Free Printable Q1 Calc Review

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27.

Given the function graphed, which is the graph of the derivative?

a)

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c)

d)

28.

Given the function graphed, which is the graph of the derivative?

a)

b)

c)

d)

29.

True or False: If $f\left(x\right)\ =\ x^2\ +\ x$ , then $f'\left(x\right)$ exists for every real number x.

a)

True

b)

False

30.

True or False: If $\lim_{x\rightarrow0^-}\ f'\left(x\right)\ =\ \lim_{x\rightarrow0^+}\ f'\left(x\right)$ , then $f$ is differentiable at 0.

a)

False

b)

True

31.

Let $f\left(x\right)\ =\ 1-3x^2$ . Which of the following is equal to $f'\left(1\right)$ ?

a)

6

b)

-5

c)

-2

d)

-6

32.

True or False: If $f$ has a derivative at $x\ =\ a$ , then $f$ is continuous at $x\ =\ a$ ?

a)

False

b)

True

33.

If $f\left(x\right)\ =\ 10x^2\ -64x$ then $f'\left(3\right)\ =\ ?$

(a)

34.

Find $\frac{dy}{\text{}dx}$ , If $\ y=\ \frac{x^3}{3}\ +\ \frac{x^2}{2}+x$ .

a)

$x^2+x$

b)

$x^2+x+1$

c)

$3x^2+2x+1$

d)

$x^3+x^2\ +1$

35.

Find $f'\left(-2\right)$ , If $f\left(x\right)=\ \frac{x^3}{3}\ +\ \frac{x^2}{2}+x$ .

(a)

36.

Let $f\left(x\right)\ =\ \left|x+1\right|$ . Which of the following statements about $f$ is or are true?
I. $f$ is continuous at x = -1
II. $f$ is differentiable at x = -1
III. $f$ has a corner at x = -1

a)

I only

b)

III only

c)

I and III only

d)

I and II only

37.

Find $\frac{dy}{\text{}dx}$ , If $\ y=\left(\ x^2+1\right)\left(x^3+3\right)$ .

a)

$2x\ +\ 3x^2$

b)

$6x^3$

c)

$2x^2+2x\ +3x^{5\ }+\ 9x^2$

d)

$5x^4+3x^2+6x$

38.

A derivative is defined as

a)

the slope of the secant line

b)

the slope of a straight line

c)

the slope of the tangent line

d)

the slope of a horizontal line

39.

Which of the following is the Power Rule?

a)

$\frac{d}{dx}\left(x^n\right)=nx^n-1$

b)

$\frac{d}{dx}\left[x^n\right]=nx^{\left(n+1\right)}$

c)

$\frac{d}{dx}\left[x^n\right]=\left(n-1\right)x^n-1$

d)

$\frac{d}{dx}\left[x^n\right]=nx^{\left(n-1\right)}$

40.

Which of the following is the Product Rule?

a)

$\frac{d}{dx}\left(fg\right)=gf'+fg'$

b)

$\frac{d}{dx}\left[fg\right]=gf^2+fg^2$

c)

$\frac{d}{dx}\left[fg\right]=f'g'+fg$

d)

$\frac{d}{dx}\left[fg\right]=gf'-fg'$

41.

Which of the following is the Quotient Rule?

a)

$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{gf'+fg'}{g^2}$

b)

$\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{gf'-fg'}{g}$

c)

$\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{fg-gf}{g^2}$

d)

$\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{gf'-fg'}{g^2}$

42.

Find the derivative of $f\left(x\right)=4x^5+2x^3+7$

a)

$f'\left(x\right)=20x^4+6x^2+7$

b)

$f'\left(x\right)=20x^4+6x^2$

c)

$f'\left(x\right)=20x^5+6x^3+7$

d)

$f'\left(x\right)=20x^5+6x^3$

43.

Find $f'\left(-1\right)$ if $f\left(x\right)=4x^5+2x^3+7$

(a)

44.

Find the derivative of $y=\frac{4}{x^5}-9x$

a)

$y'=-20x^{-6}-9$

b)

$y'=20x^4-9$

c)

$y'=-20x^{-4}-9$

d)

$y'=-20x^{-6}-9x^{-1}$

45.

Find the derivative of $h\left(x\right)=x^{\frac{3}{4}}-\sqrt[7]{x}$

a)

$h'\left(x\right)=\frac{3}{4}x^{-\frac{1}{4}}+7x^{-8}$

b)

$h'\left(x\right)=\frac{3}{4}x^{-\frac{3}{4}}-\frac{1}{7}x^{-\frac{1}{7}}$

c)

$h'\left(x\right)=\frac{3}{4}x^{-\frac{1}{4}}-\frac{1}{7}x^{-\frac{6}{7}}$

d)

$h'\left(x\right)=\frac{3}{4}x^{-\frac{1}{4}}+\frac{1}{7}x^{-\frac{8}{7}}$

46.

Find the derivative of $f\left(x\right)=\frac{8x^3+6}{7x-1}$

a)

$f'\left(x\right)=\frac{24x^2}{7}$

b)

c)

d)

47.

Find the derivative of $h\left(x\right)=\left(3x^4\right)\left(8x^4-6x\right)$

a)

$h'\left(x\right)=\left(12x^3\right)\left(32x^3-6\right)$

b)

c)

d)

48.

