Question 1

Evaluate the integral.   $\int \left( x ^ { 3 } + x ^ { 4 } \right) \sec ^ { 2 } \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) \mathrm { d } x$

Accepted Answer

A)  $\frac { 1 } { 5 } \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
B)  $\frac { 1 } { 20 } \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
C)  $\frac { 1 } { 4 } \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
D)  $\tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
E)  $20 \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
A)  $\frac { 1 } { 5 } \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
B)  $\frac { 1 } { 20 } \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
C)  $\frac { 1 } { 4 } \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
D)  $\tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ 
E)  $20 \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$ B

Question 2

Find the derivative of the function.  $p ( x ) = 3 + 2 \sin \left( \frac { \pi } { 2 } ( x - 1 ) \right)$

Accepted Answer

A)  $p ^ { \prime } ( x ) = \pi \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
B)  $p ^ { \prime } ( x ) = 2 \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
C)  $p ^ { \prime } ( x ) = \pi \cos \left( \frac { \pi } { 2 } ( x + 1 ) \right)$ 
D)  $p ^ { \prime } ( x ) = \pi \sin \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
E)  $p ^ { \prime } ( x ) = 3 + \pi \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
A)  $p ^ { \prime } ( x ) = \pi \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
B)  $p ^ { \prime } ( x ) = 2 \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
C)  $p ^ { \prime } ( x ) = \pi \cos \left( \frac { \pi } { 2 } ( x + 1 ) \right)$ 
D)  $p ^ { \prime } ( x ) = \pi \sin \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ 
E)  $p ^ { \prime } ( x ) = 3 + \pi \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$ A

Question 3

Evaluate the integral.  $\int _ { - \frac { \pi } { 3 } } ^ { \frac { \pi } { 4 } } \sin x \mathrm {~d} x$

Accepted Answer

A)  $\cos \frac { \pi } { 4 } - \cos \pi$ 
B)  $\sin \frac { \pi } { 3 } + \sin \frac { \pi } { 4 }$ 
C)  $\sin \frac { \pi } { 3 } - \cos \frac { \pi } { 5 }$ 
D)  $\cos \frac { \pi } { 3 } - \cos \frac { \pi } { 4 }$ 
E)  none of these 
A)  $\cos \frac { \pi } { 4 } - \cos \pi$ 
B)  $\sin \frac { \pi } { 3 } + \sin \frac { \pi } { 4 }$ 
C)  $\sin \frac { \pi } { 3 } - \cos \frac { \pi } { 5 }$ 
D)  $\cos \frac { \pi } { 3 } - \cos \frac { \pi } { 4 }$ 
E)  none of these D

Question 4

Find the derivative of the function.   $u ( x ) = \cos \left( x ^ { 2 } - 2 x \right)$

Accepted Answer

A)  $u ^ { \prime } ( x ) = ( 2 x - 2 ) \cos \left( x ^ { 2 } - 2 x \right)$ 
B)  $u ^ { \prime } ( x ) = - ( 2 x + 2 ) \sin \left( x ^ { 2 } - 2 x \right)$ 
C)  $u ^ { \prime } ( x ) = - ( 2 x - 2 ) \cos \left( x ^ { 2 } - 2 x \right)$ 
D)  $u ^ { \prime } ( x ) = - ( 2 x - 2 ) \sin \left( x ^ { 2 } - 2 x \right)$ 
E)  $u ^ { \prime } ( x ) = ( 2 x + 2 ) \sin \left( x ^ { 2 } - 2 x \right)$ 
A)  $u ^ { \prime } ( x ) = ( 2 x - 2 ) \cos \left( x ^ { 2 } - 2 x \right)$ 
B)  $u ^ { \prime } ( x ) = - ( 2 x + 2 ) \sin \left( x ^ { 2 } - 2 x \right)$ 
C)  $u ^ { \prime } ( x ) = - ( 2 x - 2 ) \cos \left( x ^ { 2 } - 2 x \right)$ 
D)  $u ^ { \prime } ( x ) = - ( 2 x - 2 ) \sin \left( x ^ { 2 } - 2 x \right)$ 
E)  $u ^ { \prime } ( x ) = ( 2 x + 2 ) \sin \left( x ^ { 2 } - 2 x \right)$

