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

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

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 \(x\text{.}\)

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 \(\ds \frac{d}{dx}|x^2+x|=|2x+1|\text{.}\)

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.

26.

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

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

30.

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

31.

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

32.

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

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.

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

37.

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

38.

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

39.

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

40.

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

41.

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

42.

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

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 \(a\text{.}\)

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

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 \(x\text{.}\)

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 \(x_0\text{.}\)

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.

51.

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

52.

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

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

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

55.

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

56.

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

57.

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

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 \(f^\prime\text{.}\)

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 \(L(x)=f'(a)+f(a)(x-a)\text{.}\)

64.

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

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 \(-\frac{1}{4}\text{.}\)

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

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

The last four True/False questions ALL pertain to the following function. Let

71.

\(f(3)=-1\)

72.

\(f(2)=11\)

73.

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

74.

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

75.

If a particle has a constant acceleration, then its position function is a cubic 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 \((b-a)f'(c)=f(b)-f(a)\text{.}\)

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

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

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.

83.

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

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

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

91.

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

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

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.

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.

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 \(x=2\text{.}\)

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.

108.

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

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.

110.

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

111.

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

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 \(g(g^{-1}(0))=0\text{.}\)