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

Sketch a possible graph for a function f that has the stated properties. - $\text { The domain of } \mathrm { f } \text { is } [ 0,5 ] \text { and the derivative is undefined at } \mathrm { x } = 3 \text {. }$ $Sketch a possible graph for a function f that has the stated properties. - \text { The domain of } \mathrm { f } \text { is } [ 0,5 ] \text { and the derivative is undefined at } \mathrm { x } = 3 \text {. }$

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Question 41

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Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = | \sin x | ; \left[ 0 , \frac { 3 } { 2 } \pi \right]$

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Question 42

Find the definite integral by computing an area. - $\int _ { 0 } ^ { 7 } \sqrt { 49 - x ^ { 2 } } d x$

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Question 43

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow 1^{-}} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow 1^{-}} f(x)$

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Question 44

Solve the problem. -The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with left-end point values.

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Question 45

Find the derivative of the function using the definition of derivative. - $f ( x ) = \frac { 8 } { x + 2 }$

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Question 46

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }$

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Question 47

Sketch a possible graph for a function f that has the stated properties. - $\text { The domain of } \mathrm { f } \text { is } [ - 4,4 ] , \mathrm { f } \text { is non-increasing on } [ 0,4 ] \text {, and the derivative at } \mathrm { x } = 3 \text { is } 0 \text {. }$ $Sketch a possible graph for a function f that has the stated properties. - \text { The domain of } \mathrm { f } \text { is } [ - 4,4 ] , \mathrm { f } \text { is non-increasing on } [ 0,4 ] \text {, and the derivative at } \mathrm { x } = 3 \text { is } 0 \text {. }$

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Question 48

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Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 1 } \left( x ^ { 2 } - 3 x \right) ^ { 4 }$

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Question 49

Find the indicated limit. - $\lim _ { x \rightarrow \infty } \left( e ^ { - x ^ { 2 } } \sin x \right)$

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Question 50

Find the indicated limit. - $\lim _ { x \rightarrow \infty } x ^ { 2 } e ^ { x }$

(Multiple Choice)

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Question 51

Match the function with the correct table values. - $f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$ $Match the function with the correct table values. - f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

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Question 52

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = x \sin ( 4 x ) \text { at } x = 9$

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Question 53

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = \frac { 1 } { x } ; \left[ 1 , e ^ { 6 } \right]$

(Multiple Choice)

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Question 54

Find the indicated limit. - $\lim _ { \mathrm { u } \rightarrow 2 ^ { + } } \frac { | \mathrm { u } - 2 | } { \mathrm { u } - 2 }$

(Multiple Choice)

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Question 55

Find the limit. -Let $\lim _ { x \rightarrow 8 } f ( x ) = 1$ and $\lim _ { x \rightarrow 8 } g ( x ) = - 9$ . Find $\lim _ { x \rightarrow 8 } [ f ( x ) + g ( x ) ] ^ { 2 }$ .

(Multiple Choice)

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Question 56

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

(Multiple Choice)

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Question 57

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

(Multiple Choice)

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Question 58

Find the derivative of the function using the definition of derivative. - $f ( x ) = - 5$

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Question 59

Solve the problem. -Estimate the "LRAM" area under the graph of the function above the $x$ -axis and under the graph of the function from $x = 0$ to $x = 5$ . Use 5 subintervals.

(Multiple Choice)

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Question 60

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = | \sin x | ; \left[ 0 , \frac { 3 } { 2 } \pi \right]$

Find the definite integral by computing an area. - $\int _ { 0 } ^ { 7 } \sqrt { 49 - x ^ { 2 } } d x$

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow 1^{-}} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow 1^{-}} f(x)$

Solve the problem. -The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with left-end point values.

Find the derivative of the function using the definition of derivative. - $f ( x ) = \frac { 8 } { x + 2 }$

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 1 } \left( x ^ { 2 } - 3 x \right) ^ { 4 }$

Find the indicated limit. - $\lim _ { x \rightarrow \infty } \left( e ^ { - x ^ { 2 } } \sin x \right)$

Find the indicated limit. - $\lim _ { x \rightarrow \infty } x ^ { 2 } e ^ { x }$

Match the function with the correct table values. - $f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$ $Match the function with the correct table values. - f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = x \sin ( 4 x ) \text { at } x = 9$

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = \frac { 1 } { x } ; \left[ 1 , e ^ { 6 } \right]$

Find the indicated limit. - $\lim _ { \mathrm { u } \rightarrow 2 ^ { + } } \frac { | \mathrm { u } - 2 | } { \mathrm { u } - 2 }$

Find the limit. -Let $\lim _ { x \rightarrow 8 } f ( x ) = 1$ and $\lim _ { x \rightarrow 8 } g ( x ) = - 9$ . Find $\lim _ { x \rightarrow 8 } [ f ( x ) + g ( x ) ] ^ { 2 }$ .

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

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

Find the derivative of the function using the definition of derivative. - $f ( x ) = - 5$

Solve the problem. -Estimate the "LRAM" area under the graph of the function above the $x$ -axis and under the graph of the function from $x = 0$ to $x = 5$ . Use 5 subintervals.

