## Section5.1True Or False

### Exercises5.1.1Exercises

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 $x=0\text{.}$

True.
###### 2.

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

False.
###### 3.

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

False.
###### 4.

$\ln x$ exists for any $x>1\text{.}$

True.
###### 5.

$\ln x=\pi$ has a unique solution.

True.
###### 6.

$e^{-x}$ is negative for some values of $x\text{.}$

False.
###### 7.

$\ln e^{x^2}=x^2$ for all $x\text{.}$

True.
###### 8.

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

False.
###### 9.

$\tan x$ is defined for all $x\text{.}$

False.
###### 10.

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

Hint
Take $f(x)=|x|\text{.}$
False.

Answer each of the following either TRUE or FALSE.

###### 11.

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 $x\text{.}$

Hint
Find $\ds \lim _{x\to 2}f(x)\text{.}$
False.
###### 12.

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

Hint
Take $f(x)=1$ and $g(x)=2\text{.}$
False.
###### 13.

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

True.
###### 14.

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

Hint
Apply the Mean Value Theorem.
False.
###### 15.

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

Hint
Apply the chain rule.
False.
###### 16.

$\ds \frac{d}{dx}10^x=x10^{x-1}$

False.
###### 17.

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

False.
###### 18.

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

True.
###### 19.

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

Hint
The limit equals $g'(2)\text{.}$
True.
###### 20.

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

False.
###### 21.

$\ds \frac{d}{dx}\tan ^2x=\frac{d}{dx}\sec ^2x$

Hint
$\tan ^2x-\sec ^2 x=-1\text{.}$
True.
###### 22.

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

False. $\ds y=|x^2+x|$ is not differentiable for all real numbers.
###### 23.

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

False.
###### 24.

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

False.
###### 25.

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.

Hint
Take $\ds \lim _{x\to 5}\frac{x-5}{x-5}\text{.}$
False.
###### 26.

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

Hint
Take $\ds f(x)=\frac{1}{x-1}$ if $x>1$ and $f(x)=0$ if $x\leq 1\text{.}$
False.
###### 27.

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 $x=c\text{.}$

False.
###### 28.

$\sqrt{a^2}=a$ for all $a>0\text{.}$

True.
###### 29.

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

False. $c$ might be an isolated point.
###### 30.

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

Hint
Take $f(x)=x^3\text{.}$
False.

Are the following statements TRUE or FALSE.

###### 31.

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

True.
###### 32.

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

True. $\frac{1}{\sqrt{x}}\cdot\ln 2^{\sqrt{x}}=\ln 2\text{,}$ $x>0$
###### 33.

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

True.
###### 34.

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

False. $f(x)\geq 1\text{.}$
###### 35.

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

True.
###### 36.

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

False $\ds f(x)=-\frac{x^4-256}{4}+3\text{.}$
###### 37.

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

False. Take $f(x)=x^2$ and $c=1\text{.}$
###### 38.

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

True. $\ds \frac{1}{\csc u} =\sin u$ with $\sin u\not= 0\text{.}$
###### 39.

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

Hint
Use the chain rule.
True.
###### 40.

$\sinh ^2x-\cosh ^2x=1\text{.}$

False. $\sinh ^2x-\cosh ^2x=-1\text{.}$
###### 41.

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

False. $\ds \int \frac{dx}{x^2+1}=\arctan x+C\text{.}$
###### 42.

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

False. $\ds \int \frac{dx}{3-2x}=-\frac{\ln |3-2x|}{2}+C\text{.}$

Answer each of the following either TRUE or FALSE.

###### 43.

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

False.
###### 44.

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 $a\text{.}$

Hint
Take $\ds f(x)=\frac{x^2}{x}$ and $a=0\text{.}$
False.
###### 45.

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 $f(c)=0\text{.}$

Hint
Take $\ds f(x)=\frac{x^2}{x}\text{,}$ $a=-1\text{,}$ and $b=1\text{.}$
False.
###### 46.

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

Hint
Take $f(x)=x|x|\text{.}$
False.
###### 47.

It is impossible for a function to be discontinuous at every number $x\text{.}$

Hint
Take $f(x)=1$ if $x$ is rational and $f(x)=0$ if $x$ is irrational.
False.
###### 48.

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 $x\text{.}$

Hint
Take $g(x)=0\text{.}$
False.
###### 49.

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

Hint
Take $f(x)=\frac{1}{x}$ if $x\not= 0\text{,}$ $f(0)=0\text{,}$ and $g(x)=-f(x)\text{.}$
True.
###### 50.

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.

Hint
Take $f(x)=\sin x$ and $g(x)=-\sin x\text{.}$
False.
###### 51.

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

False. The numerator is an exponential function with a base greater than 1 and the denominator is a polynomial.
###### 52.

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

Hint
Take $\ds f(x)=\tan \frac{\pi x}{2}\text{.}$
False.

Answer each of the following either TRUE or FALSE.

###### 53.

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 $f'(c)=g'(d)\text{.}$

Hint
Take $f(x)=10x$ and $g(x)=20x$ if $x\in [0,0.5]$ and $g(x)=10x$ if $x\in (0.5,1]\text{.}$
False.
###### 54.

