Question 1

Sketch the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$ using any extrema, intercepts, symmetry, and asymptotes.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E) 
  D

Question 2

Find the point on the graph of the function $f ( x ) = \sqrt { x }$ that is closest to the point $( 18,0 )$

Accepted Answer

A)  $\left( \frac { 35 } { 2 } , \sqrt { \frac { 35 } { 2 } } \right)$ 
B)  $\left( \frac { 37 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$ 
C)  $\left( \frac { 35 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$ 
D)  $\left( \sqrt { \frac { 37 } { 2 } } , \frac { 35 } { 2 } \right)$ 
E)  $\left( \sqrt { \frac { 35 } { 2 } } , \frac { 35 } { 2 } \right)$ 
A)  $\left( \frac { 35 } { 2 } , \sqrt { \frac { 35 } { 2 } } \right)$ 
B)  $\left( \frac { 37 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$ 
C)  $\left( \frac { 35 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$ 
D)  $\left( \sqrt { \frac { 37 } { 2 } } , \frac { 35 } { 2 } \right)$ 
E)  $\left( \sqrt { \frac { 35 } { 2 } } , \frac { 35 } { 2 } \right)$ A

Question 3

Match the function $f ( x ) = \frac { 3 x } { x ^ { 2 } + 2 }$ with one of the following graphs.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E) 
  E

Question 4

Determine the x-coordinate(s) of any relative extrema and inflection points of the function $y = x \ln \frac { x ^ { 5 } } { 6 }$

Accepted Answer

A) relative minimum: $x = 6 e ^ { \frac { 1 } { 5 } } ;$ inflection point $: x = 6 ^ { \frac { - 9 } { 20 } }$ 
B)  no relative extrema; inflection point: $x = 6 ^ { \frac { - 9 } { 20 } }$ 
C)  relative minimum: $x = \frac { ( 6 ) ^ { \frac { 1 } { 5 } } } { e }$; no inflection points 
D) no relative extrema; inflection point: $x = 6 e ^ { \frac { - 1 } { 5 } }$ 
E)  relative maximum: $x = \left( \frac { 6 } { e } \right) ^ { \frac { 1 } { 5 } }$ inflection point: $x = 6 e ^ { \frac { - 1 } { 5 } }$ 
A) relative minimum: $x = 6 e ^ { \frac { 1 } { 5 } } ;$ inflection point $: x = 6 ^ { \frac { - 9 } { 20 } }$ 
B)  no relative extrema; inflection point: $x = 6 ^ { \frac { - 9 } { 20 } }$ 
C)  relative minimum: $x = \frac { ( 6 ) ^ { \frac { 1 } { 5 } } } { e }$; no inflection points 
D) no relative extrema; inflection point: $x = 6 e ^ { \frac { - 1 } { 5 } }$ 
E)  relative maximum: $x = \left( \frac { 6 } { e } \right) ^ { \frac { 1 } { 5 } }$ inflection point: $x = 6 e ^ { \frac { - 1 } { 5 } }$

Question 5

Identify the open intervals on which the function $y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }$ increasing or decreasing.

Accepted Answer

A)  increasing on $\left( \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } \right) ;$ decreasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 11 \pi } { 6 } , 2 \pi \right)$ 
B)  increasing on $\left( 0 , \frac { 11 \pi } { 6 } \right)$ and $\left( \frac { 7 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ 
C)  increasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 11 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } \right)$ 
D)  increasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 5 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 7 \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$ 
E)  increasing on $\left( \frac { 7 \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$ and $\left( \frac { 5 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \right)$ 
A)  increasing on $\left( \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } \right) ;$ decreasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 11 \pi } { 6 } , 2 \pi \right)$ 
B)  increasing on $\left( 0 , \frac { 11 \pi } { 6 } \right)$ and $\left( \frac { 7 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ 
C)  increasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 11 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } \right)$ 
D)  increasing on $\left( 0 , \frac { 7 \pi } { 6 } \right)$ and $\left( \frac { 5 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 7 \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$ 
E)  increasing on $\left( \frac { 7 \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$ and $\left( \frac { 5 \pi } { 6 } , 2 \pi \right)$; decreasing on $\left( \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \right)$

Question 6

Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 3$ . Use the Second Derivative Test where applicable.

