Deck 16: Trigonometric Models

Question 1

Evaluate the integral. $\int ( 10 x + 25 ) \sin \left( x ^ { 2 } + 5 x \right) \mathrm { d } x$

A) $- 5 x \sin \left( x ^ { 2 } + 5 x \right) + C$
B) $5 \cos \left( x ^ { 2 } + 5 x \right) + C$
C) $5 x \cos \left( x ^ { 2 } + 5 x \right) + C$
D) $- 5 \cos \left( x ^ { 2 } + 5 x \right) + C$
E) $5 \sin \left( x ^ { 2 } + 5 x \right) + C$

Answer 1

$- 5 \cos \left( x ^ { 2 } + 5 x \right) + C$

Question 2

Recall that the average of a function $f ( x )$ on an interval $[ a , b ]$ is $\bar { f } = \frac { 1 } { b - a } \int _ { a } ^ { b } f ( x ) \mathrm { d } x$
Find the average of the given function.
$f ( x ) = \sin ( 5 x )$ over $[ 0,5 \pi ]$

A)Average = $\frac { 2 } { 25 \pi }$
B) Average = $\frac { 25 \pi } { 2 }$
C) Average = $\frac { \pi } { 5 }$
D) Average = $\frac { 25 \pi } { 4 }$
E) Average = $\frac { 5 } { \pi }$

Answer 2

Average = $\frac { 2 } { 25 \pi }$

Question 3

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

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

Answer 3

$\int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 6 } } 2 \sin x d x = 0$

Question 4

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

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

Question 5

Evaluate the integral. ? $\int ( 8 \cos x - 4.1 \sin x - 9.3 ) \mathrm { d } x$ ?

A) $8 \sin x + 4.1 \cos x - 9.3 x + C$
B) $- 8 \sin x + 4.1 \cos x + C$
C) $8 \sin x - 4.1 \cos x - 9.3 x + C$
D) $- 8 \sin x - 4.1 \cos x + C$
E) $8 \sin x - 4.1 \cos x + 9.3 x + C$

Question 6

Recall that the total income received from time $t = a$ to time $t = b$ from a continuous income stream of $R ( t )$ dollars per year is
Total value = TV = $\int _ { a } ^ { b } R ( t ) \mathrm { d } t$
Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate.
$R ( t ) = 800,000 \sin ( 2 \pi t )$ , $0 \leq t \leq 5$ , at 9%

A)TV = $0, FV = $72,344.91
B) TV = $0, FV = $327,074.77
C) TV = $1,600,000, FV = $834,722.23
D) TV = $0, FV = $256,372.45
E) none of these

Question 7

Evaluate the integral. $\int \left( 3.4 \sec ^ { 2 } x + \frac { \cos x } { 1.4 } - 3.6 e ^ { x } \right) \mathrm { d } x$

A) $3.4 \tan x + \frac { \sin x } { 1.4 } - 3.6 e ^ { x } + C$
B) $\frac { 3.4 } { \tan x } + \frac { \sin x } { 1.4 } - 3.6 e ^ { x } + C$
C) $3.4 \tan x - \frac { \sin x } { 1.4 } - 3.6 e ^ { x } + C$
D) $- \frac { 3.4 } { \tan x } - \frac { \sin x } { 1.4 } - 3.6 e ^ { x } + C$
E) $- \frac { 3.4 } { \tan x } + \frac { \sin x } { 1.4 } - 3.6 e ^ { x } + C$

Question 8

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

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) $20 \tan \left( 5 x ^ { 4 } + 4 x ^ { 5 } \right) + C$

Question 9

Evaluate the integral.
​

Question 10

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

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

Question 11

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

A) $\ln \left| \sec \left( \frac { \pi } { 4 } \right) \right| - \ln \left| \sec \left( - \frac { \pi } { 4 } \right) \right|$
B) $\ln \left| \sin \left( \frac { \pi } { 4 } \right) \right| - \ln \left| \sin \left( \frac { \pi } { 4 } \right) \right|$
C) $\sin \left( - \frac { \pi } { 4 } \right) - \sin \left( \frac { \pi } { 4 } \right)$
D) $\cos \left( \frac { \pi } { 4 } \right) - \cos \left( - \frac { \pi } { 4 } \right)$
E) none of these

Question 12

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.
$f ( x ) = \cos \left( \frac { \pi x } { 18 } \right)$

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 13

Evaluate the integral. $\int - \sin ( - 4 x - 7 ) \mathrm { d } x$

A) $\frac { \cos ( 4 x + 7 ) } { 4 } + C$
B) $4 \cos ( 4 x - 7 ) + C$
C) $- \frac { \cos ( 4 x + 7 ) } { 4 } + C$
D) $- 4 \cos ( 4 x + 7 ) + C$
E) $- \frac { \cos ( 4 x - 7 ) } { 4 } + C$

Question 14

Evaluate the integral.
​ ​
Use the symbol C to write the constant.