Find the derivative of $g\left(x\right)=\frac{7x^5+3x^4}{x^3}$

a)

$g'\left(x\right)=35x^4+12x^3-3x^{-4}$

b)

$g'\left(x\right)=\frac{35x^4+12x^3}{3x^2}$

c)

$g'\left(x\right)=14x+3$

d)

$g'\left(x\right)=14x+3x^{-1}$

49.

$\frac{dy}{dx}\ if\ y=e^{\left(7-9x\right)}$ Find

a)

$-9e^{\left(7-9x\right)}$

b)

$e^{-9}$

c)

$-9\ln\left(7-9x\right)$

d)

$7e^{\left(7-9x\right)}$

50.

Find the derivative of f(x) = (x⁶ + 4)⁵

a)

f '(x) = 5x⁵(x⁴ + 4)⁴

b)

f '(x) = 6x⁵(x⁶ + 4)⁴

c)

f '(x) = 30x⁵(x⁶ + 4)⁴

d)

f '(x) = 30x⁶(x⁶ + 4)⁴

51.

What rule should be used in deriving h(x) = 2(x - 4)³

a)

Power rule

b)

Product rule

c)

Quotient rule

d)

Chain rule

52.

Find the derivative: $y=3\ln\left(x^2-3\right)$

a)

$\frac{6x}{x^2-3}$

b)

$\frac{3}{x^2-3}$

c)

$\frac{3x}{x^2-3}$

d)

$\frac{9x}{x^2-3}$

53.

$f\left(x\right)=\frac{1}{x-1}$

What is the derivative of f?

a)

$\frac{1}{\left(x-1\right)^{-1}}$

b)

$-\frac{1}{\left(x-1\right)^{-2}}$

c)

$-\frac{1}{x-1}$

d)

$-\frac{1}{\left(x-1\right)^2}$

54.

Find $\frac{dy}{dx}$ for $y=e^{-5x^5}$ .

a)

$e^{-5x^5}$

b)

$-5e^{-5x^5}$

c)

$-25x^4e^{-5x^5}$

d)

$\ln\left(-5x^5\right)$

55.

Find the derivative of $f\left(x\right)=\sqrt{5x-3}$

a)

$\frac{5}{2\sqrt{5x-3}}$

b)

$\frac{1}{2\sqrt{5x-3}}$

c)

$\frac{5}{\sqrt{5x-3}}$

d)

$-\sqrt{5x-3}$

56.

a)

b)

c)

d)

57.

Find

\frac{d^2y}{dx^2}

given

y=x^3-2x^2+11x-9

a)

$3x^2-4x+11$

b)

$6x-4$

c)

$6$

d)

$0$

58.

Use the product rule to find the derivative of
$f\left(x\right)=x^2\sin x$

a)

$f'\left(x\right)=2x\cos x$

b)

$f'\left(x\right)=2x\sin x+x^2\cos x$

c)

$f'\left(x\right)=2x\sin x-x^2\cos x$

d)

$f'\left(x\right)=x\sin x+x^2\cos x$

59.

Which limit does not exist?

a)

b)

c)

None of them

d)

Both exist

60.

$y=\frac{x^2+6}{2x−7}.$

a)

$\frac{2x^2−14x−12}{(2x−7)^2}$

b)

$\frac{4x^2−12x−2}{(2x−7)^2}$

c)

$\frac{x^2−x+12}{(2x−7)^2}$

d)

$\frac{2x^2+14x+12}{(2x−7)^2}$

61.

$\frac{2x+1}{x^2−3}$

a)

$\frac{2x^2-3x+5}{\left(x^2-3\right)^2}$

b)

$\frac{-2x^2-2x-6}{\left(x^2-3\right)^2}$

c)

$\frac{-2x^2+2x+6}{\left(x^2+3\right)^2}$

d)

$\frac{x^2-2x-6}{\left(x^2+3\right)^2}$

62.

$\frac{5x+1}{x^2−3}$

a)

$\frac{5x^2-15}{\left(x-3\right)^2}$

b)

$\frac{x^2-3}{5x+1}$

c)

$\frac{-5x^2-2x-15}{\left(x^2-3\right)^2}$

d)

$\frac{5x^2-3x+1}{\left(x-3\right)^2}$

63.

$f\left(x\right)=\frac{4}{x^3+5}$

Find $f'\left(x\right)$

a)

$\frac{x^2}{\left(x^3+5\right)^2}$

b)

$\frac{-4x^2}{\left(x^3+5\right)^2}$

c)

$\frac{4}{\left(x^3+5\right)^2}$

d)

$\frac{-12x^2}{\left(x^3+5\right)^2}$

64.

What is the limit?

a)

Infinity

b)

20

c)

DNE

d)

12

65.

What is the derivative of f?

a)

1/(x - 1)^-1

b)

-1/(x - 1)^-2

c)

-1/(x - 1)

d)

-1/(x - 1)²

66.

Given that y = (x+2)/(x-3), find y'.

a)

y' = -1/(x-3)²

b)

y' = -5/(x-3)²

c)

y' = (2x-5)/(x-3)²

d)

y' = (2x-1)/(x-3)²

67.

Find the derivative.
y = x²/ (3x-1)

a)

y' = 9x²- 12
y' = (3x-1) / (3x-1)²

b)

y' = (3x² - 2x) / (3x-1)²

c)

y' = (6x+1)²/ 3

68.