Question 5

Calculate the derivative.  $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$

Accepted Answer

A)  $\frac { \cot ( 6 x - 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$ 
B)  $\frac { \cot ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$ 
C)  $- 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$ 
D)  $\frac { \cot ( 6 x + 5 ) } { x }$ 
E)  $\frac { \tan ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$ 
A)  $\frac { \cot ( 6 x - 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$ 
B)  $\frac { \cot ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$ 
C)  $- 6 \ln | x | \csc ^ { 2 } ( 6 x + 5 )$ 
D)  $\frac { \cot ( 6 x + 5 ) } { x }$ 
E)  $\frac { \tan ( 6 x + 5 ) } { x } - 6 \ln | x | \csc ^ { 2 } ( 6 x - 5 )$

Question 6

Model the curve with a cosine function.
​   ​
Note that the period of the curve is $P = 28$ , its range is $[ 0,110 ]$ and the graph of the cosine function is shifted upward 55 units and shifted to the right 14 units. Write the model function as a function of x and π.

Accepted Answer

The answer of Model the curve with a cosine function.
​...

Question 7

Evaluate the integral.   $\int 10.8 \cos ( 6 x - 1 ) \mathrm { d } x$

Accepted Answer

A)  $10.8 \sin ( 6 x - 1 ) + C$ 
B)  $1.8 \cos ( 6 x - 1 ) + C$ 
C)    $1.8 \sin ( 6 x - 1 ) + C$ 
D)  $- 1.8 \sin ( 6 x - 1 ) + C$ 
E)  $1.8 \sin \left( 6 x ^ { 2 } - x \right) + C$ 
A)  $10.8 \sin ( 6 x - 1 ) + C$ 
B)  $1.8 \cos ( 6 x - 1 ) + C$ 
C)    $1.8 \sin ( 6 x - 1 ) + C$ 
D)  $- 1.8 \sin ( 6 x - 1 ) + C$ 
E)  $1.8 \sin \left( 6 x ^ { 2 } - x \right) + C$

Question 8

Use geometry to compute the given integral.  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x$

Accepted Answer

A)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 6$ 
B)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x \mathrm {~d} x = - 4$ 
C)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 0$ 
D)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 8$ 
E)  none of these 
A)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 6$ 
B)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x \mathrm {~d} x = - 4$ 
C)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 0$ 
D)  $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 8$ 
E)  none of these

Question 9

The uninflated cost of Dugout brand snow shovels currently varies from a high of $30 on January 1 $( t = 0 )$ to a low of $6 on July 1 $( t = 0.5 )$ . Assuming this trend were to continue indefinitely, calculate the uninflated cost $u ( t )$ of Dugout snow shovels as a function of time t in years. (Use a sine function.)

Accepted Answer

A)  $u ( t ) = 15 \sin ( 2 \pi ( t - 0.75 ) ) + 3$ 
B)  $u ( t ) = 12 \sin ( 2 \pi ( t - 0.75 ) ) + 18$ 
C)  $u ( t ) = 18 \sin ( 2 \pi ( t - 0.75 ) ) + 12$ 
D)  $u ( t ) = - 12 \sin ( 2 \pi ( t - 0.75 ) ) - 18$ 
E)  $u ( t ) = 12 \sin ( 2 \pi ( t + 0.75 ) ) - 18$ 
A)  $u ( t ) = 15 \sin ( 2 \pi ( t - 0.75 ) ) + 3$ 
B)  $u ( t ) = 12 \sin ( 2 \pi ( t - 0.75 ) ) + 18$ 
C)  $u ( t ) = 18 \sin ( 2 \pi ( t - 0.75 ) ) + 12$ 
D)  $u ( t ) = - 12 \sin ( 2 \pi ( t - 0.75 ) ) - 18$ 
E)  $u ( t ) = 12 \sin ( 2 \pi ( t + 0.75 ) ) - 18$

Question 10

Decide whether the integral converges. If the integral converges, compute its value.   $\int _ { 0 } ^ { + \infty } e ^ { - 3 x } \cos ( 3 x ) \mathrm { d } x$

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 3 }$ 
E)  diverges 
A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 3 }$ 
E)  diverges

Question 11

Use the conversion formula $\cos x = \sin \left( \frac { \pi } { 2 } - x \right)$ to replace the expression   $g ( t ) = 45 - \cos ( t - 5 )$  
By a sine function.

