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

Find the indicated limit, if it exists. - $\lim _ { x \rightarrow 0 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } 8 - x ^ { 2 } & x < 0 \\ 8 & x = 0 \\ 2 x + 8 & x > 0 \end{array} \right.$

(Multiple Choice)

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Question 21

Solve the problem. -An industrial ventilation system can move 2000 cubic feet per minute. How many cubic feet of air are moved in 8 hours?

(Multiple Choice)

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Question 22

Find the derivative of the function at the specified point. - $f ( x ) = x ^ { 2 } + 7 x - 2$ at $x = 0$

(Multiple Choice)

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Question 23

Find the limit. -Let $\lim _ { x \rightarrow 6 } f ( x ) = 225$ . Find $\lim _ { x \rightarrow 6 } \sqrt { f ( x ) }$ .

(Multiple Choice)

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Question 24

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 6 } \frac { x ^ { 2 } - 36 } { x + 6 }$

(Multiple Choice)

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Question 25

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 1 } \sqrt { x - 7 }$

(Multiple Choice)

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Question 26

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

(Multiple Choice)

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Question 27

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = \ln ( 7 x )$ at $x = 1$

(Multiple Choice)

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Question 28

Find the indicated limit. - $\lim _ { x \rightarrow 0 } \frac { \sin ( x ) } { x ^ { 2 } }$

(Multiple Choice)

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Question 29

Find the derivative of the function using the definition of derivative. - $f ( x ) = x ^ { 2 } + 7 x - 2$

(Multiple Choice)

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Question 30

Find the indicated limit, if it exists. - $\lim _ { x \rightarrow 5 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } - 5 - x & x < - 5 \\ 4 & x = - 5 \\ x + 9 & x > - 5 \end{array} \right.$

(Multiple Choice)

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Question 31

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

(Multiple Choice)

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Question 32

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

(Multiple Choice)

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Question 33

Solve the problem. -An arrow is shot straight up from level ground. The distance (in meters) of the arrow above the ground (the position function) is $\mathrm { f } ( \mathrm { t } ) = 4 + 130 \mathrm { t } - 4.9 \mathrm { t } ^ { 2 }$ at any time $\mathrm { t }$ (in sec). Find $\mathrm { f } ^ { \prime } ( 1 )$ and the initial velocity of the arrow.

(Multiple Choice)

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Question 34

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

(Multiple Choice)

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Question 35

Find the derivative of the function using the definition of derivative. - $f ( x ) = 5 x + 9$

(Multiple Choice)

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Question 36

Find the definite integral by computing an area. - $\int _ { 2 } ^ { 5 } 0.5 x d x$

(Multiple Choice)

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Question 37

Use a graph of the function to find the derivative of the function at the given point, if it exists. - $f(x)=\left\{\begin{array}{cl}\frac{(x-3)(x-5)}{x-3} & x \neq 3 \\-2 & x=3\end{array} \quad \text { at } x=3\right.$

(Multiple Choice)

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Question 38

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

(Multiple Choice)

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Question 39

Find the limit. -Let $\lim _ { x \rightarrow 4 } f ( x ) = - 10$ and $\lim _ { x \rightarrow 4 } g ( x ) = 4$ . Find $\lim _ { x \rightarrow 4 } [ f ( x ) - g ( x ) ]$ .

(Multiple Choice)

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Question 40

Find the indicated limit, if it exists. - $\lim _ { x \rightarrow 0 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } 8 - x ^ { 2 } & x < 0 \\ 8 & x = 0 \\ 2 x + 8 & x > 0 \end{array} \right.$

Solve the problem. -An industrial ventilation system can move 2000 cubic feet per minute. How many cubic feet of air are moved in 8 hours?

Find the derivative of the function at the specified point. - $f ( x ) = x ^ { 2 } + 7 x - 2$ at $x = 0$

Find the limit. -Let $\lim _ { x \rightarrow 6 } f ( x ) = 225$ . Find $\lim _ { x \rightarrow 6 } \sqrt { f ( x ) }$ .

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 6 } \frac { x ^ { 2 } - 36 } { x + 6 }$

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 1 } \sqrt { x - 7 }$

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

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = \ln ( 7 x )$ at $x = 1$

Find the indicated limit. - $\lim _ { x \rightarrow 0 } \frac { \sin ( x ) } { x ^ { 2 } }$

Find the derivative of the function using the definition of derivative. - $f ( x ) = x ^ { 2 } + 7 x - 2$

Find the indicated limit, if it exists. - $\lim _ { x \rightarrow 5 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } - 5 - x & x < - 5 \\ 4 & x = - 5 \\ x + 9 & x > - 5 \end{array} \right.$

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

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

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Find the derivative of the function using the definition of derivative. - $f ( x ) = 5 x + 9$

Find the definite integral by computing an area. - $\int _ { 2 } ^ { 5 } 0.5 x d x$

Use a graph of the function to find the derivative of the function at the given point, if it exists. - $f(x)=\left\{\begin{array}{cl}\frac{(x-3)(x-5)}{x-3} & x \neq 3 \\-2 & x=3\end{array} \quad \text { at } x=3\right.$

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

Find the limit. -Let $\lim _ { x \rightarrow 4 } f ( x ) = - 10$ and $\lim _ { x \rightarrow 4 } g ( x ) = 4$ . Find $\lim _ { x \rightarrow 4 } [ f ( x ) - g ( x ) ]$ .