Find the derivative of f(x) = (x⁶ + 4)⁵

a)

f '(x) = 5x⁵(x⁴ + 4)⁴

b)

f '(x) = 6x⁵(x⁶ + 4)⁴

c)

f '(x) = 30x⁵(x⁶ + 4)⁴

d)

f '(x) = 30x⁶(x⁶ + 4)⁴

69.

Find the derivative of f(x)=(x³-2x)²

a)

6x⁵ - 12x³+8x

b)

6x⁵ - 16x³+8x

c)

x⁶-4x⁴+4x²

d)

6x⁵ - 16x^3-8x

70.

find y' for y= (2x+1)¹⁰

a)

10(2x+1)⁹

b)

20(2x-1)⁹

c)

20(2x+1)¹⁰

d)

20(2x+1)⁹

71.

a)

b)

c)

d)

72.

find y' for y= (2x+1)¹⁰

a)

10(2x+1)⁹

b)

20(2x-1)⁹

c)

20(2x+1)¹⁰

d)

20(2x+1)⁹

73.

a)

b)

c)

d)

74.

a)

b)

c)

d)

75.

a)

b)

c)

d)

76.

Find the derivative of the given equation
f(x) = x³ + x² + 3

a)

3x² + 2x

b)

3x + 2x

c)

3x + 2x + 3

d)

x³ + x²

77.

Find the derivative of the given equation
f(x) = 1/x²

a)

1/2x

b)

-2x

c)

2x

d)

-2x^-3

78.

Differentiate f(x) =(2/ x⁵) - 5.

a)

x⁵-3

b)

-10x⁶

c)

x⁵-5

d)

-10x^-6

79.

Find the slope of the tangent line to f(x) = -3x²-6x at x = 1.

a)

m = 0

b)

f'(x) = -6x - 6

c)

f'(x) = 6x

d)

m = -12

80.

What is the derivative of xⁿ?

a)

(n-1)xⁿ

b)

nxⁿ⁺¹

c)

(n+1)x^n-1

d)

nx^n-1

81.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

82.

Find the derivative of the given equation
f(x) = -4x

a)

4

b)

x

c)

-4

d)

0

83.

Set up the derivative of y=(3x⁴- 7)*(5x²+ 1)

a)

(12x³)*(10x)

b)

(3x⁴ - 7)*(10x) + (12x³)*(5x² + 1)

c)

(3x⁴ - 7)*(10x) - (12x³)*(5x² + 1)

d)

(3x⁴ - 7)/(10x) + (12x³)/(5x² + 1)

84.

What is the derivative of cos(x)?

a)

sin(x)

b)

-sin(x)

c)

cos(x)

d)

-cos(x)

85.

Let f(x) = (x - 1)(x + 2). Find f'(0)

a)

0

b)

1

c)

2

d)

3

86.

Find the derivative.
y = x²/ (3x-1)

a)

y' = 9x²- 12
y' = (3x-1) / (3x-1)²

b)

y' = (3x² - 2x) / (3x-1)²

c)

y' = (6x+1)²/ 3

87.

Which rule would you need to use?

a)

Power Rule

b)

Product Rule

c)

Quotient Rule

d)

Chain Rule

88.

Find the derivative of $\sin\left(x^3-2x\right)$

a)

$\left(3x^2-2\right)\cos\left(x^3-2x\right)$

b)

$-\left(3x^2-2\right)\cos\left(x^3-2x\right)$

c)

$\cos\left(2x^2-2\right)$

d)

$\sin\left(2x^2-2\right)$

89.

If $y=\ln\left(\ln2x\right)$ , find y'

a)

$\frac{1}{2x}$

b)

$\frac{1}{\ln2x}$

c)

$\frac{1}{x\ln2x}$

d)

$\frac{1}{2x\ln x}$

90.

a)

A

b)

B

c)

C

d)

D

91.

a)

A

b)

B

c)

C

d)

D

92.

a)

A

b)

B

c)

C

d)

D

93.

(a)

94.

Find y' if y = tan(3x²+2).

a)

sec²(3x²+2)

b)

6xsec²(3x²+2)

c)

6xsec²x(3x²+2)

d)

sec²(6x)

95.

$\frac{dy}{dx}\ if\ y=e^{\left(7-9x\right)}$ Find

a)

$-9e^{\left(7-9x\right)}$

b)

$e^{-9}$

c)

$-9\ln\left(7-9x\right)$

d)

$7e^{\left(7-9x\right)}$

96.

The graph of f is shown. Which of the following statements is false?

a)

$f\left(1\right)=\lim_{x\rightarrow1}f\left(x\right)$

b)

$f\left(2\right)=\lim_{x\rightarrow2}f\left(x\right)$

c)

$f\left(x\right)$ has removable discontinuity at $x=2$

d)

$f\left(x\right)$ has a jump discontinuity at $x=4$

97.

$\lim_{x\rightarrow-2}\ \frac{x^2+5x+6}{x+2}$ is

a)

0

b)

-1

c)

1

d)

nonexistent

98.

Let $f\left(x\right)$ be the piecewise function defined above. For what value of k is f(x) continuous at $x=-2$

a)

-1

b)

0

c)

1

d)

2

99.