Accepted Answer

A)  relative max: $( 1 , - 2 )$; no relative min 
B)  no relative max or min 
C)  relative min: $( 0 , - 3 )$; no relative $\max$ 
D)  relative max: $( 1 , - 2 )$; relative min: $( 0 , - 3 )$ 
E)  relative max: $( 0 , - 3 )$; no relative min 
A)  relative max: $( 1 , - 2 )$; no relative min 
B)  no relative max or min 
C)  relative min: $( 0 , - 3 )$; no relative $\max$ 
D)  relative max: $( 1 , - 2 )$; relative min: $( 0 , - 3 )$ 
E)  relative max: $( 0 , - 3 )$; no relative min

Question 7

A meteorologist measures the atmospheric pressure P (in kilograms per square meter) at altitude h (in kilometers). The data are shown below. Use the regression capabilities of the Graphing utility to find a linear model for the revised data points obtained by plotting the points $( h , \ln P )$ $\begin{array}{|c|c|c|c|c|c|}
\hline h & 0 & 5 & 10 & 15 & 20 \
\hline p & 28,085 & 15169 & 6,458 & 3,371 & 1,405 \
\hline
\end{array}$

Accepted Answer

A)  $\ln P = - 0.1499 h + 10.3018$ 
B)  $\ln P = - 0.1944 h - 14.3601$ 
C)  $\ln P = 0.1944 h + 10.3601$ 
D)  $\ln P = - 0.1499 h - 14.3016$ 
E)  $\ln P = - 0.1499 h - 10.3018$ 
A)  $\ln P = - 0.1499 h + 10.3018$ 
B)  $\ln P = - 0.1944 h - 14.3601$ 
C)  $\ln P = 0.1944 h + 10.3601$ 
D)  $\ln P = - 0.1499 h - 14.3016$ 
E)  $\ln P = - 0.1499 h - 10.3018$

Question 8

$\text { Locate the absolute extrema of the function } f ( x ) = \cos ( \pi x ) \text { on the closed interval }$ $\left[ 0 , \frac { 1 } { 2 } \right]$

Accepted Answer

A)  absolute max: $( 0,1 ) ;$ absolute min: $\left( \left( \frac { 1 } { 2 } \right) , \cos \left( \frac { \pi } { 2 } \right) \right)$ 
B)  absolute max: $\left( \left( \frac { 1 } { 2 } \right) , \cos \left( \frac { \pi } { 2 } \right) \right) ;$ absolute min: $( 0,1 )$ 
C)  absolute max: $\left( \left( \frac { 1 } { 2 } \right) , \cos \left( \frac { \pi } { 2 } \right) \right) ;$ no absolute min 
D)  no absolute max; absolute min: $( 0,1 )$ 
E)  absolute max: $( 0,1 )$; no absolute min 
A)  absolute max: $( 0,1 ) ;$ absolute min: $\left( \left( \frac { 1 } { 2 } \right) , \cos \left( \frac { \pi } { 2 } \right) \right)$ 
B)  absolute max: $\left( \left( \frac { 1 } { 2 } \right) , \cos \left( \frac { \pi } { 2 } \right) \right) ;$ absolute min: $( 0,1 )$ 
C)  absolute max: $\left( \left( \frac { 1 } { 2 } \right) , \cos \left( \frac { \pi } { 2 } \right) \right) ;$ no absolute min 
D)  no absolute max; absolute min: $( 0,1 )$ 
E)  absolute max: $( 0,1 )$; no absolute min

Question 9

$\text { Find the value of the derivative (if it exists) of the function } f ( x ) = 15 - | x | \text { at the }$ extremum point $( 0,15 )$