Question 15

Evaluate the integral.
​ ​
Use the symbol C to write the constant.

Question 16

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

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

Question 17

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

A) $2 \ln | \tan ( 3 x - 7 ) | + C$
B) $2 \ln | \sec ( 3 x - 7 ) + \tan ( 3 x - 7 ) | + C$
C) $6 \ln | \sec ( 3 x - 7 ) + \tan ( 3 x - 7 ) | + C$
D) $\ln | \sec ( 3 x - 7 ) + \tan ( 3 x - 7 ) | + C$
E) $2 \ln | \sec ( 3 x - 7 ) | + C$

Question 18

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

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 19

Evaluate the integral. $\int _ { 1 } ^ { \sqrt { \frac { \pi } { 2 } + 1 } } 4 x \cos \left( x ^ { 2 } - 1 \right) \mathrm { d } x$

A)2
B) 10
C) 1
D) 6
E) 4

Question 20

Use geometry to compute the given integral. $\int _ { 0 } ^ { 5 \pi } 6 \cos x d x$

A) $\int _ { 0 } ^ { 5 \pi } 6 \cos x d x = 5$
B) $\int _ { 0 } ^ { 5 \pi } 6 \cos x d x = 12$
C) $\int _ { 0 } ^ { 5 \pi } 6 \cos x d x = 1$
D) $\int _ { 0 } ^ { 5 \pi } 6 \cos x d x = 0$
E) none of these

Question 21

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

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 22

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

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 23

Calculate the derivative. $\frac { d } { d x } \left[ \sec \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \right]$

A) $\frac { 6 x ^ { 4 } - 18 x ^ { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } \csc \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \tan \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right)$
B) $\frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \csc \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \tan \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right)$
C) $\frac { 6 x ^ { 4 } - 18 x ^ { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } \sec \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \tan \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right)$
D) $\frac { 6 x ^ { 3 } - 18 x ^ { 2 } } { x ^ { 2 } - 1 } \sec \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \tan \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right)$
E) $\frac { 6 x ^ { 4 } - 18 x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } \sec \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } - 1 } \right) \tan \left( \frac { 6 x ^ { 3 } } { x ^ { 2 } + 1 } \right)$

Question 24

Find the derivative of the function. $t ( x ) = \frac { \cot x } { 4 + 8 \sec x }$

A) $t ^ { \prime } ( x ) = - \frac { ( 4 - 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 - 8 \sec x ) ^ { 2 } }$
B) $t ^ { \prime } ( x ) = - \frac { ( 4 + 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 + 8 \sec x ) ^ { 2 } }$
C) $t ^ { \prime } ( x ) = \frac { ( 4 + 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 + 8 \sec x ) ^ { 2 } }$
D) $t ^ { \prime } ( x ) = - \frac { ( 4 + 8 \sec x ) \csc ^ { 2 } x + 8 \sec x } { ( 4 - 8 \sec x ) ^ { 2 } }$
E) $t ^ { \prime } ( x ) = - \frac { ( 4 - 8 \sec x ) \csc ^ { 2 } x - 8 \sec x } { ( 4 - 8 \sec x ) ^ { 2 } }$

Question 25

Calculate the derivative. $\frac { d } { d x } [ \sin ( 7 x ) ] ^ { 0.5 }$

A) $3.5 \sin ( 7 x ) \cdot \cos ( 7 x )$
B) $3.5 ( \sin ( 7 x ) ) ^ { - 0.5 } \cdot \cos ( 7 x )$
C) $( \sin ( 7 x ) ) ^ { - 0.5 } \cdot \cos ( 7 x )$
D) $7 ( \sin ( 7 x ) ) ^ { - 0.5 } \cdot \cos ( 7 x )$
E) $7 ( \sin ( 7 x ) ) ^ { 0.5 } \cdot \cos ( 7 x )$

Question 26

Find the derivative of the function. $g ( x ) = 9 \sin ( x ) \cdot \tan x$

A) $g' ( x ) = 9 \cos ( x ) \cdot \tan x + 9 \sin ( x ) \cdot \csc ^ { 2 } x$
B) $g '( x ) = 9 \cos ( x ) \cdot \tan x + 9 \sin ( x ) \cdot \sec ^ { 2 } x$
C) $g '( x ) = 9 \cos ( x ) \cdot \tan x - 9 \sin ( x ) \cdot \cot x$
D) $g' ( x ) = 9 \cos ( x ) \cdot \tan x + 9 \cos ( x ) \cdot \cot x$
E) $g '( x ) = 9 \cos ( x ) \cdot \tan x - 9 \sin ( x ) \cdot \sec ^ { 2 } x$