How many removable and non-removable discontinuities does the graph of $y=\frac{\left(x+2\right)\left(x-6\right)}{\left(x^2+8x+12\right)\left(x-3\right)}$ have?

a)

0 removable discontinuities, 1 non-removable discontinuity.

b)

1 removable discontinuity, 1 non-removable discontinuity.

c)

2 removable discontinuities, 1 non-removable discontinuity.

d)

1 removable discontinuity, 2 non-removable discontinuities.

100.

If $\lim_{x\rightarrow3^-}\ f\left(x\right)=2$ and $\lim_{x\rightarrow3^+}\ f\left(x\right)=4$ , which of the following must be true about $f\left(x\right)$ ?

a)

f(x) is not continuous at $x=3$

b)

$\lim_{x\rightarrow3}\ f\left(x\right)$ exists

c)

$f\left(3\right)=2$

d)

$f\left(x\right)$ has removable discontinuity at $x=3$

101.

If f is the function defined by $f\left(x\right)=\frac{x^2+5x+6}{x+3}$ then $\lim_{x\rightarrow-3}\ f\left(x\right)$ is

a)

0

b)

-1

c)

1

d)

nonexistent

102.

For which of the following does $\lim_{x\rightarrow\infty}\ f\left(x\right)=0$ ?

a)

II only

b)

III only

c)

I and II only

d)

I, II, and III

103.

Let f be a function that is continuous on the closed interval $\left[1,3\right]$ with $f\left(1\right)=4$ and $f\left(3\right)=10$ . Which of the following is guaranteed by the Intermediate Value Theorem?

a)

$f\left(2\right)=7$

b)

$f\left(x\right)=2$ has at least one solution in the open interval (1, 3)

c)

$f\left(x\right)=8$ has at least one solution in the open interval (1, 3)

d)

None of the above are guaranteed by the Intermediate Value Theorem

104.

If the function f is continuous for all real numbers and $f\left(x\right)=\frac{x^2-1}{x-1}$ when $x\ne\ 1$ , then $f\left(1\right)$ is

a)

0

b)

1

c)

2

d)

undefined

105.

For which of the following does $\lim_{x\rightarrow4}\ f\left(x\right)$ exist?

a)

I only

b)

III only

c)

I and II only

d)

I and III only

106.

$\lim_{x\rightarrow0}\ \frac{x^3-2x^2}{x^2+x}$ is

a)

-2

b)

0

c)

1

d)

2

107.

Let f, g, and h be continuous functions on their domain except at $x=5$ . If $g\left(x\right)\le f\left(x\right)\le h\left(x\right)$ for all $x$ and $\lim_{x\rightarrow5}\ f\left(x\right)=2$ , which of the following must be false?

a)

$\lim_{x\rightarrow5}\ g\left(x\right)=2$

b)

$\lim_{x\rightarrow5}\ h\left(x\right)=2$

c)

$f\left(5\right)=2$

d)

$\lim_{x\rightarrow5^-}\ f\left(x\right)=\lim_{x\rightarrow5^+}\ f\left(x\right)$

108.

Which of the following statements about f(x), shown in the graph to the left, is true?

a)

$\lim_{x\rightarrow2}\ f\left(x\right)$ does not exist

b)

$\lim_{x\rightarrow3}\ f\left(x\right)$ does not exist

c)

$\lim_{x\rightarrow4}\ f\left(x\right)$ does not exist

d)

$\lim_{x\rightarrow5}\ f\left(x\right)$ does not exist

109.

Selected values of $f\left(x\right)$ are shown in the table. According to the table, which of the following is the best estimate of $\lim_{x\rightarrow2}\ \cos\left(f\left(x\right)\right)$ ?

a)

-1

b)

0

c)

1

d)

2

110.

$\lim_{x\rightarrow\infty}\ \frac{7x-3x^2}{4x^2+2x-1}$ is

a)

$-\frac{3}{4}$

b)

$\frac{3}{4}$

c)

$-\frac{7}{4}$

d)

$\frac{7}{4}$

111.

If $f\left(x\right)$ is a rational function and has a vertical asymptote at $x=1$ , which of the following statements must be false?

a)

$\lim_{x\rightarrow1^-}\ f\left(x\right)=0$

b)

$\lim_{x\rightarrow1^+}\ f\left(x\right)=-\infty$

c)

$\lim_{x\rightarrow\infty}\ f\left(x\right)=1$

d)

$\lim_{x\rightarrow-\infty}\ f\left(x\right)=0$

112.

If $f\left(1\right)=\lim_{x\rightarrow1^+}\ f\left(x\right)$ and $f\left(x\right)$ is not continuous at $x=1$ , which of the following statements must be true?

a)

$\lim_{x\rightarrow1^-}\ f\left(x\right)=\lim_{x\rightarrow1^+}\ f\left(x\right)$

b)

$\lim_{x\rightarrow1}\ f\left(x\right)$ does not exist

c)

$\lim_{x\rightarrow1^-}\ f\left(x\right)=f\left(1\right)$

d)

$f\left(x\right)$ has a removable discontinuity at $x=1$

113.

$\lim_{x\rightarrow3}\ \frac{x-3}{\sqrt{\ x+1}-2}$ is

a)

-4

b)

-1

c)

1

d)

4

114.

Let f be a continuous function with selected values given in the table to the left. What is the minimum number of times that $f\left(c\right)=8$ on the interval $\left[0,\ 10\right]$

a)

0

b)

1

c)

2

d)

3

115.