Accepted Answer

A)  $g ( t ) = 45 - \sin \left( \frac { \pi } { 2 } - t + 5 \right)$ 
B)  $g ( t ) = 45 - \sin \left( \frac { \pi } { 2 } - t - 5 \right)$ 
C)  $g ( t ) = 45 - \sin ( \pi - t + 5 )$ 
D)  $g ( t ) = 45 - \sin \left( \frac { \pi } { 2 } + t + 5 \right)$ 
E)  $g ( t ) = \frac { \pi } { 2 } - 45 - \sin ( t + 5 )$ 
A)  $g ( t ) = 45 - \sin \left( \frac { \pi } { 2 } - t + 5 \right)$ 
B)  $g ( t ) = 45 - \sin \left( \frac { \pi } { 2 } - t - 5 \right)$ 
C)  $g ( t ) = 45 - \sin ( \pi - t + 5 )$ 
D)  $g ( t ) = 45 - \sin \left( \frac { \pi } { 2 } + t + 5 \right)$ 
E)  $g ( t ) = \frac { \pi } { 2 } - 45 - \sin ( t + 5 )$

Question 12

The depth of water $d ( t )$ at my favorite surfing spot varies from 5 to 15 feet, depending on the time. Last Sunday high tide occurred at 5:00 A.M. and the next high tide occurred at 6:30 P.M. Use a sine function to model to the depth of water as a function of time t in hours since midnight in Sunday morning.

Accepted Answer

The answer of The depth of water \(d ( t...

Question 13

Recall that the average of a function $f ( x )$ on an interval $[ a , b ]$ is   $\bar { f } = \frac { 1 } { b - a } \int _ { a } ^ { b } f ( x ) \mathrm { d } x$  
Calculate the 9-unit moving average of the function.
  $f ( x ) = \cos \left( \frac { \pi x } { 18 } \right)$

Accepted Answer

A)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) - \cos \left( \frac { \pi x } { 18 } \right) \right)$ 
B)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) - \cos \left( \frac { \pi x } { 3 } \right) \right)$ 
C)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) + \cos \left( \frac { \pi x } { 2 } \right) \right)$ 
D)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \cos \left( \frac { \pi x } { 18 } \right) - \sin \left( \frac { \pi x } { 18 } \right) \right)$ 
E)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) + \cos \left( \frac { \pi x } { 18 } \right) \right)$ 
A)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) - \cos \left( \frac { \pi x } { 18 } \right) \right)$ 
B)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) - \cos \left( \frac { \pi x } { 3 } \right) \right)$ 
C)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) + \cos \left( \frac { \pi x } { 2 } \right) \right)$ 
D)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \cos \left( \frac { \pi x } { 18 } \right) - \sin \left( \frac { \pi x } { 18 } \right) \right)$ 
E)  $\bar { f } ( x ) = \frac { 2 } { \pi } \left( \sin \left( \frac { \pi x } { 18 } \right) + \cos \left( \frac { \pi x } { 18 } \right) \right)$

Question 14

Find the derivative of the function.  $y ( x ) = 7 \cos \left( e ^ { x } \right) + 9 e ^ { x } \cos x$