Accepted Answer

A)  13 
B)  0 
C)  15 
D)  $- 15$ 
E)  does not exist 
A)  13 
B)  0 
C)  15 
D)  $- 15$ 
E)  does not exist

Question 10

Find the points of inflection and discuss the concavity of the function $f ( x ) = x \sqrt { x + 16 }$

Accepted Answer

A)  no inflection points; concave up on $( - 16 , \infty )$ 
B)  no inflection points; concave down on $( - 16 , \infty )$ 
C)  inflection point at $x = 16$; concave up on $( - 16 , \infty )$ 
D)  inflection point at $x = 0$; concave up on $( - 16,0 )$; concave down on $( 0 , \infty )$ 
E)  inflection point at $x = 16$; concave down on $( - 16 , \infty )$ 
A)  no inflection points; concave up on $( - 16 , \infty )$ 
B)  no inflection points; concave down on $( - 16 , \infty )$ 
C)  inflection point at $x = 16$; concave up on $( - 16 , \infty )$ 
D)  inflection point at $x = 0$; concave up on $( - 16,0 )$; concave down on $( 0 , \infty )$ 
E)  inflection point at $x = 16$; concave down on $( - 16 , \infty )$

Question 11

Find any critical numbers of the function $g ( t ) = t \sqrt { 2 - t } , \mathrm { t } < 2$

Accepted Answer

A)  0 
B)  $- \frac { 2 } { 3 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 4 } { 3 }$ 
A)  0 
B)  $- \frac { 2 } { 3 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 4 } { 3 }$

Question 12

Find the differential dy of the function $$y = - x ^ { 2 } - 3 x - 2$$

Accepted Answer

A)  $( - 2 x - 3 ) d x$ 
B)  $\left( - x ^ { 3 } - 3 x ^ { 2 } - 2 x \right) d x$ 
C)  $\left( - 2 x ^ { 2 } - 3 x - 2 \right) d x$ 
D)  $\left( - \frac { 1 } { 3 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } - 2 x \right) d x$ 
E)  $( - x - 3 ) d x$ 
A)  $( - 2 x - 3 ) d x$ 
B)  $\left( - x ^ { 3 } - 3 x ^ { 2 } - 2 x \right) d x$ 
C)  $\left( - 2 x ^ { 2 } - 3 x - 2 \right) d x$ 
D)  $\left( - \frac { 1 } { 3 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } - 2 x \right) d x$ 
E)  $( - x - 3 ) d x$

Question 13

Find all points of inflection of the graph of the function $f ( x ) = 2 \sin 7 x + \sin 14 x \text { on }$ the interval . Round your answer to three decimal places wherever applicable.

Accepted Answer

A)  $( 0.26,1.452 ) , ( 0.449,0 ) , ( 0.637 , - 1.452 )$ 
B)  $( 0.282,1.11 ) , ( 0.669 , - 0.004 )$ 
C)  $( 0.449,0 ) , ( 0.637 , - 1.452 )$ 
D)  $( 0.26,1.452 ) , ( 0.449,0 ) , ( 0.659 , - 1.794 )$ 
E)  $( 0.26,1.452 ) , ( 0.637 , - 1.452 )$ 
A)  $( 0.26,1.452 ) , ( 0.449,0 ) , ( 0.637 , - 1.452 )$ 
B)  $( 0.282,1.11 ) , ( 0.669 , - 0.004 )$ 
C)  $( 0.449,0 ) , ( 0.637 , - 1.452 )$ 
D)  $( 0.26,1.452 ) , ( 0.449,0 ) , ( 0.659 , - 1.794 )$ 
E)  $( 0.26,1.452 ) , ( 0.637 , - 1.452 )$

Question 14

Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum.