The graph of $f\left(x\right)$ is shown above. For which value(s) of x does $f\left(x\right)$ have a removable discontinuity?

a)

$x=2$

b)

$x=2$ and $x=3$

c)

$x=3$ and $x=5$

d)

$x=2,\ x=3,\$ and $x=5$

116.

TRUE OR FALSE? If f is continuous on [-1,1], f(-1)=4 and f(1)= -2, then there is a zero (a y-value of zero) between -1 and 1.

a)

TRUE

b)

FALSE

c)

CANNOT BE DETERMINED

117.

TRUE OR FALSE? There is a zero on the interval [1,3].

a)

True because the function is continuous and there is a sign change on the interval [1,3]

b)

We cannot be certain since the function is not continuous on the interval [1,3]

c)

False because there is no sign change on the interval [1,3]

d)

False because the graph is not continuous on the interval [1,3]

118.

Given a continuous function on the interval [4,7]. Let's say that f(4) = 10 and f(7) = 20. Is there a value on the interval [4,7] where f(x) = 15?

a)

Yes - if the function is continuous, then there must be an x value on that interval that passes through 15 on its way from 10 to 20.

b)

No because there is no sign change on the interval [4,7].

c)

No because that would be just too good to be true.

d)

Yes - at x = 5.5.

119.

Let f be a function that is continuous on the closed interval [1, 3] with f(1) = 10 and f(3) = 18. Which of the following statements MUST be true?

a)

10 ≤ f(2) ≤ 18

b)

f is increasing on the interval [1, 3]

c)

f(x) = 17 has at least one solution in the interval [1, 3]

d)

f(2) = 14

120.

Let f be a function that is continuous on the closed interval [2,4] with f(2) = 10 and f(4) = 20. Which of the following is guaranteed by the Intermediate Value Theorem?

a)

f(x) = 13 has at least one solution in the open interval (2,4)

b)

f(3) = 15

c)

f attains a maximum on the open interval (2,4)

d)

f(x) = 10 at some other value(s) of x other than x = 2

121.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

122.

Find the limit as x approaches 3 from the left

a)

4

b)

3

c)

2

d)

DNE

123.

a)

Does not exist

b)

2

c)

0

d)

1

124.

a)

0

b)

1

c)

2

d)

DNE

125.

a)

2

b)

4

c)

32

d)

DNE

126.

The graph of f is shown in the figure. Which of the following statements is false?

a)

b)

c)

d)

127.

Let f be a continuous function for which f(-2)=1 and f(5)=-3. The Intermediate Value Theorem guarantees that

a)

f(c)=2 for at least one c between -3 and 1

b)

f(c)=0 for at least one c between -2 and 5

c)

f(c)=0 for at least one c between -3 and 1

d)

f(c)=2 for at least one c between -2 and 5

128.

The line x=c is a vertical asymptote of the graph of the function f. Which of the following statements cannot be true?

a)

b)

c)

f(c) is undefined

d)

f is continuous at x=c

129.

a)

I only

b)

II only

c)

I, II, III

d)

None

130.

a)

none

b)

-3 only

c)

3 only

d)

-3 and 3

131.

a)

7

b)

1/4

c)

infinty

d)

-8

132.

a)

-1/4

b)

-3/10

c)

-9/5

d)

DNE

133.

a)

2

b)

-3/4

c)

-1/3

d)

1/4

134.

a)

0

b)

3

c)

1

d)

Does not exist

135.

Given the function f(x) = x³ + x - 3, which of the intervals below contains a zero?

a)

[-1,1]

b)

[0,1]

c)

[1,2]

d)

None of these intervals

136.

Given the function g(x) = 2x⁴ - x³ + 8/x which of the intervals must contain a zero?

a)

[-1,1]

b)

[0,2]

c)

[2,4]

d)

None of these intervals

137.

TRUE OR FALSE? There is a zero on the interval [1,3].

a)

True because the function is continuous and there is a sign change on the interval [1,3]

b)

We cannot be certain since the function is not continuous on the interval [1,3]

c)

False because there is no sign change on the interval [1,3]

d)

False because the graph is not continuous on the interval [1,3]

138.

TRUE OR FALSE? There is NO zero on the interval [-4,0].

a)

True - the function is not continuous on the interval

b)

True - there is no sign change on the interval [-4,0]

c)

False - the function is continuous and there is a sign change on the interval [-4,0]

d)

False - the IVT is an existence theorem. It can tell us when we can be certain there IS a zero but not when to be certain when there is NOT.

139.

$\lim_{x\rightarrow-2^-}\left(\frac{2x^2+x+1}{x+2}\right)$

a)

0

b)

$\infty$

c)

$-\infty$

d)

2

140.

Find the limit of the function as x approaches 2.

a)

1

b)

-1

c)

5

d)

DNE

141.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

142.

What is the limit?

a)

DNE

b)

2/3

c)

1/4

d)

Infinity

143.

What is the limit?

a)

Infinity

b)

20

c)

DNE

d)

12

144.

a)

0

b)

∞

c)

- ∞

d)

DNE

145.

What is the limit?

a)

5/2

b)

-2/3

c)

Infinity

d)

17/3

146.

a)

-2

b)

Does not exist

c)

120

d)

10

147.

a)

0

b)

2

c)

Does not exist

d)

-2

148.

a)

Does not exist

b)

-7/5

c)

-5/9

d)

-1/2

149.