Accepted Answer

A)  $y' ( x ) = e ^ { x } \left( - 7 \sin \left( e ^ { x } \right) + 9 \cos x - 9 \sin x \right)$ 
B)  $y' ( x ) = - 7 \sin \left( e ^ { x } \right) + 9 \sin x - 9 \cos x$ 
C)  $y' ( x ) = - 7 \sin \left( e ^ { x } \right) + 9 \cos x - 9 \sin x$ 
D)  $y '( x ) = e ^ { x } \left( - 7 \sin \left( e ^ { x } \right) - 9 \cos x + 9 \sin x \right)$ 
E)  $y ^ { \prime } ( x ) = e ^ { x ^ { 2 } } \left( - 7 \sin \left( e ^ { x } \right) - 9 \cos x + 9 \sin x \right)$ 
A)  $y' ( x ) = e ^ { x } \left( - 7 \sin \left( e ^ { x } \right) + 9 \cos x - 9 \sin x \right)$ 
B)  $y' ( x ) = - 7 \sin \left( e ^ { x } \right) + 9 \sin x - 9 \cos x$ 
C)  $y' ( x ) = - 7 \sin \left( e ^ { x } \right) + 9 \cos x - 9 \sin x$ 
D)  $y '( x ) = e ^ { x } \left( - 7 \sin \left( e ^ { x } \right) - 9 \cos x + 9 \sin x \right)$ 
E)  $y ^ { \prime } ( x ) = e ^ { x ^ { 2 } } \left( - 7 \sin \left( e ^ { x } \right) - 9 \cos x + 9 \sin x \right)$

Question 15

Evaluate the integral.   $\int 6 \sec ( 3 x - 7 ) \mathrm { d } x$

Accepted Answer

Question 16

Sketch the curves without any technological help. $f ( t ) = 2 \cos t$ ; $g ( t ) = 3.3 \cos ( 2 t )$

Accepted Answer

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

Question 17

The cost of Dig-It brand snow shovels is given by  $c ( t ) = 3 \sin ( 2 \pi ( t - 0.75 ) )$ 
Where t is time in years since January 1, 1997. How fast, in dollars per year, is the cost increasing on October 30, 1997

Accepted Answer

A) $21.85 per year 
B)  $18.85 per year 
C)  $9.42 per year 
D)  $20.85 per year 
E)  $6.00 per year 
A) $21.85 per year 
B)  $18.85 per year 
C)  $9.42 per year 
D)  $20.85 per year 
E)  $6.00 per year

Question 18

Find the derivative of the function.   $w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)$

Accepted Answer

A)   $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 2 \right)$ 
B)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$ 
C)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } + 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$ 
D)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right)$ 
E)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 4 \right)$ 
A)   $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 2 \right)$ 
B)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$ 
C)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } + 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } ( 2 x - 2 )$ 
D)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right)$ 
E)  $w ^ { \prime } ( x ) = 2 \sec ^ { 2 } x \cdot \sin x \cdot \tan \left( x ^ { 2 } - 2 \right) + 4 x \cdot \sec x \cdot \sec ^ { 2 } \left( x ^ { 2 } - 4 \right)$

Question 19

Evaluate the integral   $\int _ { \frac { 1 } { \pi } } ^ { \frac { 3 } { \pi } } 9 \frac { \sin \left( \frac { 1 } { x } \right) } { x ^ { 2 } } \mathrm {~d} x$

Accepted Answer

A) 9.5 
B)  9 
C)  31.5 
D)  22.5 
E)  13.5 
A) 9.5 
B)  9 
C)  31.5 
D)  22.5 
E)  13.5

Question 20

Evaluate the integral.
​ $\int - \sin ( - 7 x - 8 ) \mathrm { d } x$

Accepted Answer

The answer of Evaluate the integral.
​ \(\int - \sin (...

Evaluate the integral. $\int \left( x ^ { 3 } + x ^ { 4 } \right) \sec ^ { 2 } \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) \mathrm { d } x$

Find the derivative of the function. $p ( x ) = 3 + 2 \sin \left( \frac { \pi } { 2 } ( x - 1 ) \right)$

Evaluate the integral. $\int _ { - \frac { \pi } { 3 } } ^ { \frac { \pi } { 4 } } \sin x \mathrm {~d} x$

Find the derivative of the function. $u ( x ) = \cos \left( x ^ { 2 } - 2 x \right)$

Calculate the derivative. $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$

Model the curve with a cosine function. Note that the period of the curve is $P = 28$ , its range is $[ 0,110 ]$ and the graph of the cosine function is shifted upward 55 units and shifted to the right 14 units. Write the model function as a function of x and π.