Question 27

Calculate the derivative. $\frac { d } { d x } \left[ e ^ { - 3 x } \sin ( 3 \pi x ) \right]$

A) $e ^ { 3 x } ( - 3 \sin ( 3 \pi x ) - 3 \pi \cos ( 3 \pi x ) )$
B) $e ^ { - 3 x } ( - 3 \sin ( 3 \pi x ) + 3 \pi \cos ( 3 \pi x ) )$
C) $e ^ { - 3 x } ( - 3 \sin ( 3 \pi x ) - 3 \pi \cos ( 3 \pi x ) )$
D) $e ^ { - 3 x } ( \sin ( 3 \pi x ) + \cos ( 3 \pi x ) )$
E) $e ^ { - 3 x } ( \sin ( 3 \pi x ) - 3 \pi \cos ( 3 \pi x ) )$

Question 28

Find the derivative of the function. $u ( x ) = \sec \left( x ^ { 5.2 } + 2.8 x - 9 \right)$

A) $u ^ { \prime } ( x ) = \left( 5.2 x ^ { 4.2 } + 2.8 \right) \sec \left( x ^ { 5.2 } - 2.8 x - 9 \right) \tan \left( x ^ { 5.2 } - 2.8 x - 9 \right)$
B) $u ^ { \prime } ( x ) = \left( 5.2 x ^ { 4.2 } - 2.8 \right) \sec \left( x ^ { 5.2 } + 2.8 x - 9 \right) \tan \left( x ^ { 5.2 } + 2.8 x - 9 \right)$
C) $u ^ { \prime } ( x ) = \left( 5.2 x ^ { 4.2 } - 2.8 \right) \sec \left( x ^ { 5.2 } - 2.8 x - 9 \right) \tan \left( x ^ { 5.2 } + 2.8 x - 9 \right)$
D) $u ^ { \prime } ( x ) = \left( 5.2 x ^ { 4.2 } + 2.8 \right) \sec \left( x ^ { 5.2 } + 2.8 x - 9 \right) \tan \left( x ^ { 5.2 } + 2.8 x - 9 \right)$
E) $u ^ { \prime } ( x ) = \left( 5.2 x ^ { 4.2 } - 2.8 \right) \sec \left( x ^ { 5.2 } - 2.8 x - 9 \right) \tan \left( x ^ { 5.2 } - 2.8 x - 9 \right)$

Question 29

Find the derivative of the function. $s ( x ) = 2 \cos ^ { 2 } x$

A) $s '( x ) = - 4 \cos ( x ) \cdot \cos x$
B) $s '( x ) = - 4 \cos ( x ) \cdot \sin x$
C) $s ^ { \prime } ( x ) = - 2 \cos ^ { 2 } ( x ) \cdot \sin x$
D) $s' ( x ) = - 2 \cos ( x ) \cdot \sin x$
E) $s' ( x ) = - 4 \csc x$

Question 30

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

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 31

Find the derivative of the function. $z ( x ) = 3 \ln \mid 4 ( \sec x + \tan x )$

A) $z ( x ) = 9 \sec x$
B) $z ^ { \prime } ( x ) = \frac { 3 \sec x } { 4 }$
C) $z ^ { \prime } ( x ) = 3 \left( 1 + \tan ^ { 2 } x \right)$
D) $z ^ { \prime } ( x ) = \sec x$
E) $z ( x ) = 3 \sec x$

Question 32

Find the derivative of the function. $r ( x ) = 4 x \cos x + 9 x ^ { 2 } + 8$

A) $r'( x ) = - 4 x \cos x + 4 \cos x + 18 x$
B) $r' ( x ) = - 4 x \sin x + 4 \cos x + 18 x$
C) $r '( x ) = - 4 x \sin x + 4 \cos x - 18 x$
D) $r '( x ) = - 4 x \sin x + 4 \sin x + 18 x$
E) $r '( x ) = 4 x \sin x - 4 \cos x + 18 x$

Question 33

Find the derivative of the function. $f ( x ) = 9 \sin x - 8 \cos x$

A) $f ^ { \prime } ( x ) = 9 \cos x + 8 \sin x$
B) $f ^ { \prime } ( x ) = 8 \cos x - 9 \sin x$
C) $f ^ { \prime } ( x ) = 9 \cos x - 8 \sin x$
D) $f ^ { \prime } ( x ) = - 9 \cos x - 8 \sin x$
E) $f ^ { \prime } ( x ) = 8 \cos x + 9 \sin x$

Question 34

Decide whether each integral converges. If the integral converges, compute its value.
Choose the correct letter for each question.