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

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

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - f(x)=∣sin⁡x∣;[0,32π]f ( x ) = | \sin x | ; \left[ 0 , \frac { 3 } { 2 } \pi \right]f(x)=∣sinx∣;[0,23​π]

Find the definite integral by computing an area. - ∫0749−x2dx\int _ { 0 } ^ { 7 } \sqrt { 49 - x ^ { 2 } } d x∫07​49−x2​dx

Use the given graph to determine the limit, if it exists. - lim⁡x→1−f(x)\lim _{x \rightarrow 1^{-}} f(x)limx→1−​f(x)

Solve the problem. -The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with left-end point values.

Find the derivative of the function using the definition of derivative. - f(x)=8x+2f ( x ) = \frac { 8 } { x + 2 }f(x)=x+28​

Determine the limit algebraically, if possible. - lim⁡x→0x2cot⁡x2\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }limx→0​x2cotx2

Find the limit of the function by using direct substitution. - lim⁡x→1(x2−3x)4\lim _ { x \rightarrow 1 } \left( x ^ { 2 } - 3 x \right) ^ { 4 }limx→1​(x2−3x)4

Find the indicated limit. - lim⁡x→∞(e−x2sin⁡x)\lim _ { x \rightarrow \infty } \left( e ^ { - x ^ { 2 } } \sin x \right)limx→∞​(e−x2sinx)

Find the indicated limit. - lim⁡x→∞x2ex\lim _ { x \rightarrow \infty } x ^ { 2 } e ^ { x }limx→∞​x2ex

Match the function with the correct table values. - f(x)=x3−6x+8x−2f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }f(x)=x−2x3−6x+8​

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - f(x)=xsin⁡(4x) at x=9f ( x ) = x \sin ( 4 x ) \text { at } x = 9f(x)=xsin(4x) at x=9

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - f(x)=1x;[1,e6]f ( x ) = \frac { 1 } { x } ; \left[ 1 , e ^ { 6 } \right]f(x)=x1​;[1,e6]

Find the indicated limit. - lim⁡u→2+∣u−2∣u−2\lim _ { \mathrm { u } \rightarrow 2 ^ { + } } \frac { | \mathrm { u } - 2 | } { \mathrm { u } - 2 }limu→2+​u−2∣u−2∣​

Find the limit. -Let lim⁡x→8f(x)=1\lim _ { x \rightarrow 8 } f ( x ) = 1limx→8​f(x)=1 and lim⁡x→8g(x)=−9\lim _ { x \rightarrow 8 } g ( x ) = - 9limx→8​g(x)=−9 . Find lim⁡x→8[f(x)+g(x)]2\lim _ { x \rightarrow 8 } [ f ( x ) + g ( x ) ] ^ { 2 }limx→8​[f(x)+g(x)]2 .

Find the limit of the function by using direct substitution. - lim⁡x→0x3−6x+8x−2\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }limx→0​x−2x3−6x+8​

Find the limit. -Let lim⁡x→2f(x)=2\lim _ { x \rightarrow 2 } f ( x ) = 2limx→2​f(x)=2 and lim⁡x→2g(x)=8\lim _ { x \rightarrow 2 } g ( x ) = 8limx→2​g(x)=8 . Find lim⁡x→2[f(x)⋅g(x)]\lim _ { x \rightarrow 2 } [ f ( x ) \cdot g ( x ) ]limx→2​[f(x)⋅g(x)] .

Find the derivative of the function using the definition of derivative. - f(x)=−5f ( x ) = - 5f(x)=−5

Solve the problem. -Estimate the "LRAM" area under the graph of the function above the xxx -axis and under the graph of the function from x=0x = 0x=0 to x=5x = 5x=5 . Use 5 subintervals.

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

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = | \sin x | ; \left[ 0 , \frac { 3 } { 2 } \pi \right]$

Find the definite integral by computing an area. - $\int _ { 0 } ^ { 7 } \sqrt { 49 - x ^ { 2 } } d x$

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow 1^{-}} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow 1^{-}} f(x)$

Find the derivative of the function using the definition of derivative. - $f ( x ) = \frac { 8 } { x + 2 }$

Determine the limit algebraically, if possible. - $\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }$

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 1 } \left( x ^ { 2 } - 3 x \right) ^ { 4 }$

Find the indicated limit. - $\lim _ { x \rightarrow \infty } \left( e ^ { - x ^ { 2 } } \sin x \right)$

Find the indicated limit. - $\lim _ { x \rightarrow \infty } x ^ { 2 } e ^ { x }$

Match the function with the correct table values. - $f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$ $Match the function with the correct table values. - f ( x ) = \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = x \sin ( 4 x ) \text { at } x = 9$

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = \frac { 1 } { x } ; \left[ 1 , e ^ { 6 } \right]$

Find the indicated limit. - $\lim _ { \mathrm { u } \rightarrow 2 ^ { + } } \frac { | \mathrm { u } - 2 | } { \mathrm { u } - 2 }$

Find the limit. -Let $\lim _ { x \rightarrow 8 } f ( x ) = 1$ and $\lim _ { x \rightarrow 8 } g ( x ) = - 9$ . Find $\lim _ { x \rightarrow 8 } [ f ( x ) + g ( x ) ] ^ { 2 }$ .

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } - 6 x + 8 } { x - 2 }$

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

Find the derivative of the function using the definition of derivative. - $f ( x ) = - 5$

Solve the problem. -Estimate the "LRAM" area under the graph of the function above the $x$ -axis and under the graph of the function from $x = 0$ to $x = 5$ . Use 5 subintervals.