Evaluate the integral. $\int 10.8 \cos ( 6 x - 1 ) \mathrm { d } x$

Use geometry to compute the given integral. $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { + \infty } e ^ { - 3 x } \cos ( 3 x ) \mathrm { d } x$

Use the conversion formula $\cos x = \sin \left( \frac { \pi } { 2 } - x \right)$ to replace the expression $g ( t ) = 45 - \cos ( t - 5 )$ By a sine function.

Recall that the average of a function $f ( x )$ on an interval $[ a , b ]$ is $\bar { f } = \frac { 1 } { b - a } \int _ { a } ^ { b } f ( x ) \mathrm { d } x$ Calculate the 9-unit moving average of the function. $f ( x ) = \cos \left( \frac { \pi x } { 18 } \right)$

Find the derivative of the function. $y ( x ) = 7 \cos \left( e ^ { x } \right) + 9 e ^ { x } \cos x$

Evaluate the integral. $\int 6 \sec ( 3 x - 7 ) \mathrm { d } x$

Sketch the curves without any technological help. $f ( t ) = 2 \cos t$ ; $g ( t ) = 3.3 \cos ( 2 t )$

The cost of Dig-It brand snow shovels is given by $c ( t ) = 3 \sin ( 2 \pi ( t - 0.75 ) )$ Where t is time in years since January 1, 1997. How fast, in dollars per year, is the cost increasing on October 30, 1997

Find the derivative of the function. $w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)$

Evaluate the integral $\int _ { \frac { 1 } { \pi } } ^ { \frac { 3 } { \pi } } 9 \frac { \sin \left( \frac { 1 } { x } \right) } { x ^ { 2 } } \mathrm {~d} x$

Evaluate the integral. $\int - \sin ( - 7 x - 8 ) \mathrm { d } x$

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Exam 16: Trigonometric Models

Evaluate the integral. ∫(x3+x4)sec⁡2(5x4+4x5)dx\int \left( x ^ { 3 } + x ^ { 4 } \right) \sec ^ { 2 } \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) \mathrm { d } x∫(x3+x4)sec2(5x4+4x5)dx

Find the derivative of the function. p(x)=3+2sin⁡(π2(x−1))p ( x ) = 3 + 2 \sin \left( \frac { \pi } { 2 } ( x - 1 ) \right)p(x)=3+2sin(2π​(x−1))

Evaluate the integral. ∫−π3π4sin⁡x dx\int _ { - \frac { \pi } { 3 } } ^ { \frac { \pi } { 4 } } \sin x \mathrm {~d} x∫−3π​4π​​sinx dx

Find the derivative of the function. u(x)=cos⁡(x2−2x)u ( x ) = \cos \left( x ^ { 2 } - 2 x \right)u(x)=cos(x2−2x)

Calculate the derivative. ddx([ln⁡∣x∣][cot⁡(6x+5)])\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )dxd​([ln∣x∣][cot(6x+5)])

Model the curve with a cosine function. ​ ​ Note that the period of the curve is P=28P = 28P=28 , its range is [0,110][ 0,110 ][0,110] and the graph of the cosine function is shifted upward 55 units and shifted to the right 14 units. Write the model function as a function of x and π.

Evaluate the integral. ∫10.8cos⁡(6x−1)dx\int 10.8 \cos ( 6 x - 1 ) \mathrm { d } x∫10.8cos(6x−1)dx

Use geometry to compute the given integral. ∫−π6π62sin⁡xdx\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x∫−6π​6π​​2sinxdx

Decide whether the integral converges. If the integral converges, compute its value. ∫0+∞e−3xcos⁡(3x)dx\int _ { 0 } ^ { + \infty } e ^ { - 3 x } \cos ( 3 x ) \mathrm { d } x∫0+∞​e−3xcos(3x)dx

Use the conversion formula cos⁡x=sin⁡(π2−x)\cos x = \sin \left( \frac { \pi } { 2 } - x \right)cosx=sin(2π​−x) to replace the expression g(t)=45−cos⁡(t−5)g ( t ) = 45 - \cos ( t - 5 )g(t)=45−cos(t−5) By a sine function.

