Exam 11: An Introduction to Calculus: Limits, Derivatives, and Integrals

Compute the average of the RRAM and LRAM approximations to estimate the area between the graph of the function and the x-axis over the given interval using the indicated number of subintervals. (The function is non-negative on the given interval). - $f ( x ) = x ^ { 2 } + 4 ; [ 0,5 ] ; 5$ subintervals

(Multiple Choice)

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Question 81

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 6 } \frac { x + 6 } { ( x - 6 ) ^ { 2 } }$

(Multiple Choice)

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Question 82

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { - 5 } ^ { \pi - 5 } \sin ( x + 5 ) d x$

(Multiple Choice)

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Question 83

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

(Multiple Choice)

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Question 84

Find the limit. -Let $\lim _ { x \rightarrow 8 } f ( x ) = - 8$ and $\lim _ { x \rightarrow 8 } g ( x ) = - 2$ . Find $\lim _ { x \rightarrow 8 } \frac { [ f ( x ) ] ^ { 2 } } { - 7 + g ( x ) }$ .

(Multiple Choice)

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Question 85

Solve the problem. -A ball is tossed straight up from level ground. The velocity of the ball at any time t (sec) is v(t) = 68 - 32t ft/sec. Find how far the ball has traveled at its maximum height.

(Multiple Choice)

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Question 86

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = x ^ { 2 } + 5 x ; x = 4$

(Multiple Choice)

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Question 87

Use graphs and tables to find the limit and identify any vertical asymptotes. - $\lim _ { x \rightarrow 7 } \frac { 1 } { ( x - 7 ) ^ { 2 } }$

(Multiple Choice)

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Question 88

Find the indicated limit. - $\lim _ { x \rightarrow 10 ^ { - } } \text {int } x$

(Multiple Choice)

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Question 89

Find the definite integral by computing an area. - $\int _ { 1 } ^ { 5 } ( 2 x + 7 ) d x$

(Multiple Choice)

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Question 90

Estimate the slope of the tangent line at the indicated point. -

(Multiple Choice)

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Question 91

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = \frac { 4 } { x } ; x = 5$

(Multiple Choice)

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Question 92

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow0 } \frac { 10 \sin x } { 6 x }$

(Multiple Choice)

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Question 93

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow 2^{+}} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow 2^{+}} f(x)$

(Multiple Choice)

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Question 94

Match the function with the correct table values. - $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$ $Match the function with the correct table values. - f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

(Multiple Choice)

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Question 95

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow0 } \frac { 6 x } { 2 \sin x }$

(Multiple Choice)

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Question 96

Use the given graph to determine the limit, if it exists. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the given graph to determine the limit, if it exists. - \lim _ { x \rightarrow 0 } f ( x )$

(Multiple Choice)

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Question 97

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { \pi } ^ { 2 \pi } 2 | \cos x | d x$

(Multiple Choice)

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Question 98

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = \frac { 6 } { x + 3 } ; x = 2$

(Multiple Choice)

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Question 99

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 4 } \sqrt [ 3 ] { 9 x ^ { 3 } - 64 }$

(Multiple Choice)

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Question 100

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 6 } \frac { x + 6 } { ( x - 6 ) ^ { 2 } }$

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { - 5 } ^ { \pi - 5 } \sin ( x + 5 ) d x$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Find the limit. -Let $\lim _ { x \rightarrow 8 } f ( x ) = - 8$ and $\lim _ { x \rightarrow 8 } g ( x ) = - 2$ . Find $\lim _ { x \rightarrow 8 } \frac { [ f ( x ) ] ^ { 2 } } { - 7 + g ( x ) }$ .

Solve the problem. -A ball is tossed straight up from level ground. The velocity of the ball at any time t (sec) is v(t) = 68 - 32t ft/sec. Find how far the ball has traveled at its maximum height.

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = x ^ { 2 } + 5 x ; x = 4$

Use graphs and tables to find the limit and identify any vertical asymptotes. - $\lim _ { x \rightarrow 7 } \frac { 1 } { ( x - 7 ) ^ { 2 } }$

Find the indicated limit. - $\lim _ { x \rightarrow 10 ^ { - } } \text {int } x$

Find the definite integral by computing an area. - $\int _ { 1 } ^ { 5 } ( 2 x + 7 ) d x$

Estimate the slope of the tangent line at the indicated point. -

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = \frac { 4 } { x } ; x = 5$

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow0 } \frac { 10 \sin x } { 6 x }$

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow 2^{+}} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow 2^{+}} f(x)$

Match the function with the correct table values. - $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$ $Match the function with the correct table values. - f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow0 } \frac { 6 x } { 2 \sin x }$

Use the given graph to determine the limit, if it exists. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the given graph to determine the limit, if it exists. - \lim _ { x \rightarrow 0 } f ( x )$

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { \pi } ^ { 2 \pi } 2 | \cos x | d x$