What is the limit of the function as x approaches -4 from the left?

a)

2

b)

-4

c)

DNE

d)

-2

150.

Find the limit.

a)

6

b)

3

c)

5/6

d)

-1/6

151.

a)

∞

b)

DNE

c)

3

d)

1

152.

a)

10

b)

4

c)

3

d)

-2

153.

a)

2

b)

-3/4

c)

-1/3

d)

1/4

154.

a)

Does not exist

b)

0

c)

5

d)

-1

155.

a)

A

b)

B

c)

C

d)

D

156.

Evaluate

a)

-7

b)

0

c)

1

d)

DNE

157.

Use the table of values to evaluate the limit:
$\lim_{x\rightarrow0^+}f\left(x\right)=$

a)

0

b)

$\infty$

c)

$-\infty$

d)

10000

e)

-1000

158.

Evaluate:
$\lim_{x\rightarrow1^+}a\left(x\right)$

a)

1

b)

-1

c)

-3

d)

$\infty$

e)

DNE

159.

Is the function continuous at x = 2, and why?

a)

Yes, $\lim_{x\rightarrow2}f\left(x\right)=f\left(2\right)$

b)

Yes, $\lim_{x\rightarrow2^-}f\left(x\right)=\lim_{x\rightarrow2^+}f\left(x\right)$

c)

No, $\lim_{x\rightarrow2^-}f\left(x\right)\ne\lim_{x\rightarrow2^+}f\left(x\right)$

d)

No, $\lim_{x\rightarrow2}f\left(x\right)\ne f\left(2\right)$

160.

Find $\lim_{x\rightarrow0}\ f\left(x\right)$

a)

-1

b)

3

c)

0

d)

DNE

161.

Given this table, what limits are most accurate?

a)

As x approaches 3, the limit is infinity

b)

As x approaches 3, the limit is negative infinity

c)

As x approaches 3 from the left, the limit is negative infinity & as x approaches 3 from the right, the limit is infinity

d)

As x approaches 3 from the left, the limit is infinity and as x approaches 3 from the right, the limit is negative infinity

162.

Given the graph of g(x),

\lim_{x\rightarrow\left(-3^+\right)}g\left(x\right)\ =

a)

-4

b)

-3

c)

0

d)

6

163.

The graph of f is shown in the figure. If f is defined at k, but the limit of f(x) as x approaches k DNE, then k =

a)

a

b)

b

c)

c

d)

0

164.

The graph of f is shown in the figure. Which of the following statements are true? (Check all that are true.)

a)

$\lim_{x\rightarrow a}f\left(x\right)\ exists$

b)

$\lim_{x\rightarrow b}\ f\left(x\right)\ exists$

c)

$\lim_{x\rightarrow c}\ f\left(x\right)\ exists$

d)

$\lim_{x\rightarrow b}\ f\left(x\right)\ =\ f\left(b\right)$

e)

$\lim_{x\rightarrow a}\ f\left(x\right)\ =\ f\left(a\right)$

165.

a)

Does not exist

b)

2

c)

0

d)

1

166.

Find dy/dx
xy+y²=2

a)

-y/(x+2y)

b)

y/(x+2y)

c)

-3y/x

d)

-3x/y

167.

Find $\frac{dy}{dx}$ : $y^2=10x$

a)

$\frac{5}{y}$

b)

$\frac{10}{y}$

c)

$\frac{y^2}{5}$

d)

$\frac{y^2}{10}$

168.

Find $\frac{dy}{dx}$ : $2x^2-3y^2=4$

a)

$\frac{2x}{y}$

b)

$\frac{x}{3y}$

c)

$\frac{3y}{2x}$

d)

$\frac{2x}{3y}$

169.

Find the derivative of $x^2+xy+y^3=0$

a)

$-\ \frac{2x}{1+3y^2}$

b)

$-\frac{x+3y^2}{2x+y}$

c)

$-\frac{2x+y}{x+3y^2}$

d)

$-\frac{2x}{x+3y^2}$

170.

Find dy/dx for
y = (x² + 1)³

a)

dy/dx = 3(x² + 1)²

b)

dy/dx = 3(2x)²

c)

dy/dx = 6x(x² + 1)²

d)

dy/dx = 2x(x² + 1)²

171.

y=e^sin3

y'=?

(a)

172.

a)

-3

b)

8

c)

0

d)

9

173.

$\lim_{h\rightarrow0}\left(\frac{5\left(x+h\right)^2-5x^2}{h}\right)$

a)

$5x^2$

b)

$10x$

c)

10

d)

DNE

174.

a)

A

b)

B

c)

C

d)

D

e)

E

175.

a)

A

b)

B

c)

C

d)

D

e)

E

176.

a)

A

b)

B

c)

C

d)

D

e)

E

177.

a)

A

b)

B

c)

C

d)

D

e)

E

178.

f'(x)=

a)

$\lim_{h\rightarrow0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$

b)

$\lim_{x\rightarrow c}\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}\right)$

c)

$\frac{\text{d}y}{\text{d}x}$

d)

an equation for the slope of the tangent line

179.

$\lim_{x\rightarrow2}\left(\frac{x^2-4}{x-2}\right)$

a)

2

b)

4

c)

6

d)

8

180.

Find the slope at x = 3 of the equation f(x) = 3x²

a)

6x

b)

27

c)

18

d)

3x

181.