Find the derivative of the function. y(x)=7cos⁡(ex)+9excos⁡xy ( x ) = 7 \cos \left( e ^ { x } \right) + 9 e ^ { x } \cos xy(x)=7cos(ex)+9excosx

Evaluate the integral. ∫6sec⁡(3x−7)dx\int 6 \sec ( 3 x - 7 ) \mathrm { d } x∫6sec(3x−7)dx

Sketch the curves without any technological help. f(t)=2cos⁡tf ( t ) = 2 \cos tf(t)=2cost ; g(t)=3.3cos⁡(2t)g ( t ) = 3.3 \cos ( 2 t )g(t)=3.3cos(2t)

The cost of Dig-It brand snow shovels is given by c(t)=3sin⁡(2π(t−0.75))c ( t ) = 3 \sin ( 2 \pi ( t - 0.75 ) )c(t)=3sin(2π(t−0.75)) Where t is time in years since January 1, 1997. How fast, in dollars per year, is the cost increasing on October 30, 1997

Find the derivative of the function. w(x)=2sec⁡(x)⋅tan⁡(x2−2)w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)w(x)=2sec(x)⋅tan(x2−2)

Evaluate the integral ∫1π3π9sin⁡(1x)x2 dx\int _ { \frac { 1 } { \pi } } ^ { \frac { 3 } { \pi } } 9 \frac { \sin \left( \frac { 1 } { x } \right) } { x ^ { 2 } } \mathrm {~d} x∫π1​π3​​9x2sin(x1​)​ dx

Evaluate the integral. ​ ∫−sin⁡(−7x−8)dx\int - \sin ( - 7 x - 8 ) \mathrm { d } x∫−sin(−7x−8)dx

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Evaluate the integral. $\int \left( x ^ { 3 } + x ^ { 4 } \right) \sec ^ { 2 } \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) \mathrm { d } x$

Find the derivative of the function. $p ( x ) = 3 + 2 \sin \left( \frac { \pi } { 2 } ( x - 1 ) \right)$

Evaluate the integral. $\int _ { - \frac { \pi } { 3 } } ^ { \frac { \pi } { 4 } } \sin x \mathrm {~d} x$

Find the derivative of the function. $u ( x ) = \cos \left( x ^ { 2 } - 2 x \right)$

Calculate the derivative. $\frac { d } { d x } ( [ \ln | x | ] [ \cot ( 6 x + 5 ) ] )$

Model the curve with a cosine function. Note that the period of the curve is $P = 28$ , its range is $[ 0,110 ]$ and the graph of the cosine function is shifted upward 55 units and shifted to the right 14 units. Write the model function as a function of x and π.

Evaluate the integral. $\int 10.8 \cos ( 6 x - 1 ) \mathrm { d } x$

Use geometry to compute the given integral. $\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { + \infty } e ^ { - 3 x } \cos ( 3 x ) \mathrm { d } x$

Use the conversion formula $\cos x = \sin \left( \frac { \pi } { 2 } - x \right)$ to replace the expression $g ( t ) = 45 - \cos ( t - 5 )$ By a sine function.

Find the derivative of the function. $y ( x ) = 7 \cos \left( e ^ { x } \right) + 9 e ^ { x } \cos x$

Evaluate the integral. $\int 6 \sec ( 3 x - 7 ) \mathrm { d } x$

Sketch the curves without any technological help. $f ( t ) = 2 \cos t$ ; $g ( t ) = 3.3 \cos ( 2 t )$

The cost of Dig-It brand snow shovels is given by $c ( t ) = 3 \sin ( 2 \pi ( t - 0.75 ) )$ Where t is time in years since January 1, 1997. How fast, in dollars per year, is the cost increasing on October 30, 1997

Find the derivative of the function. $w ( x ) = 2 \sec ( x ) \cdot \tan \left( x ^ { 2 } - 2 \right)$

Evaluate the integral $\int _ { \frac { 1 } { \pi } } ^ { \frac { 3 } { \pi } } 9 \frac { \sin \left( \frac { 1 } { x } \right) } { x ^ { 2 } } \mathrm {~d} x$

Evaluate the integral. $\int - \sin ( - 7 x - 8 ) \mathrm { d } x$