Accepted Answer

A)  28 and 28 
B)  28 and 14 
C)  22 and 17 
D)  34 and 11 
E)  24 and 16 
A)  28 and 28 
B)  28 and 14 
C)  22 and 17 
D)  34 and 11 
E)  24 and 16

Question 15

Determine whether Rolle's Theorem can be applied to $f ( x ) = \frac { x ^ { 2 } - 13 } { x } \text { on the closed }$ interval $[ - 13,13 ]$. If Rolle's Theorem can be applied, find all values of $c$ in the open interval $( - 13,13 )$ such that $f ^ {t } ( c ) = 0$

Accepted Answer

A)  $c = 8$ 
B)  $c = 12 , c = 11$ 
C)  $c = 11 , c = 8$ 
D)  $c = 12$ 
E)  Rolle's Theorem does not apply 
A)  $c = 8$ 
B)  $c = 12 , c = 11$ 
C)  $c = 11 , c = 8$ 
D)  $c = 12$ 
E)  Rolle's Theorem does not apply

Question 16

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = ( x - 5 ) ( x - 6 ) ( x - 7 )$ on the closed interval $[ 5,7 ]$. If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $( 5,7 )$ such that $f ^ { t } ( c ) = 0$.

Accepted Answer

A) Rolle's Theorem applies; $c = 5 + \frac { \sqrt { 3 } } { 3 }$ 
B) Rolle's Theorem applies; $c = 6 + \frac { \sqrt { 3 } } { 3 }$ 
C) Rolle's Theorem does apply; $c = 7 - \frac { \sqrt { 3 } } { 3 }$ 
D) Rolle's Theorem applies; $c = 6 + \frac { \sqrt { 3 } } { 3 } , 6 - \frac { \sqrt { 3 } } { 3 }$ 
E)  Rolle's Theorem does not apply 
A) Rolle's Theorem applies; $c = 5 + \frac { \sqrt { 3 } } { 3 }$ 
B) Rolle's Theorem applies; $c = 6 + \frac { \sqrt { 3 } } { 3 }$ 
C) Rolle's Theorem does apply; $c = 7 - \frac { \sqrt { 3 } } { 3 }$ 
D) Rolle's Theorem applies; $c = 6 + \frac { \sqrt { 3 } } { 3 } , 6 - \frac { \sqrt { 3 } } { 3 }$ 
E)  Rolle's Theorem does not apply

Question 17

Match the function $f ( x ) = \frac { - 2 x } { \sqrt { 64 x ^ { 2 } + 2 } }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. $\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \
\hline f(x) & & & & & & & \
\hline
\end{array}$

Accepted Answer

A)  $- 0.2500$ 
B)  $1.0000$ 
C)  $2.1800$ 
D)  $- 0.9000$ 
E)  does not exist 
A)  $- 0.2500$ 
B)  $1.0000$ 
C)  $2.1800$ 
D)  $- 0.9000$ 
E)  does not exist

Question 18

$\text { Locate the absolute extrema of the function } y = x ^ { 2 } - 2 \cos x \text { on the closed interval }$ $[ - 3,5 ] \text {. Round your answers to four decimal places. }$

Accepted Answer

A)  left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum 
critical points: $( 0,0.0000 )$ absolute minimum 
B)  left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
 right endpoint: $( 5,8.0822 )$ absolute maximum
 critical points: none 
C)  left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
 right endpoint: $( 5,8.0822 )$ absolute maximum
 critical points: $( 0,0.0000 )$ 
D)  left endpoint: $( - 3,10.9800 )$ absolute minimum
 right endpoint: $( 5,24.4327 )$ absolute maximum
 critical points: none 
E)  left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum
 critical points: $( 0 , - 2.0000 )$ absolute minimum 
A)  left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum 
critical points: $( 0,0.0000 )$ absolute minimum 
B)  left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
 right endpoint: $( 5,8.0822 )$ absolute maximum
 critical points: none 
C)  left endpoint: $( - 3 , - 6.2822 )$ absolute minimum
 right endpoint: $( 5,8.0822 )$ absolute maximum
 critical points: $( 0,0.0000 )$ 
D)  left endpoint: $( - 3,10.9800 )$ absolute minimum
 right endpoint: $( 5,24.4327 )$ absolute maximum
 critical points: none 
E)  left endpoint: $( - 3,10.9800 )$
right endpoint: $( 5,24.4327 )$ absolute maximum
 critical points: $( 0 , - 2.0000 )$ absolute minimum

Question 19

A sector with central angle $\theta \text { is }$ is cut from a circle of radius 10 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of The cone is a maximum.

