Question 1

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

Accepted Answer

A)  0 
B)  Does not exist 
C)  $- 2$ 
D)  2 
A)  0 
B)  Does not exist 
C)  $- 2$ 
D)  2

Question 2

Find the derivative of the function at the specified point.
-$f ( x ) = \frac { - 5 } { x }$ at $x = 2$

Accepted Answer

A)  $\frac { 5 } { 4 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 2 } { 5 }$ 
D)  $\frac { 5 } { 2 }$ 
A)  $\frac { 5 } { 4 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 2 } { 5 }$ 
D)  $\frac { 5 } { 2 }$

Question 3

Sketch a possible graph for a function f that has the stated properties.
-$$\text { The domain of } \mathrm { f } \text { is } [ 0,5 ] \text { and the derivative is } 2 \text { at } x = 2 \text { and } x = 4 \text {. }$$

Accepted Answer

The answer of Sketch a possible graph for a function...

Question 4

Find the derivative of the function using the definition of derivative.
-$$f ( x ) = 3 x ^ { 2 } - 4 x$$

Accepted Answer

A)  $f ^ { \prime } ( x ) = 2 x - 4$ 
B)  $f ^ { \prime } ( x ) = 6 x - 4$ 
C)  $f ^ { \prime } ( x ) = 3 x - 4$ 
D)  $f ^ { \prime } ( x ) = 6 x$ 
A)  $f ^ { \prime } ( x ) = 2 x - 4$ 
B)  $f ^ { \prime } ( x ) = 6 x - 4$ 
C)  $f ^ { \prime } ( x ) = 3 x - 4$ 
D)  $f ^ { \prime } ( x ) = 6 x$

Question 5

Find the definite integral by computing an area.
-$$\int _ { 1 } ^ { 8 } ( 5 x - 4 ) d x$$

Accepted Answer

A)  $\frac { 259 } { 2 }$ 
B)  128 
C)  $\frac { 371 } { 2 }$ 
D)  $\frac { 315 } { 2 }$ 
A)  $\frac { 259 } { 2 }$ 
B)  128 
C)  $\frac { 371 } { 2 }$ 
D)  $\frac { 315 } { 2 }$

Question 6

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).
-$f ( x ) = 16 - 3 x ; [ 0,5 ] ; 5$ subintervals

Accepted Answer

A)  $\frac { 85 } { 2 }$ 
B)  $\frac { 69 } { 2 }$ 
C)  85 
D)  $\frac { 101 } { 2 }$ 
A)  $\frac { 85 } { 2 }$ 
B)  $\frac { 69 } { 2 }$ 
C)  85 
D)  $\frac { 101 } { 2 }$

Question 7

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

Accepted Answer

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

Question 8

Find the definite integral by computing an area.
-$$\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x$$

Accepted Answer

A)  36 
B)  18 
C)  12 
D)  72 
A)  36 
B)  18 
C)  12 
D)  72

Question 9

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

Accepted Answer

A)  $f ^ { \prime } ( x ) = - 8 x ^ { 2 }$ 
B)  $f ^ { \prime } ( x ) = 8$ 
C)  $f ^ { \prime } ( x ) = - \frac { 8 } { x ^ { 2 } }$ 
D)  $f ^ { \prime } ( x ) = \frac { 8 } { x ^ { 2 } }$ 
A)  $f ^ { \prime } ( x ) = - 8 x ^ { 2 }$ 
B)  $f ^ { \prime } ( x ) = 8$ 
C)  $f ^ { \prime } ( x ) = - \frac { 8 } { x ^ { 2 } }$ 
D)  $f ^ { \prime } ( x ) = \frac { 8 } { x ^ { 2 } }$

Question 10

Find the indicated limit.
-$\lim _ { x \rightarrow \infty } \frac { \ln x ^ { 3 } } { \ln x ^ { 6 } }$

Accepted Answer

A)  The limit does not exist. 
B)  $\frac { 1 } { 2 }$ 
C)  2 
D)  0 
A)  The limit does not exist. 
B)  $\frac { 1 } { 2 }$ 
C)  2 
D)  0

Question 11

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

Accepted Answer

A)  $5.043 ; 5.364 ; 5.396 ; 5.404 ; 5.436 ; 5.763$ 
B)  $16.692 ; 17.592 ; 17.689 ; 17.710 ; 17.808 ; 18.789$ 
C)  $16.810 ; 17.880 ; 17.988 ; 18.012 ; 18.120 ; 19.210$ 
A)  $5.043 ; 5.364 ; 5.396 ; 5.404 ; 5.436 ; 5.763$ 
B)  $16.692 ; 17.592 ; 17.689 ; 17.710 ; 17.808 ; 18.789$ 
C)  $16.810 ; 17.880 ; 17.988 ; 18.012 ; 18.120 ; 19.210$