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 $f'(c)=g'(c)\text{.}$

Hint
Take $F(x)=f(x)-g(x)$ and apply Rolle's Theorem.
True.
###### 55.

For all $x$ in the domain of $\sec ^{-1}x\text{,}$

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

Answer each of the following either TRUE or FALSE.

###### 56.

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

False. The limit is missing.
###### 57.

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

False. The Squeeze Theorem.
###### 58.

The graph below exhibits three types of discontinuities.

True.
###### 59.

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

True.
###### 60.

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 $f^{-1}\text{.}$

True.
###### 61.

If the graph of $f$ is on the Figure below, to the left, the graph to the right must be that of $f^\prime\text{.}$

False. For $x\lt 3$ the function is decreasing.
###### 62.

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 $[a,b]\text{.}$

True .
###### 63.

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

False. It should be $L(x)=f(a)+f'(a)(x-a)\text{.}$
###### 64.

The graphs below are labeled correctly with possible eccentricities for the given conic sections:

False. The eccentricity of a circle is $e=0\text{.}$
###### 65.

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

Hint
Note that $g'(x)=-0.5$ and $f'(3)\approx 0.5\text{.}$
True.

Answer TRUE or FALSE to the following questions.

###### 66.

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

True.
###### 67.

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

True.
###### 68.

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

False. $f(g(x))=(x+1)^2\text{.}$
###### 69.

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

True.
###### 70.

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*}
True.
###### 71.

$f(3)=-1$

False. $f(3)=16\text{.}$
###### 72.

$f(2)=11$

False.
###### 73.

$f$ is continuous at $x=3\text{.}$

True.
###### 74.

$f$ is continuous at $x=2\text{.}$

False.

Answer TRUE or FALSE to the following questions.

###### 75.

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

False. It is a quadratic polynomial.
###### 76.

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 $(b-a)f'(c)=f(b)-f(a)\text{.}$

False. The function should be also continuous on $[a,b]\text{.}$
###### 77.

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

Hint
Take $f(x)=-x\text{.}$
False.
###### 78.

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

Hint
Take $f(x)=-|x|\text{.}$
False.

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

###### 79.

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 $x\text{.}$

Hint
Use the Mean Value Theorem.
False.
###### 80.

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

Hint
Take $y=(x-5)^4\text{.}$
False.
###### 81.

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

Hint
Take $f(x)=x^3\text{,}$ $c=0\text{.}$
False.
###### 82.

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.

True. Since $f$ is differentiable, by Rolle's Theorem there is a local extremum between any two isolated solutions of $f(x)=0\text{.}$
###### 83.

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

Hint
Take $f(x)=x-1\text{.}$
False.

Answer the following questions TRUE or False.

###### 84.

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.

False.
###### 85.

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

Hint
Take $f(x)=\sin x\text{.}$
False.
###### 86.

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

False. $g^\prime(2)=4\text{.}$
###### 87.

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

False.
###### 88.

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

False.
###### 89.

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

Hint
Take $f(x)=|x|\text{.}$
False.
###### 90.

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

True. If if is differentiable at $c$ then $f$ is continuous at $c\text{.}$
###### 91.

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

False. It is not given that $f$ is continuous.
###### 92.

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

True.
###### 93.

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 $y=\frac{1}{9}g+\frac{1}{3}\text{.}$

True.

Are the following statements true or false?

###### 94.

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

True.
###### 95.

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.

Hint
Take functions $\displaystyle f(x)=xe^{-1/x^2}\sin (x^{-4})$ and $\displaystyle g(x)=e^{-1/x^2}\text{.}$
False.
###### 96.

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

Hint
Take $f(x)=x^4\text{.}$
False.
###### 97.

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

Hint
Take $f(x)=|x|$ and $x=0\text{.}$
False.
###### 98.

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 $(c,f(c))\text{.}$

True.
###### 99.

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

Hint
Take $f(x)=e^x\text{.}$
False.
###### 100.

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

True.
###### 101.

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

True.

For each statement indicate whether is True or False.

###### 102.

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

Hint
Use the Mean Value Theorem.
False.
###### 103.

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

Hint
Take $\displaystyle f(x)=(x-2)^2\text{.}$
False.
###### 104.

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

True.
###### 105.

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 $r\text{.}$

Hint
Take $f(x)=\sqrt[3]{x}\text{.}$
False.
###### 106.

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

True.
###### 107.

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

False.

For each statement indicate whether is True or False.

###### 108.

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

Hint
Take $f(x)=x^2-2$ and $a=0\text{.}$
False.
###### 109.

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.

Hint
Take $\displaystyle f(x)=g(x)=x\text{.}$
False.
###### 110.

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

False. Recall, $\ds \sin^{-1}x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ for all $x\in[-1,1]\text{.}$
###### 111.

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

Hint
Take $\displaystyle f(x)=\frac{1}{x-3}$ if $x\not= 3$ and $f(3)=2\text{.}$
False.
###### 112.

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

True.
###### 113.

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

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