What is the slope of the line normal to the curve y = x² + x at x = 1?

a)

-1

b)

-1/2

c)

-1/3

d)

-1/4

182.

What is the equation of the line tangent to the curve y = x² at x = 1?

a)

y = 2x + 1

b)

y = 2x - 1

c)

y = -2x + 1

d)

y = x - 1

183.

What is the equation of the tangent line at x = 1 $of$ $f\left(x\right)=\ \sqrt{x}$

a)

$y=\frac{1}{2}x-\frac{1}{2}$

b)

$y=\frac{1}{2}x+\frac{1}{2}$

c)

$y=2x-1$

d)

$y=2x$

184.

What is the equation of the line tangent to f(x)= 4x²+2x-1 at x=0?

a)

y+1=2(x+1)

b)

y=2x-1

c)

y=2x+2

d)

y= -x-1

185.

When we "take the derivative" of a function what are we finding?

a)

What's a derivative?

b)

The rate at which our struggles in Calculus are increasing.

c)

The slope of the secant line

d)

The slope of the tangent line

186.

a)

a

b)

b

c)

c

d)

d

e)

e

187.

Write the equation of the normal line of: $f\left(x\right)=2x^2-\frac{1}{x}$ at x = 1

a)

$y=-\frac{1}{5}x+\frac{4}{5}$

b)

$y=5x-6$

c)

$y-1=5\left(x-1\right)$

d)

$y-1=-\frac{1}{5}\left(x-1\right)$

188.

Find the slope of the Normal line to: $f\left(x\right)\ =\ \frac{4\sqrt{x}}{5}$
at x = 16

a)

$\frac{2}{5}$

b)

$-\frac{1}{160}$

c)

$\frac{1}{10}$

d)

$-10$

189.

Find the equation of the tangent line at the point (-1,1) of: $f\left(x\right)\ =\ x^4$

a)

$y=-\frac{1}{4}x-3$

b)

$y=-4x-3$

c)

$y=\frac{1}{4}x+3$

d)

$y=4x+3$

190.

Find the equation of the normal line at x = 0 of:
$f\left(x\right)\ =\ x^3+e^x$

a)

$y=-x+1$

b)

$y=\ x+1$

c)

$y=-x\ -1$

d)

$y=x\ -\ 1$

191.

Find the equation of the normal line at (3,5) of

y=x^2-3x+5

a)

$y-5=3\left(x-3\right)$

b)

$y-5=\frac{1}{3}\left(x-3\right)$

c)

$y-5=-\frac{1}{3}\left(x-3\right)$

d)

$y-3=\frac{1}{3}\left(x-5\right)$

192.

Find the equation of the tangent line at (1,-2) of

y=x^3-3x^2

a)

$y+2=3\left(x+1\right)$

b)

$y-2=3\left(x-1\right)$

c)

$y-2=-3\left(x+1\right)$

d)

$y+2=-3\left(x-1\right)$

193.

Find the slope of the tangent line at x = 9 of

f\left(x\right)=\sqrt{x}

a)

$\frac{1}{6}$

b)

$-\frac{1}{6}$

c)

$6$

d)

$-6$

194.

Find the slope of the normal line at x = 9 of

f\left(x\right)=\sqrt{x}

a)

$\frac{1}{6}$

b)

$-\frac{1}{6}$

c)

$6$

d)

$-6$

195.

Find where the function has a horizontal tangent line.

y=2x^2-8x+3

a)

x = -2

b)

x = 1

c)

x = 2

d)

x = 1/2

196.

Find where the function has a horizontal tangent line.

y=2x^3-6x^{ }+5

a)

$x=2$

b)

$x=0$

c)

$x=\sqrt{2}$

d)

$x=\sqrt{6}$

197.

Find the equation of the tangent line to the curve

y=-2x^3+8x^2-19\ at\ x=2.

a)

$y-2=8\left(x-2\right)$

b)

$y+3=8\left(x-2\right)$

c)

$y+3=8\left(x+2\right)$

d)

$y+2=8\left(x-3\right)$

198.

When will the slope of the tangent line equal 4 of

f\left(x\right)=x^2-2

a)

$x=2$

b)

$x=-2$

c)

Never

d)

$x=0$

199.

What is the equation of the normal line at x = -3 of

y=x^2+12x+11

a)

$y-16=-2\left(x-3\right)$

b)

$y-16=6\left(x-3\right)$

c)

$y+16=6\left(x+3\right)$

d)

$y+16=6\left(x-3\right)$

200.

What is the derivative of

y=\frac{2}{\sqrt{x}}

a)

$y'=2\sqrt{x}$

b)

$y'=\frac{2}{\sqrt{x}}$

c)

$y'=\frac{1}{x^{\frac{3}{2}}}$

d)

$y'=-\frac{1}{x^{\frac{3}{2}}}$

201.

$\lim_{x\rightarrow-4}f\left(x\right)=$
If the limit does not exist, write "DNE."

(a)

202.

The table above gives values of a function $f$ at selected values of $x.$ Which of the following conclusions is supported by the data in the table?

a)

$\lim_{x\rightarrow3}f\left(x\right)=0$

b)

$\lim_{x\rightarrow3}f\left(x\right)=3$

c)

$\lim_{x\rightarrow3}f\left(x\right)\ =10$

d)

$\lim_{x\rightarrow3}f\left(x\right)$ does not exist

203.