Question 12

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

Accepted Answer

A)  $- 2$ 
B)  $- \frac { 1 } { 20 }$ 
C)  2 
D)  $\frac { 1 } { 20 }$ 
A)  $- 2$ 
B)  $- \frac { 1 } { 20 }$ 
C)  2 
D)  $\frac { 1 } { 20 }$

Question 13

Use NDER on a calculator to find the numerical derivative of the function at the specified point.
-$$f ( x ) = x ^ { 2 } + 5 x \text { at } x = 4$$

Accepted Answer

A)  13 
B)  21 
C)  9 
D)  3 
A)  13 
B)  21 
C)  9 
D)  3

Question 14

Use NINT on a calculator to find the numerical integral of the function over the specified interval.
-$$f ( x ) = x ^ { 2 } \cos x ; [ 0,6 \pi ]$$
Round to three decimal places.

Accepted Answer

A)  $- 38.699$ 
B)  $38.699$ 
C)  $37.699$ 
D)  $19.85$ 
A)  $- 38.699$ 
B)  $38.699$ 
C)  $37.699$ 
D)  $19.85$

Question 15

Solve the problem.
-Estimate the "RRAM" 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.

Accepted Answer

A)  15.5 
B)  12.5 
C)  16.5 
D)  10.0 
A)  15.5 
B)  12.5 
C)  16.5 
D)  10.0

Question 16

Find the limit of the function algebraically.
-$$\lim _ { x \rightarrow \frac { 1 } { 4 } } \frac { x ^ { 4 } - 1 } { x - 1 }$$

Accepted Answer

A)  4 
B)  2 
C)  Does not exist 
D)  0 
A)  4 
B)  2 
C)  Does not exist 
D)  0

Question 17

Find the derivative of the function at the specified point.
-$f ( x ) = \frac { 8 } { x }$ at $x = - 1$

Accepted Answer

A)  0 
B)  8 
C)  4 
D)  $- 8$ 
A)  0 
B)  8 
C)  4 
D)  $- 8$

Question 18

Solve the problem.
-A piece of tissue paper is picked up in gusty wind. The table shows the velocity of the paper at 2 second intervals. Estimate the distance the paper traveled using right-endpoints.
Timd Velocity

Accepted Answer

A)  $221 \mathrm { ft }$ 
B)  $244 \mathrm { ft }$ 
C)  $224 \mathrm { ft }$ 
D)  $122 \mathrm { ft }$ 
A)  $221 \mathrm { ft }$ 
B)  $244 \mathrm { ft }$ 
C)  $224 \mathrm { ft }$ 
D)  $122 \mathrm { ft }$

Question 19

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

Accepted Answer

A)  $y = 13 x - 16$ 
B)  $y = - \frac { 4 x } { 25 } + \frac { 8 } { 5 }$ 
C)  $y = \frac { x } { 20 } + \frac { 1 } { 5 }$ 
D)  $y = - 39 x - 80$ 
A)  $y = 13 x - 16$ 
B)  $y = - \frac { 4 x } { 25 } + \frac { 8 } { 5 }$ 
C)  $y = \frac { x } { 20 } + \frac { 1 } { 5 }$ 
D)  $y = - 39 x - 80$

Question 20

Find the indicated limit.
-$$\lim _ { x \rightarrow 2 } \frac { 2 ^ { x } + x - 7 } { x - 2 }$$

Accepted Answer

A)  4 
B)  Does not exist 
C)  $- 1$ 
D)  2 
A)  4 
B)  Does not exist 
C)  $- 1$ 
D)  2

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

Find the derivative of the function at the specified point. - $f ( x ) = \frac { - 5 } { x }$ at $x = 2$

Find the derivative of the function using the definition of derivative. - $f ( x ) = 3 x ^ { 2 } - 4 x$

Find the definite integral by computing an area. - $\int _ { 1 } ^ { 8 } ( 5 x - 4 ) d x$

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). - $f ( x ) = 16 - 3 x ; [ 0,5 ] ; 5$ subintervals

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

Find the definite integral by computing an area. - $\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x$

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

Find the indicated limit. - $\lim _ { x \rightarrow \infty } \frac { \ln x ^ { 3 } } { \ln x ^ { 6 } }$

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

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. - $f ( x ) = x ^ { 2 } + 5 x \text { at } x = 4$

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = x ^ { 2 } \cos x ; [ 0,6 \pi ]$ Round to three decimal places.

Solve the problem. -Estimate the "RRAM" 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.

Find the limit of the function algebraically. - $\lim _ { x \rightarrow \frac { 1 } { 4 } } \frac { x ^ { 4 } - 1 } { x - 1 }$

Find the derivative of the function at the specified point. - $f ( x ) = \frac { 8 } { x }$ at $x = - 1$

Solve the problem. -A piece of tissue paper is picked up in gusty wind. The table shows the velocity of the paper at 2 second intervals. Estimate the distance the paper traveled using right-endpoints. Timd Velocity

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

Find the indicated limit. - $\lim _ { x \rightarrow 2 } \frac { 2 ^ { x } + x - 7 } { x - 2 }$

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 by using direct substitution. - lim⁡x→0(x−2)\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )limx→0​(x​−2)

Find the derivative of the function at the specified point. - f(x)=−5xf ( x ) = \frac { - 5 } { x }f(x)=x−5​ at x=2x = 2x=2

Find the derivative of the function using the definition of derivative. - f(x)=3x2−4xf ( x ) = 3 x ^ { 2 } - 4 xf(x)=3x2−4x

Find the definite integral by computing an area. - ∫18(5x−4)dx\int _ { 1 } ^ { 8 } ( 5 x - 4 ) d x∫18​(5x−4)dx

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

Find the definite integral by computing an area. - ∫−33(2x+6)dx\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x∫−33​(2x+6)dx

Find the derivative of the function using the definition of derivative. - f(x)=8xf ( x ) = \frac { 8 } { x }f(x)=x8​

Find the indicated limit. - lim⁡x→∞ln⁡x3ln⁡x6\lim _ { x \rightarrow \infty } \frac { \ln x ^ { 3 } } { \ln x ^ { 6 } }limx→∞​lnx6lnx3​

Match the function with the correct table values. - f(x)=x2+8x−2f ( x ) = x ^ { 2 } + 8 x - 2f(x)=x2+8x−2

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)=x2+5x at x=4f ( x ) = x ^ { 2 } + 5 x \text { at } x = 4f(x)=x2+5x at x=4

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - f(x)=x2cos⁡x;[0,6π]f ( x ) = x ^ { 2 } \cos x ; [ 0,6 \pi ]f(x)=x2cosx;[0,6π] Round to three decimal places.

Solve the problem. -Estimate the "RRAM" 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.

Find the limit of the function algebraically. - lim⁡x→14x4−1x−1\lim _ { x \rightarrow \frac { 1 } { 4 } } \frac { x ^ { 4 } - 1 } { x - 1 }limx→41​​x−1x4−1​

Find the derivative of the function at the specified point. - f(x)=8xf ( x ) = \frac { 8 } { x }f(x)=x8​ at x=−1x = - 1x=−1

Solve the problem. -A piece of tissue paper is picked up in gusty wind. The table shows the velocity of the paper at 2 second intervals. Estimate the distance the paper traveled using right-endpoints. Timd Velocity

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

Find the indicated limit. - lim⁡x→22x+x−7x−2\lim _ { x \rightarrow 2 } \frac { 2 ^ { x } + x - 7 } { x - 2 }limx→2​x−22x+x−7​

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 by using direct substitution. - $\lim _ { x \rightarrow 0 } ( \sqrt { x } - 2 )$

Find the derivative of the function at the specified point. - $f ( x ) = \frac { - 5 } { x }$ at $x = 2$

Find the derivative of the function using the definition of derivative. - $f ( x ) = 3 x ^ { 2 } - 4 x$

Find the definite integral by computing an area. - $\int _ { 1 } ^ { 8 } ( 5 x - 4 ) d x$

Find the definite integral by computing an area. - $\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x$

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

Find the indicated limit. - $\lim _ { x \rightarrow \infty } \frac { \ln x ^ { 3 } } { \ln x ^ { 6 } }$

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

Use NDER on a calculator to find the numerical derivative of the function at the specified point. - $f ( x ) = x ^ { 2 } + 5 x \text { at } x = 4$

Use NINT on a calculator to find the numerical integral of the function over the specified interval. - $f ( x ) = x ^ { 2 } \cos x ; [ 0,6 \pi ]$ Round to three decimal places.

Solve the problem. -Estimate the "RRAM" 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.

Find the limit of the function algebraically. - $\lim _ { x \rightarrow \frac { 1 } { 4 } } \frac { x ^ { 4 } - 1 } { x - 1 }$

Find the derivative of the function at the specified point. - $f ( x ) = \frac { 8 } { x }$ at $x = - 1$

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

Find the indicated limit. - $\lim _ { x \rightarrow 2 } \frac { 2 ^ { x } + x - 7 } { x - 2 }$