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = \frac { 6 } { x + 3 } ; x = 2$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 4 } \sqrt [ 3 ] { 9 x ^ { 3 } - 64 }$

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

Prerequisites

Filters

Exam 11: An Introduction to Calculus: Limits, Derivatives, and Integrals

Find the limit of the function algebraically. - lim⁡x→6x+6(x−6)2\lim _ { x \rightarrow 6 } \frac { x + 6 } { ( x - 6 ) ^ { 2 } }limx→6​(x−6)2x+6​

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - ∫−5π−5sin⁡(x+5)dx\int _ { - 5 } ^ { \pi - 5 } \sin ( x + 5 ) d x∫−5π−5​sin(x+5)dx

Find the limit of the function by using direct substitution. - lim⁡x→2(x3+5x2−7x+1)\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)limx→2​(x3+5x2−7x+1)

Solve the problem. -A ball is tossed straight up from level ground. The velocity of the ball at any time t (sec) is v(t) = 68 - 32t ft/sec. Find how far the ball has traveled at its maximum height.

Find the equation of the tangent line to the curve when x has the given value. - f(x)=x2+5x;x=4f ( x ) = x ^ { 2 } + 5 x ; x = 4f(x)=x2+5x;x=4

Use graphs and tables to find the limit and identify any vertical asymptotes. - lim⁡x→71(x−7)2\lim _ { x \rightarrow 7 } \frac { 1 } { ( x - 7 ) ^ { 2 } }limx→7​(x−7)21​

Find the indicated limit. - lim⁡x→10−int x\lim _ { x \rightarrow 10 ^ { - } } \text {int } xlimx→10−​int x

Find the definite integral by computing an area. - ∫15(2x+7)dx\int _ { 1 } ^ { 5 } ( 2 x + 7 ) d x∫15​(2x+7)dx

Estimate the slope of the tangent line at the indicated point. -

Find the equation of the tangent line to the curve when x has the given value. - f(x)=4x;x=5f ( x ) = \frac { 4 } { x } ; x = 5f(x)=x4​;x=5

Determine the limit algebraically, if possible. - lim⁡x→010sin⁡x6x\lim _ { x \rightarrow0 } \frac { 10 \sin x } { 6 x }limx→0​6x10sinx​

Use the given graph to determine the limit, if it exists. - lim⁡x→2+f(x)\lim _{x \rightarrow 2^{+}} f(x)limx→2+​f(x)

Match the function with the correct table values. - f(x)=x−1x2+4x−5f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }f(x)=x2+4x−5x−1​

Determine the limit algebraically, if possible. - lim⁡x→06x2sin⁡x\lim _ { x \rightarrow0 } \frac { 6 x } { 2 \sin x }limx→0​2sinx6x​

Use the given graph to determine the limit, if it exists. - lim⁡x→0f(x)\lim _ { x \rightarrow 0 } f ( x )limx→0​f(x)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - ∫π2π2∣cos⁡x∣dx\int _ { \pi } ^ { 2 \pi } 2 | \cos x | d x∫π2π​2∣cosx∣dx

Find the equation of the tangent line to the curve when x has the given value. - f(x)=6x+3;x=2f ( x ) = \frac { 6 } { x + 3 } ; x = 2f(x)=x+36​;x=2

Find the limit of the function by using direct substitution. - lim⁡x→49x3−643\lim _ { x \rightarrow 4 } \sqrt [ 3 ] { 9 x ^ { 3 } - 64 }limx→4​39x3−64​

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

Prerequisites

Filters

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 6 } \frac { x + 6 } { ( x - 6 ) ^ { 2 } }$

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { - 5 } ^ { \pi - 5 } \sin ( x + 5 ) d x$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = x ^ { 2 } + 5 x ; x = 4$

Use graphs and tables to find the limit and identify any vertical asymptotes. - $\lim _ { x \rightarrow 7 } \frac { 1 } { ( x - 7 ) ^ { 2 } }$

Find the indicated limit. - $\lim _ { x \rightarrow 10 ^ { - } } \text {int } x$

Find the definite integral by computing an area. - $\int _ { 1 } ^ { 5 } ( 2 x + 7 ) d x$

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = \frac { 4 } { x } ; x = 5$

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow0 } \frac { 10 \sin x } { 6 x }$

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow 2^{+}} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow 2^{+}} f(x)$

Match the function with the correct table values. - $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$ $Match the function with the correct table values. - f ( x ) = \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow0 } \frac { 6 x } { 2 \sin x }$

Use the given graph to determine the limit, if it exists. - $\lim _ { x \rightarrow 0 } f ( x )$ $Use the given graph to determine the limit, if it exists. - \lim _ { x \rightarrow 0 } f ( x )$

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact to compute the definite integral. - $\int _ { \pi } ^ { 2 \pi } 2 | \cos x | d x$

Find the equation of the tangent line to the curve when x has the given value. - $f ( x ) = \frac { 6 } { x + 3 } ; x = 2$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 4 } \sqrt [ 3 ] { 9 x ^ { 3 } - 64 }$