Accepted Answer

A)  $\sqrt { \frac { 4 } { 3 } } \pi$ radians 
B)  $2 \left( 1 - \sqrt { \frac { 2 } { 3 } } \right) \pi$ radians 
C)  $\frac { 3 } { 10 } \pi$ radians 
D)  $\frac { 1 } { 10 } \pi$ radians 
E)  $\frac { 1 } { 5 } \pi$ radians 
A)  $\sqrt { \frac { 4 } { 3 } } \pi$ radians 
B)  $2 \left( 1 - \sqrt { \frac { 2 } { 3 } } \right) \pi$ radians 
C)  $\frac { 3 } { 10 } \pi$ radians 
D)  $\frac { 1 } { 10 } \pi$ radians 
E)  $\frac { 1 } { 5 } \pi$ radians

Question 20

$\text { Find the critical number of the function } f ( x ) = - 6 x ^ { 2 } + 108 x + 7$

Accepted Answer

A)  $x = 18$ 
B)  $x = 9$ 
C)  $x = 84$ 
D)  $x = 14$ 
E)  $x = 7$ 
A)  $x = 18$ 
B)  $x = 9$ 
C)  $x = 84$ 
D)  $x = 14$ 
E)  $x = 7$

Sketch the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$ using any extrema, intercepts, symmetry, and asymptotes.

Find the point on the graph of the function $f ( x ) = \sqrt { x }$ that is closest to the point $( 18,0 )$

Match the function $f ( x ) = \frac { 3 x } { x ^ { 2 } + 2 }$ with one of the following graphs.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function $y = x \ln \frac { x ^ { 5 } } { 6 }$

Identify the open intervals on which the function $y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }$ increasing or decreasing.

Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 3$ . Use the Second Derivative Test where applicable.

$\text { Locate the absolute extrema of the function } f ( x ) = \cos ( \pi x ) \text { on the closed interval }$ $\left[ 0 , \frac { 1 } { 2 } \right]$

$\text { Find the value of the derivative (if it exists) of the function } f ( x ) = 15 - | x | \text { at the }$ extremum point $( 0,15 )$ $\text { Find the value of the derivative (if it exists) of the function } f ( x ) = 15 - | x | \text { at the } extremum point ( 0,15 )$

Find the points of inflection and discuss the concavity of the function $f ( x ) = x \sqrt { x + 16 }$

Find any critical numbers of the function $g ( t ) = t \sqrt { 2 - t } , \mathrm { t } < 2$

Find the differential dy of the function $y = - x ^ { 2 } - 3 x - 2$

Find all points of inflection of the graph of the function $f ( x ) = 2 \sin 7 x + \sin 14 x \text { on }$ the interval . Round your answer to three decimal places wherever applicable.

Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum.

Determine whether Rolle's Theorem can be applied to $f ( x ) = \frac { x ^ { 2 } - 13 } { x } \text { on the closed }$ interval $[ - 13,13 ]$ . If Rolle's Theorem can be applied, find all values of $c$ in the open interval $( - 13,13 )$ such that $f ^ {t } ( c ) = 0$

Determine whether Rolle's Theorem can be applied to the function $f ( x ) = ( x - 5 ) ( x - 6 ) ( x - 7 )$ on the closed interval $[ 5,7 ]$ . If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $( 5,7 )$ such that $f ^ { t } ( c ) = 0$ .

Match the function $f ( x ) = \frac { - 2 x } { \sqrt { 64 x ^ { 2 } + 2 } }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 1 f(x)

$\text { Locate the absolute extrema of the function } y = x ^ { 2 } - 2 \cos x \text { on the closed interval }$ $[ - 3,5 ] \text {. Round your answers to four decimal places. }$

A sector with central angle $\theta \text { is }$ is cut from a circle of radius 10 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of The cone is a maximum.

