

no match.

\(\displaystyle i.\)

no match.

\(\displaystyle ix. \)