You are given a table containing some values of differentiable functions and their derivatives. Use the table data and the rules of differentiation to solve the following problem.

If $h\left(x\right)=f\left(x\right).g\left(x\right)$ Find $h'\left(1\right)$

a)

$\frac{3}{2}$

b)

$2$

c)

$\frac{-1}{2}$

d)

$-3$

204.

You are given a table containing some values of differentiable functions and their derivatives. Use the table data and the rules of differentiation to solve the following problem.
If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ Find $h'\left(4\right)$

a)

$0$

b)

$2$

c)

$\frac{5}{6}$

d)

$-3$

205.

Evaluate:

$\lim_{x\rightarrow\infty}\ \frac{3x^2+1}{2x^3+x-1}$

a)

$\frac{3}{2}$

b)

3

c)

0

d)

D.N.E

206.

What is the equation of the tangent to the curve f(x)= 4x²+2x-1 at the point x=0?

a)

y+1=2(x+1)

b)

y = 2x - 1

c)

y=2x+2

d)

y = 2x -3

207.

Write the slope of the line tangent to the graph of y=x² – 2 at the point x = -8.

a)

2

b)

-4

c)

-8

d)

-16

208.

Find the equation of the normal line at x = 0 of:
$f\left(x\right)\ =\ x^3+e^x$

a)

$y=-x+1$

b)

$y=\ x+1$

c)

$y=-x\ -1$

d)

$y=x\ -\ 1$

209.

Find the equation of the tangent line at the point (-1,1) of: $f\left(x\right)\ =\ x^4$

a)

$y=-\frac{1}{4}x-3$

b)

$y=-4x-3$

c)

$y=\frac{1}{4}x+3$

d)

$y=4x+3$

210.

Find the slope of the Normal line to: $f\left(x\right)\ =\ \frac{4\sqrt{x}}{5}$
at x = 16

a)

$\frac{2}{5}$

b)

$-\frac{1}{160}$

c)

$\frac{1}{10}$

d)

$-10$

211.

Find the equation of the normal line at (3,5) of

y=x^2-3x+5

a)

$y-5=3\left(x-3\right)$

b)

$y-5=\frac{1}{3}\left(x-3\right)$

c)

$y-5=-\frac{1}{3}\left(x-3\right)$

d)

$y-3=\frac{1}{3}\left(x-5\right)$

212.

Find where the function has a horizontal tangent line.

y=2x^2-8x+3

a)

x = -2

b)

x = 1

c)

x = 2

d)

x = 1/2

213.

Find dy/dx at a given point.

a)

5/4

b)

4/5

c)

1

d)

-5

214.

Which of the following would be a valid reason the above function is non-differentiable at x = 0?

a)

The graph contains a corner.

b)

The graph contains a cusp.

c)

The graph contains a discontinuity.

d)

The graph contains a vertical tangent.

215.

Can a function be continuous but not differentiable?

a)

Yes

b)

No

216.

If a function is differentiable, it is also continuous.

a)

Yes

b)

No

c)

It all depends on the function in question.

217.

The function shown

a)

is continuous at x = 2

b)

is differentiable at x = 2

c)

has a limit that exists at x = 2

d)

exists at x = 2

218.

In which quadrant of the graph above will there be a value of x that is non-differentiable?

a)

1

b)

2

c)

3

d)

4

219.

For what value(s) of x is the function continuous but not differentiable?

a)

x = -1

b)

x = 0

c)

x = 2

d)

x = 3; x = -2

220.

At which point(s) will the slopes of the tangent line be zero?

a)

at C only

b)

at points A, C and E only

c)

at point B and D only

d)

at points A and E only

221.

$If\ y=\frac{3}{4+x^2},\ then\ \frac{dy}{dx}=$

a)

$\frac{3}{2x}$

b)

$\frac{3x}{\left(1+x^2\right)^2}$

c)

$\frac{6x}{\left(4+x^2\right)^2}$

d)

$-\frac{6x}{\left(4+x^2\right)^2}$

e)

$-\frac{3}{\left(4+x^2\right)^2}$

222.

Find the slope of the tangent line to
f(x) = -3x²- 6x
at x = 1.

a)

m = 0

b)

f'(x) = -6x - 6

c)

f'(x) = 6x

d)

m = -12

223.

What is the slope of the tangent line to $f\left(x\right)=7\cos\left(x\right)$ at $x=\frac{\pi}{4}$

a)

$-\frac{7}{2}$

b)

$\frac{7\sqrt{3}}{2}$

c)

$-\frac{7\sqrt{2}}{2}$

d)

$\frac{7\sqrt{2}}{2}$

224.

Find the equation for the tangent line of the function $y=-x^2+4\sqrt{x}$ at $x=4$

a)

$y+8=-7\left(x-4\right)$

b)

$y-7=-8\left(x-4\right)$

c)

$y-24=4\left(x-8\right)$

d)

$y=-2x+\frac{2}{\sqrt{x}}$

225.

Find an equation of the tangent line to the curve $y=2x\sin\left(x\right)$ at the point $\left(\frac{\pi}{2},\pi\right)$

a)

$y=-2x$

b)

$y=2x+2\pi$

c)

$y=-2x+2\pi$

d)

$y=2x$

Q1 Calc Review

Total questions: 225

Worksheet time: 15hrs 44mins

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