If the Function F is Continuous at the Point X f X Then It is Diffrentiable True or False

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Exercises 5.1.1 Exercises

Answer the following questions.

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 \(f'(x)=0\text{.}\)

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 \(\ds \lim _{x\to 2}f(x)\text{.}\)

Answer

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

Answer

False. \(\ds y=|x^2+x|\) is not differentiable for all real numbers.

23.

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

24.

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

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 \(\ds \lim _{x\to 5}\frac{x-5}{x-5}\text{.}\)

Answer

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 \(x\leq 1\text{.}\)

Answer

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 \(x=c\text{.}\)

28.

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

29.

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

Answer

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 \(x=c\text{.}\)

Are the following statements TRUE or FALSE.

31.

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

32.

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

Answer

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 \(f'(0)=1\text{.}\)

34.

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

Answer

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

35.

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

36.

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

Answer

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 \(x=c\text{.}\)

Answer

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

38.

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

Answer

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 \(0\lt x\lt \pi\text{.}\)

40.

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

Answer

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

41.

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

Answer

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

42.

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

Answer

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 \(x_0\text{.}\)

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 \(a=0\text{.}\)

Answer

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 \(b=1\text{.}\)

Answer

46.

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

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.

Answer

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 \(x\text{.}\)

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 \(g(x)=-f(x)\text{.}\)

Answer

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 \(g(x)=-\sin x\text{.}\)

Answer

51.

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

Answer

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 \(\ds f(x)=\tan \frac{\pi x}{2}\text{.}\)

Answer

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 \(x\in (0.5,1]\text{.}\)

Answer

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.

Answer

55.

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

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

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 \(\ds \frac{f(a+h)-f(a)}{h}\text{.}\)

Answer

False. The limit is missing.

57.

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

Answer

False. The Squeeze Theorem.

58.

The graph below exhibits three types of discontinuities.

59.

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

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 \(f^{-1}\text{.}\)

61.

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

Answer

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 \([a,b]\text{.}\)

63.

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

Answer

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:

Answer

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 \(f'(3)\approx 0.5\text{.}\)

Answer

Answer TRUE or FALSE to the following questions.

66.

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

67.

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

68.

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

Answer

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 \(2(f(3.5)-f(3))\text{.}\)

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*}

71.

\(f(3)=-1\)

Answer

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

72.

\(f(2)=11\)

73.

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

74.

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

Answer TRUE or FALSE to the following questions.

75.

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

Answer

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

Answer

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 \(\ds \lim _{x\to \infty }f(x)=\infty\text{.}\)

78.

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

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 \(x\text{.}\)

80.

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

81.

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

Hint

Take \(f(x)=x^3\text{,}\) \(c=0\text{.}\)

Answer

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.

Answer

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 \([0,1]\text{.}\)

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.

85.

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

86.

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

Answer

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

87.

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

88.

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

89.

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

90.

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

Answer

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

Answer

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 \((-\infty ,3)\cup (3,\infty )\text{.}\)

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

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.

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 \(\displaystyle g(x)=e^{-1/x^2}\text{.}\)

Answer

96.

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

97.

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

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

99.

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

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.

101.

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

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 \(x\text{.}\)

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 \(\displaystyle f(x)=(x-2)^2\text{.}\)

Answer

104.

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

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 \(r\text{.}\)

106.

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

107.

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

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 \(a=0\text{.}\)

Answer

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 \(\displaystyle f(x)=g(x)=x\text{.}\)

Answer

110.

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

Answer

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 \(f(3)=2\text{.}\)

Answer

112.

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

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 \(f(x)=3^x+x-4\text{.}\)

114.

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

115.

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

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