Section5.2Multiple Choice

Exercises5.2.1Exercises

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.

B.
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.

C.
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.

C.
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.

C.

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. $y=e^x$

B.
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. $\ds \frac{3\pi }{4}$

C. The range of $y=\arcsin x$ is $\ds\left[ -\frac{\pi }{2}, \frac{\pi }{2}\right]\text{.}$
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. $(0,\infty )$

C. $\ds \frac{10-2}{4-2}=4\text{.}$
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. $\ln 2$

C. Use $\ds \frac{dP}{dt}=kP\text{.}$
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 $(a,b)$

B. $f$ is increasing.

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

B. Consider $f(x)=x^5+10x+3$ and its first derivative.
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. $\ds \frac{5}{3}$

E. $\ds \cosh (\ln 3)=\frac{3+\frac{1}{3}}{2}\text{.}$
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. $2.4$

B. $f(2.9)\approx 2+4(2.9-3)\text{.}$
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. $1$

E. $F(x)=\frac{3}{4}x^{\frac{4}{3}}+\frac{1}{4}\text{.}$
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 $x=0$

B.

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. $2x\cos (2x)$

A.
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

E. $\ds \lim _{x\to 0^+}\frac{\ln x}{x}=-\infty\text{.}$
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. $\sin e$

B. Use L'Hospital's rule.
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 $\sqrt{5}$

C. $(x-1)^2+y^2=5\text{.}$
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

B. $\ds \frac{dV}{dt}=3x^2\frac{dx}{dt}\text{.}$
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. $-\cot \theta$

E. $\ds \frac{dy}{dt}=\frac{\frac{dy}{d\theta }}{\frac{dy}{d\theta }}\text{.}$
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. $4$

D. $\ds A(t)=16\left( \frac{3}{4}\right) ^t\text{.}$
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.

D. For (1) take $f(x)=x^3$ on $(0,1)\text{.}$ For (2) take $f(x)=\sqrt[3]{x}\text{.}$ For (3) take $f(x)=x^4\text{.}$
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.

B. For (1) take $g(x)=0\text{.}$ For (3) take $f(x)=|x|\text{,}$ $g(x)=-|x|\text{,}$ and $a=0\text{.}$
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

A.

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

C.
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

B. Note $\ds y^\prime \sinh y=1+3x^2y+x^3y^\prime\text{.}$
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

D.
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

D.
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. $\ds \sqrt[3]{x}-2$

B.

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

C.
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

B.
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

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

E.
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

B.
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 $F"(x)=f(x)$

A.
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 $\ds (c,f(c))$

C.

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

37.

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

$\pi\text{.}$
38.

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

$f'(x)=\ln |x|\text{.}$
39.

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

0. $\ds \frac{dy}{dt}=2\sin x\cdot \cos x\cdot \frac{dx}{dt}\text{.}$
40.

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

$F(x)=e^{x^2}+C\text{.}$
41.

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

$0\text{.}$ $\ds \lim _{t\to \infty }\ln \frac{t+1}{t}\text{.}$
42.

Does differentiability imply continuity?

Yes.
43.

Convert the Cartesian equation $x^2+y^2=25$ into a polar equation.

$r=5\text{.}$
44.

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

1.

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.

$\ds F=x\cdot \sin \frac{1}{x}$ and $a=0\text{.}$
46.

Function that is continuous but not differentiable at a point.

$f(x)=|x|\text{.}$
47.

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

$f(x)=x^3\text{.}$
48.

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

$f(x)=x^4\text{.}$

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”.

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 $\lim_{x\to a}g(x)=L\text{.}$

1. $\displaystyle ii.$

2. $\displaystyle viii.$

3. $\displaystyle v.$

4. $\displaystyle iv.$

5. no match.

6. no match.

7. $\displaystyle vii.$

8. $\displaystyle vi.$

9. $\displaystyle iii.$

10. no match.

11. $\displaystyle i.$

12. no match.

13. $\displaystyle ix.$