$\text { Find the critical number of the function } f ( x ) = - 6 x ^ { 2 } + 108 x + 7$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 4: Extrema on an Interval

Sketch the graph of the function f(x)=x2x2−1f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }f(x)=x2−1x2​ using any extrema, intercepts, symmetry, and asymptotes.

Find the point on the graph of the function f(x)=xf ( x ) = \sqrt { x }f(x)=x​ that is closest to the point (18,0)( 18,0 )(18,0)

Match the function f(x)=3xx2+2f ( x ) = \frac { 3 x } { x ^ { 2 } + 2 }f(x)=x2+23x​ with one of the following graphs.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function y=xln⁡x56y = x \ln \frac { x ^ { 5 } } { 6 }y=xln6x5​

Identify the open intervals on which the function y=9x−18cos⁡x,0<x<2π is y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }y=9x−18cosx,0<x<2π is increasing or decreasing.

Find all relative extrema of the function f(x)=x8/9−3f ( x ) = x ^ { 8 / 9 } - 3f(x)=x8/9−3 . Use the Second Derivative Test where applicable.

Find the points of inflection and discuss the concavity of the function f(x)=xx+16f ( x ) = x \sqrt { x + 16 }f(x)=xx+16​

Find any critical numbers of the function g(t)=t2−t,t<2g ( t ) = t \sqrt { 2 - t } , \mathrm { t } < 2g(t)=t2−t​,t<2

Find the differential dy of the function y=−x2−3x−2y = - x ^ { 2 } - 3 x - 2y=−x2−3x−2

Find all points of inflection of the graph of the function f(x)=2sin⁡7x+sin⁡14x on f ( x ) = 2 \sin 7 x + \sin 14 x \text { on }f(x)=2sin7x+sin14x on the interval . Round your answer to three decimal places wherever applicable.

Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum.

Match the function f(x)=−2x64x2+2f ( x ) = \frac { - 2 x } { \sqrt { 64 x ^ { 2 } + 2 } }f(x)=64x2+2​−2x​ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 1 f(x)

A sector with central angle θ is \theta \text { is }θ is is cut from a circle of radius 10 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of The cone is a maximum.

Find the critical number of the function f(x)=−6x2+108x+7\text { Find the critical number of the function } f ( x ) = - 6 x ^ { 2 } + 108 x + 7 Find the critical number of the function f(x)=−6x2+108x+7

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Sketch the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$ using any extrema, intercepts, symmetry, and asymptotes.

Find the point on the graph of the function $f ( x ) = \sqrt { x }$ that is closest to the point $( 18,0 )$

Match the function $f ( x ) = \frac { 3 x } { x ^ { 2 } + 2 }$ with one of the following graphs.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function $y = x \ln \frac { x ^ { 5 } } { 6 }$

Identify the open intervals on which the function $y = 9 x - 18 \cos x , 0 < x < 2 \pi \text { is }$ increasing or decreasing.

Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 3$ . Use the Second Derivative Test where applicable.

Find the points of inflection and discuss the concavity of the function $f ( x ) = x \sqrt { x + 16 }$

Find any critical numbers of the function $g ( t ) = t \sqrt { 2 - t } , \mathrm { t } < 2$

Find the differential dy of the function $y = - x ^ { 2 } - 3 x - 2$

Find all points of inflection of the graph of the function $f ( x ) = 2 \sin 7 x + \sin 14 x \text { on }$ the interval . Round your answer to three decimal places wherever applicable.

Match the function $f ( x ) = \frac { - 2 x } { \sqrt { 64 x ^ { 2 } + 2 } }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. x 1 1 1 1 1 1 1 f(x)

A sector with central angle $\theta \text { is }$ is cut from a circle of radius 10 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of The cone is a maximum.

$\text { Find the critical number of the function } f ( x ) = - 6 x ^ { 2 } + 108 x + 7$