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 indicated limit, if it exists. - lim⁡x→0f(x),f(x)={8−x2x<08x=02x+8x>0\lim _ { x \rightarrow 0 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } 8 - x ^ { 2 } & x < 0 \\ 8 & x = 0 \\ 2 x + 8 & x > 0 \end{array} \right.limx→0​f(x),f(x)=⎩⎨⎧​8−x282x+8​x<0x=0x>0​

Solve the problem. -An industrial ventilation system can move 2000 cubic feet per minute. How many cubic feet of air are moved in 8 hours?

Find the derivative of the function at the specified point. - f(x)=x2+7x−2f ( x ) = x ^ { 2 } + 7 x - 2f(x)=x2+7x−2 at x=0x = 0x=0

Find the limit. -Let lim⁡x→6f(x)=225\lim _ { x \rightarrow 6 } f ( x ) = 225limx→6​f(x)=225 . Find lim⁡x→6f(x)\lim _ { x \rightarrow 6 } \sqrt { f ( x ) }limx→6​f(x)​ .

Find the limit of the function algebraically. - lim⁡x→6x2−36x+6\lim _ { x \rightarrow 6 } \frac { x ^ { 2 } - 36 } { x + 6 }limx→6​x+6x2−36​

Find the limit of the function algebraically. - lim⁡x→1x−7\lim _ { x \rightarrow 1 } \sqrt { x - 7 }limx→1​x−7​

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

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - f(x)=ln⁡(7x)f ( x ) = \ln ( 7 x )f(x)=ln(7x) at x=1x = 1x=1

Find the indicated limit. - lim⁡x→0sin⁡(x)x2\lim _ { x \rightarrow 0 } \frac { \sin ( x ) } { x ^ { 2 } }limx→0​x2sin(x)​

Find the derivative of the function using the definition of derivative. - f(x)=x2+7x−2f ( x ) = x ^ { 2 } + 7 x - 2f(x)=x2+7x−2

Determine the limit algebraically, if possible. - lim⁡x→0sin⁡6xx6\lim _ { x \rightarrow0 } \frac { \sin ^ { 6 } x } { x ^ { 6 } }limx→0​x6sin6x​

Determine the limit algebraically, if possible. - lim⁡x→0sin⁡2x2x\lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } x } { 2 x }limx→0​2xsin2x​

Find the limit of the function by using direct substitution. - lim⁡x→0(x2−5)\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)limx→0​(x2−5)

Find the derivative of the function using the definition of derivative. - f(x)=5x+9f ( x ) = 5 x + 9f(x)=5x+9

Find the definite integral by computing an area. - ∫250.5xdx\int _ { 2 } ^ { 5 } 0.5 x d x∫25​0.5xdx

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

Find the limit. -Let lim⁡x→4f(x)=−10\lim _ { x \rightarrow 4 } f ( x ) = - 10limx→4​f(x)=−10 and lim⁡x→4g(x)=4\lim _ { x \rightarrow 4 } g ( x ) = 4limx→4​g(x)=4 . Find lim⁡x→4[f(x)−g(x)]\lim _ { x \rightarrow 4 } [ f ( x ) - g ( x ) ]limx→4​[f(x)−g(x)] .

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 indicated limit, if it exists. - $\lim _ { x \rightarrow 0 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } 8 - x ^ { 2 } & x < 0 \\ 8 & x = 0 \\ 2 x + 8 & x > 0 \end{array} \right.$

Find the derivative of the function at the specified point. - $f ( x ) = x ^ { 2 } + 7 x - 2$ at $x = 0$

Find the limit. -Let $\lim _ { x \rightarrow 6 } f ( x ) = 225$ . Find $\lim _ { x \rightarrow 6 } \sqrt { f ( x ) }$ .

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 6 } \frac { x ^ { 2 } - 36 } { x + 6 }$

Find the limit of the function algebraically. - $\lim _ { x \rightarrow 1 } \sqrt { x - 7 }$

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = \ln ( 7 x )$ at $x = 1$

Find the indicated limit. - $\lim _ { x \rightarrow 0 } \frac { \sin ( x ) } { x ^ { 2 } }$

Find the derivative of the function using the definition of derivative. - $f ( x ) = x ^ { 2 } + 7 x - 2$

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

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

Find the limit of the function by using direct substitution. - $\lim _ { x \rightarrow 0 } \left( x ^ { 2 } - 5 \right)$

Find the derivative of the function using the definition of derivative. - $f ( x ) = 5 x + 9$

Find the definite integral by computing an area. - $\int _ { 2 } ^ { 5 } 0.5 x d x$

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

Find the limit. -Let $\lim _ { x \rightarrow 4 } f ( x ) = - 10$ and $\lim _ { x \rightarrow 4 } g ( x ) = 4$ . Find $\lim _ { x \rightarrow 4 } [ f ( x ) - g ( x ) ]$ .