Question 1

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

Accepted Answer

A)   -1.0000
B)    1.00001.00001.0000 
C)    10.000010.000010.0000 
D)    1945.90961945.90961945.9096

Question 2

Find the indicated limit, if it exists. - - 5 \end{array} ight.">lim⁡x→5f(x) ,f(x) ={−5−xx<−54x=−5x+9x>−5\lim _ { x ightarrow 5 } f ( x ) , f ( x ) = \left\{ \begin{array} { l l } - 5 - x & x < - 5 \ 4 & x = - 5 \ x + 9 & x > - 5 \end{array} ight.limx→5f(x) ,f(x) =⎩⎨⎧−5−x4x+9x<−5x=−5x>−5

Accepted Answer

A)   0
B)   The limit does not exist.
C)   9
D)   4

Question 3

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

Accepted Answer

A)    34\frac { 3 } { 4 }43​ 
B)    214\frac { 21 } { 4 }421​ 
C)    294\frac { 29 } { 4 }429​ 
D)    10.510.510.5

Question 4

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

Accepted Answer

A)    f′(x) =2x−2f ^ { \prime } ( x )  = 2 x - 2f′(x) =2x−2 
B)    f′(x) =2x+7f ^ { \prime } ( x )  = 2 x + 7f′(x) =2x+7 
C)    f′(x) =2xf ^ { \prime } ( x )  = 2 xf′(x) =2x 
D)    f′(x) =x+7f ^ { \prime } ( x )  = x + 7f′(x) =x+7

Question 5

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​

Accepted Answer

A)    −6- 6−6 
B)    −12- 12−12 
C)   Does not exist
D)   1

Question 6

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?

Accepted Answer

A)    2000ft32000 \mathrm { ft } ^ { 3 }2000ft3 
B)    96,000ft396,000 \mathrm { ft } ^ { 3 }96,000ft3 
C)    16,000ft316,000 \mathrm { ft } ^ { 3 }16,000ft3 
D)    960,000ft3960,000 \mathrm { ft } ^ { 3 }960,000ft3

Question 7

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) ​ .

Accepted Answer

A)   15
B)    3.87303.87303.8730 
C)   6
D)   225

Question 8

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) ] .

Accepted Answer

A)    −4- 4−4 
B)    −14- 14−14 
C)    −10- 10−10 
D)    −6- 6−6

Question 9

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

Accepted Answer

A)   Does not exist
B)    2.449489742.449489742.44948974 
C)   0
D)    −2.4494897- 2.4494897−2.4494897

Question 10

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

Accepted Answer

A)   Does not exist
B)   0
C)    −1- 1−1 
D)   1

Question 11

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

Accepted Answer

A)    12\frac { 1 } { 2 }21​ 
B)   1
C)   Does not exist
D)   0

Question 12

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

Accepted Answer

A)    0.50.50.5 
B)   0
C)   The limit does not exist.
D)   1

Question 13

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

Accepted Answer

A)   0
B)   Undefined
C)    −1- 1−1 
D)   1

Question 14

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)

Accepted Answer

A)   Does not exist
B)    −5- 5−5 
C)   5
D)   0

Question 15

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

Accepted Answer

A)   0
B)   9
C)   7
D)    −2- 2−2

Question 16

Find the indicated limit, if it exists. - 0 \end{array} ight.">lim⁡x→0f(x) ,f(x) ={8−x2x<08x=02x+8x>0\lim _ { x ightarrow 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} ight.limx→0f(x) ,f(x) =⎩⎨⎧8−x282x+8x<0x=0x>0

Accepted Answer

A)   The limit does not exist.
B)   8
C)   2
D)    −6- 6−6

Question 17

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

Accepted Answer

A)    f′(x) =9f ^ { \prime } ( x )  = 9f′(x) =9 
B)    f′(x) =0f ^ { \prime } ( x )  = 0f′(x) =0 
C)    f′(x) =5xf ^ { \prime } ( x )  = 5 xf′(x) =5x 
D)    f′(x) =5f ^ { \prime } ( x )  = 5f′(x) =5

Question 18

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  f(t) =4+130t−4.9t2\mathrm { f } ( \mathrm { t } )  = 4 + 130 \mathrm { t } - 4.9 \mathrm { t } ^ { 2 }f(t) =4+130t−4.9t2  at any time  t\mathrm { t }t  (in sec) . Find  f′(1) \mathrm { f } ^ { \prime } ( 1 ) f′(1)   and the initial velocity of the arrow.

Accepted Answer

A)    f′(1) =120.2\mathrm { f } ^ { \prime } ( 1 )  = 120.2f′(1) =120.2 ; initial velocity  =130 m/s= 130 \mathrm {~m} / \mathrm { s }=130 m/s 
B)    f′(1) =124.2\mathrm { f } ^ { \prime } ( 1 )  = 124.2f′(1) =124.2 ; initial velocity  =130 m/s= 130 \mathrm {~m} / \mathrm { s }=130 m/s 
C)    f′(1) =120.2f ^ { \prime } ( 1 )  = 120.2f′(1) =120.2 ; initial velocity  =4 m/s= 4 \mathrm {~m} / \mathrm { s }=4 m/s 
D)    f′(1) =−9.8f ^ { \prime } ( 1 )  = - 9.8f′(1) =−9.8 ; initial velocity  =130 m/s= 130 \mathrm {~m} / \mathrm { s }=130 m/s

Question 19

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

Accepted Answer

A)   6
B)   1
C)   Does not exist
D)   0

Question 20

Use a graph of the function to find the derivative of the function at the given point, if it exists.
- f(x) ={(x−3) (x−5) x−3x≠3−2x=3 at x=3f(x) =\left\{\begin{array}{cl}\frac{(x-3) (x-5) }{x-3} & x 
eq 3 \-2 & x=3\end{array} \quad 	ext { at } x=3ight.f(x) ={x−3(x−3) (x−5) ​−2​x=3x=3​ at x=3

Accepted Answer

A)   0
B)   1
C)   Does not exist
D)    −1- 1−1

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

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?

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow-\theta} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow-\theta} f(x) A) Does not exist B) 0 C) - 1 D) 1$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

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?

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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)

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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​

Correct AnswerverifiedShow Answer

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) ] .

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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)

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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) ​ .

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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​

Correct AnswerverifiedShow Answer

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Use the given graph to determine the limit, if it exists. - $\lim _{x \rightarrow-\theta} f(x)$ $Use the given graph to determine the limit, if it exists. - \lim _{x \rightarrow-\theta} f(x) A) Does not exist B) 0 C) - 1 D) 1$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified