Question 1

Estimate $\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = 2 }$ of the function $y = 4 x ^ { 2 }$ .

Accepted Answer

A) 20 
B)  17 
C)  14 
D)  16 
E)  18 
A) 20 
B)  17 
C)  14 
D)  16 
E)  18

Question 2

Estimate $f ^ { \prime } ( 5 )$ of the function $f ( x ) = 5 \ln ( x )$ .

Accepted Answer

A)  $\ln 5$ 
B)  $\frac { 1 } { \ln 5 }$ 
C)  0 
D)  5 
E)  1 
A)  $\ln 5$ 
B)  $\frac { 1 } { \ln 5 }$ 
C)  0 
D)  5 
E)  1

Question 3

At which labeled point is the slope of the tangent line greatest

Accepted Answer

A) P 
B)  Q 
C)  R 
A) P 
B)  Q 
C)  R

Question 4

Estimate the derivative of the function $f ( x ) = 8 - 8 x$ at the point $x = 9$ .  
Select the answer rounded to the nearest whole number.

Accepted Answer

A) -9 
B)  -10 
C)  -8 
D)  -7 
E)  -13 
A) -9 
B)  -10 
C)  -8 
D)  -7 
E)  -13

Question 5

Estimate the derivative of the function $f ( x ) = 7 - 3 x$ at the point $x = 2$ .
​
Please round the answer to the nearest whole number.

Accepted Answer

The answer of Estimate the derivative of the function \(f...

Question 6

Estimate the limit numerically.   $$\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- \infty$ 
C)  0 
D)  $+ \infty$ 
E)  $- \frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $- \infty$ 
C)  0 
D)  $+ \infty$ 
E)  $- \frac { 1 } { 2 }$

Question 7

Calculate the limit algebraically.   $\lim _ { x \rightarrow 9 } \frac { x ^ { 3 } - 729 } { x - 9 }$

Accepted Answer

A) -54 
B)  243 
C)  82 
D)  81 
E)  324 
A) -54 
B)  243 
C)  82 
D)  81 
E)  324

Question 8

The graph of a function f is given. Determine whether f is continuous on its domain. ​   ​

Accepted Answer

A) Discontinuous 
B)  Continuous 
A) Discontinuous 
B)  Continuous

Question 9

Estimate $\left. \frac { \mathrm { d } s } { \mathrm {~d} t } \right| _ { t = 1 }$ of the function $s = 2 t + t ^ { 2 }$ .

Accepted Answer

A) 4 
B)  3 
C)  6 
D)  9 
E)  1 
A) 4 
B)  3 
C)  6 
D)  9 
E)  1

Question 10

Find an equation of the tangent line to the graph of the function $f ( x ) = x + \frac { 1 } { x }$ at the point that has x-coordinate $x = \frac { 1 } { 4 }$ .

Accepted Answer

A)  $y = - 15 x + 8$ 
B)  $y = - 15 x - 120$ 
C)  $y = 8 x - 15$ 
D)  $y = x - 15$ 
E)  $y = 15 x + 8$ 
A)  $y = - 15 x + 8$ 
B)  $y = - 15 x - 120$ 
C)  $y = 8 x - 15$ 
D)  $y = x - 15$ 
E)  $y = 15 x + 8$

Question 11

Compute $f ^ { \prime } ( 3 )$ .  
  $f ( x ) = - 3.3 x ^ { 2 } + 3.9$

Accepted Answer

A)   $f ^ { \prime } ( 3 ) = - 3.3$ 
B)    $f ^ { \prime } ( 3 ) = - 6$ 
C)    $f ^ { \prime } ( 3 ) = - 15.9$ 
D)    $f ^ { \prime } ( 3 ) = - 19.8$ 
E)    $f ^ { \prime } ( 3 ) = - 9.9$ 
A)   $f ^ { \prime } ( 3 ) = - 3.3$ 
B)    $f ^ { \prime } ( 3 ) = - 6$ 
C)    $f ^ { \prime } ( 3 ) = - 15.9$ 
D)    $f ^ { \prime } ( 3 ) = - 19.8$ 
E)    $f ^ { \prime } ( 3 ) = - 9.9$

Question 12

If a stone is dropped from a height of 472 feet, its height after t seconds is given by $S = - 73 - 12 t ^ { 2 }$ . Find the stone's average velocity over the period $[ 2,4 ]$ .

Accepted Answer

A) -73 
B)  -69 
C)  -72 
D)  -70 
E)  -74 
A) -73 
B)  -69 
C)  -72 
D)  -70 
E)  -74

Question 13

Calculate the average rate of change of the given function over the interval $[ - 4 , - 2 ]$ . $$\begin{array} { | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 4 & - 3 & - 2 & - 1 \
\hline f ( x ) & - 8.1 & - 1.7 & - 1.7 & - 1.3 \
\hline
\end{array}$$ ​

Accepted Answer

The answer of Calculate the average rate of change of...

Question 14

Estimate the slope of the tangent line to the graph of the following function at the point $x = - 5$ .   $f ( x ) = x ^ { 4 }$

Accepted Answer

A) -505 
B)  505 
C)  500 
D)  2,500 
E)  -500 
A) -505 
B)  505 
C)  500 
D)  2,500 
E)  -500

Question 15

Estimate $g ^ { \prime } ( 6 )$ of the function $g ( t ) = \frac { 1 } { t ^ { 5 } }$ .

Accepted Answer

A)  $- \frac { 5 } { 46,656 }$ 
B)  $- \frac { 1 } { 46,676 }$ 
C)  $- \frac { 1 } { 46,646 }$ 
D)  $- \frac { 1 } { 46,651 }$ 
E)  $\frac { 1 } { 46,656 }$ 
A)  $- \frac { 5 } { 46,656 }$ 
B)  $- \frac { 1 } { 46,676 }$ 
C)  $- \frac { 1 } { 46,646 }$ 
D)  $- \frac { 1 } { 46,651 }$ 
E)  $\frac { 1 } { 46,656 }$

Question 16

Match each function with the corresponding graph.

Accepted Answer

A) $\begin{array} { l } 
f ( x ) = 3 x \
( - 3 \leq x \leq 3 )
\end{array}$ 
B) $\begin{array} { l } 
f ( x ) = - 3 x \
( - 3 \leq x \leq 3 )
\end{array}$ 
C) $\begin{array} { l } 
f ( x ) = | 3 x | \
( - 3 \leq x \leq 3 )
\end{array}$   
A) $\begin{array} { l } 
f ( x ) = 3 x \
( - 3 \leq x \leq 3 )
\end{array}$ 
B) $\begin{array} { l } 
f ( x ) = - 3 x \
( - 3 \leq x \leq 3 )
\end{array}$ 
C) $\begin{array} { l } 
f ( x ) = | 3 x | \
( - 3 \leq x \leq 3 )
\end{array}$

Question 17

At which labeled point is the slope of the tangent line least (in the sense that -7 is less than 1)

Accepted Answer

A) P 
B)  R 
C)  Q 
A) P 
B)  R 
C)  Q

Question 18

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.  
  $f ( x ) = x ^ { 2 } - 9$

Accepted Answer

A)   $f ^ { \prime } ( x ) = 2 x ^ { 2 }$ 
B)    $f ^ { \prime } ( x ) = 2$ 
C)    $f ^ { \prime } ( x ) = x - 9$ 
D)    $f ^ { \prime } ( x ) = 2 x - 9$ 
E)    $f ^ { \prime } ( x ) = 2 x$ 
A)   $f ^ { \prime } ( x ) = 2 x ^ { 2 }$ 
B)    $f ^ { \prime } ( x ) = 2$ 
C)    $f ^ { \prime } ( x ) = x - 9$ 
D)    $f ^ { \prime } ( x ) = 2 x - 9$ 
E)    $f ^ { \prime } ( x ) = 2 x$

Question 19

A) $f ( x ) = 3 x$ , $( - 3 \leq x \leq 3 )$ 
B) $f ( x ) = - 3 x$ , $( - 3 \leq x \leq 3 )$ 
C) $f ( x ) = | 3 x |$ , $( - 3 \leq x \leq 3 )$

Accepted Answer

The answer of    
A) \(f ( x...

Question 20

Estimate the limit numerically.   $\lim_ { x \rightarrow + \infty } 10 x e ^ { - x }$

Accepted Answer

A)  1 
B)  10 
C)  $+ \infty$ 
D)  0 
E)  $- \infty$ 
A)  1 
B)  10 
C)  $+ \infty$ 
D)  0 
E)  $- \infty$

Estimate $\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = 2 }$ of the function $y = 4 x ^ { 2 }$ .

Estimate $f ^ { \prime } ( 5 )$ of the function $f ( x ) = 5 \ln ( x )$ .

At which labeled point is the slope of the tangent line greatest

Estimate the derivative of the function $f ( x ) = 8 - 8 x$ at the point $x = 9$ . Select the answer rounded to the nearest whole number.

Estimate the derivative of the function $f ( x ) = 7 - 3 x$ at the point $x = 2$ . Please round the answer to the nearest whole number.

Estimate the limit numerically. $\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 9 } \frac { x ^ { 3 } - 729 } { x - 9 }$

The graph of a function f is given. Determine whether f is continuous on its domain.

Estimate $\left. \frac { \mathrm { d } s } { \mathrm {~d} t } \right| _ { t = 1 }$ of the function $s = 2 t + t ^ { 2 }$ .

Find an equation of the tangent line to the graph of the function $f ( x ) = x + \frac { 1 } { x }$ at the point that has x-coordinate $x = \frac { 1 } { 4 }$ .

Compute $f ^ { \prime } ( 3 )$ . $f ( x ) = - 3.3 x ^ { 2 } + 3.9$

If a stone is dropped from a height of 472 feet, its height after t seconds is given by $S = - 73 - 12 t ^ { 2 }$ . Find the stone's average velocity over the period $[ 2,4 ]$ .

Calculate the average rate of change of the given function over the interval $[ - 4 , - 2 ]$ . -4 -3 -2 -1 f(x) -8.1 -1.7 -1.7 -1.3

Estimate the slope of the tangent line to the graph of the following function at the point $x = - 5$ . $f ( x ) = x ^ { 4 }$

Estimate $g ^ { \prime } ( 6 )$ of the function $g ( t ) = \frac { 1 } { t ^ { 5 } }$ .

Match each function with the corresponding graph.

At which labeled point is the slope of the tangent line least (in the sense that -7 is less than 1)

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = x ^ { 2 } - 9$

A) $f ( x ) = 3 x$ , $( - 3 \leq x \leq 3 )$ B) $f ( x ) = - 3 x$ , $( - 3 \leq x \leq 3 )$ C) $f ( x ) = | 3 x |$ , $( - 3 \leq x \leq 3 )$ $A) f ( x ) = 3 x , ( - 3 \leq x \leq 3 ) B) f ( x ) = - 3 x , ( - 3 \leq x \leq 3 ) C) f ( x ) = | 3 x | , ( - 3 \leq x \leq 3 )$

Estimate the limit numerically. $\lim_ { x \rightarrow + \infty } 10 x e ^ { - x }$

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Exam 10: Introduction to the Derivative

Estimate dy dx∣x=2\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = 2 } dxdy​​x=2​ of the function y=4x2y = 4 x ^ { 2 }y=4x2 .

Estimate f′(5)f ^ { \prime } ( 5 )f′(5) of the function f(x)=5ln⁡(x)f ( x ) = 5 \ln ( x )f(x)=5ln(x) .

At which labeled point is the slope of the tangent line greatest

Estimate the derivative of the function f(x)=8−8xf ( x ) = 8 - 8 xf(x)=8−8x at the point x=9x = 9x=9 . Select the answer rounded to the nearest whole number.

Estimate the derivative of the function f(x)=7−3xf ( x ) = 7 - 3 xf(x)=7−3x at the point x=2x = 2x=2 . ​ Please round the answer to the nearest whole number.

Estimate the limit numerically. lim⁡x→−∞x6+7,000x3+800,0002x6+1,269x3\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }limx→−∞​2x6+1,269x3x6+7,000x3+800,000​

Calculate the limit algebraically. lim⁡x→9x3−729x−9\lim _ { x \rightarrow 9 } \frac { x ^ { 3 } - 729 } { x - 9 }limx→9​x−9x3−729​

The graph of a function f is given. Determine whether f is continuous on its domain. ​ ​

Estimate ds dt∣t=1\left. \frac { \mathrm { d } s } { \mathrm {~d} t } \right| _ { t = 1 } dtds​​t=1​ of the function s=2t+t2s = 2 t + t ^ { 2 }s=2t+t2 .

Find an equation of the tangent line to the graph of the function f(x)=x+1xf ( x ) = x + \frac { 1 } { x }f(x)=x+x1​ at the point that has x-coordinate x=14x = \frac { 1 } { 4 }x=41​ .

Compute f′(3)f ^ { \prime } ( 3 )f′(3) . f(x)=−3.3x2+3.9f ( x ) = - 3.3 x ^ { 2 } + 3.9f(x)=−3.3x2+3.9

If a stone is dropped from a height of 472 feet, its height after t seconds is given by S=−73−12t2S = - 73 - 12 t ^ { 2 }S=−73−12t2 . Find the stone's average velocity over the period [2,4][ 2,4 ][2,4] .

Calculate the average rate of change of the given function over the interval [−4,−2][ - 4 , - 2 ][−4,−2] . -4 -3 -2 -1 f(x) -8.1 -1.7 -1.7 -1.3 ​

Estimate the slope of the tangent line to the graph of the following function at the point x=−5x = - 5x=−5 . f(x)=x4f ( x ) = x ^ { 4 }f(x)=x4

Estimate g′(6)g ^ { \prime } ( 6 )g′(6) of the function g(t)=1t5g ( t ) = \frac { 1 } { t ^ { 5 } }g(t)=t51​ .

Match each function with the corresponding graph.

At which labeled point is the slope of the tangent line least (in the sense that -7 is less than 1)

Compute the derivative function f′(x)f ^ { \prime } ( x )f′(x) algebraically. f(x)=x2−9f ( x ) = x ^ { 2 } - 9f(x)=x2−9

A) f(x)=3xf ( x ) = 3 xf(x)=3x , (−3≤x≤3)( - 3 \leq x \leq 3 )(−3≤x≤3) B) f(x)=−3xf ( x ) = - 3 xf(x)=−3x , (−3≤x≤3)( - 3 \leq x \leq 3 )(−3≤x≤3) C) f(x)=∣3x∣f ( x ) = | 3 x |f(x)=∣3x∣ , (−3≤x≤3)( - 3 \leq x \leq 3 )(−3≤x≤3)

Estimate the limit numerically. lim⁡x→+∞10xe−x\lim_ { x \rightarrow + \infty } 10 x e ^ { - x }limx→+∞​10xe−x

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

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Estimate $\left. \frac { \mathrm { d } y } { \mathrm {~d} x } \right| _ { x = 2 }$ of the function $y = 4 x ^ { 2 }$ .

Estimate $f ^ { \prime } ( 5 )$ of the function $f ( x ) = 5 \ln ( x )$ .

Estimate the derivative of the function $f ( x ) = 8 - 8 x$ at the point $x = 9$ . Select the answer rounded to the nearest whole number.

Estimate the derivative of the function $f ( x ) = 7 - 3 x$ at the point $x = 2$ . Please round the answer to the nearest whole number.

Estimate the limit numerically. $\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 9 } \frac { x ^ { 3 } - 729 } { x - 9 }$

The graph of a function f is given. Determine whether f is continuous on its domain.

Estimate $\left. \frac { \mathrm { d } s } { \mathrm {~d} t } \right| _ { t = 1 }$ of the function $s = 2 t + t ^ { 2 }$ .

Find an equation of the tangent line to the graph of the function $f ( x ) = x + \frac { 1 } { x }$ at the point that has x-coordinate $x = \frac { 1 } { 4 }$ .

Compute $f ^ { \prime } ( 3 )$ . $f ( x ) = - 3.3 x ^ { 2 } + 3.9$

If a stone is dropped from a height of 472 feet, its height after t seconds is given by $S = - 73 - 12 t ^ { 2 }$ . Find the stone's average velocity over the period $[ 2,4 ]$ .

Calculate the average rate of change of the given function over the interval $[ - 4 , - 2 ]$ . -4 -3 -2 -1 f(x) -8.1 -1.7 -1.7 -1.3

Estimate the slope of the tangent line to the graph of the following function at the point $x = - 5$ . $f ( x ) = x ^ { 4 }$

Estimate $g ^ { \prime } ( 6 )$ of the function $g ( t ) = \frac { 1 } { t ^ { 5 } }$ .

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = x ^ { 2 } - 9$

A) $f ( x ) = 3 x$ , $( - 3 \leq x \leq 3 )$ B) $f ( x ) = - 3 x$ , $( - 3 \leq x \leq 3 )$ C) $f ( x ) = | 3 x |$ , $( - 3 \leq x \leq 3 )$ $A) f ( x ) = 3 x , ( - 3 \leq x \leq 3 ) B) f ( x ) = - 3 x , ( - 3 \leq x \leq 3 ) C) f ( x ) = | 3 x | , ( - 3 \leq x \leq 3 )$

Estimate the limit numerically. $\lim_ { x \rightarrow + \infty } 10 x e ^ { - x }$