-converges to $\frac { 1 } { 4 }$

A) $\int _ { 0 } ^ { + \infty } \sin ( 2 x ) \mathrm { d } x$
B) $\int _ { 0 } ^ { + \infty } e ^ { - 2 x } \sin ( 2 x ) \mathrm { d } x$

Question 35

Decide whether each integral converges. If the integral converges, compute its value.
Choose the correct letter for each question.


-diverges

A) $\int _ { 0 } ^ { + \infty } \sin ( 2 x ) \mathrm { d } x$
B) $\int _ { 0 } ^ { + \infty } e ^ { - 2 x } \sin ( 2 x ) \mathrm { d } x$

Question 36

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

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 37

Find the derivative of the function. $j ( x ) = 7 \sec ^ { 2 } x$

A) $j '( x ) = 14 \sec ^ { 2 } x \cdot \tan x$
B) $j '( x ) = 7 \sec ^ { 2 } x \cdot \tan x$
C) $j '( x ) = 14 \sec ^ { 2 } x \cdot \tan ^ { 2 } x$
D) $j ^ { \prime } ( x ) = - 14 \sec ^ { 2 } x \cdot \tan x$
E) $j '( x ) = 14 \sec x \cdot \tan x$

Question 38

Find the derivative of the function. $s ( x ) = \left( 4 x ^ { 2 } - 2 x + 2 \right) \tan x$

A) $s ^ { \prime } ( x ) = ( 8 x - 2 ) \tan x + \left( 4 x ^ { 2 } - 2 x - 2 \right) \sec ^ { 2 } x$
B) $s ^ { \prime } ( x ) = ( 8 x - 2 ) \tan x + \left( 4 x ^ { 2 } - 2 x + 2 \right) \sec x$
C) $s ^ { \prime } ( x ) = ( 8 x - 2 ) \tan x + \left( 4 x ^ { 2 } + 2 x + 2 \right) \sec ^ { 2 } x$
D) $s ^ { \prime } ( x ) = ( 8 x - 2 ) \tan x + \left( 4 x ^ { 2 } - 2 x + 2 \right) \sec ^ { 2 } x$
E) $s ^ { \prime } ( x ) = ( 8 x - 2 ) \tan x - \left( 4 x ^ { 2 } - 2 x + 2 \right) \sec ^ { 2 } x$

Question 39

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

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) $p ^ { \prime } ( x ) = 3 + \pi \cos \left( \frac { \pi } { 2 } ( x - 1 ) \right)$

Question 40

Find the derivative of the function. $y ( x ) = \sec \left( e ^ { 4 x } \right)$

A) $y ^ { \prime } ( x ) = 4 e ^ { 4 x } \csc \left( e ^ { 4 x } \right) \tan ^ { 2 } \left( e ^ { 4 x } \right)$
B) $y ^ { \prime } ( x ) = 16 e ^ { 4 x } \sec \left( e ^ { 4 x } \right) \tan \left( e ^ { 4 x } \right)$
C) $y ^ { \prime } ( x ) = 4 e ^ { 4 x } \sec \left( e ^ { 4 x } \right) \tan \left( e ^ { 4 x } \right)$
D) $y ^ { \prime } ( x ) = 4 e ^ { 4 x } \csc ^ { 2 } \left( e ^ { 4 x } \right) \tan ^ { 2 } \left( e ^ { 4 x } \right)$
E) $y ^ { \prime } ( x ) = 4 e ^ { 4 x } \csc \left( e ^ { 4 x } \right) \tan \left( e ^ { 4 x } \right)$

Question 41

Model the curve with a cosine function.
Note that the period of the curve is $P = \frac { 1 } { 6 }$ and its range is $[ - 1,1 ]$ .

A) $f ( x ) = \cos ( 12 x )$
B) $f ( x ) = \cos \left( \frac { \pi x } { 12 } \right)$
C) $f ( x ) = \cos ( 12 \pi x )$
D) $f ( x ) = 12 \cos ( \pi x )$
E) $f ( x ) = \cos \left( \frac { x } { 12 } \right)$

Question 42

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

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 43

Starting with the identity $\sin ^ { 2 } x + \cos ^ { 2 } x = 1$ , choose the right trigonometric identity.

A) $\sec ^ { 2 } x = 1 + \tan ^ { 2 } x$
B) $\sec ^ { 2 } x = 1 - \tan ^ { 2 } x$
C) $\cot ^ { 2 } x = 1 - \tan ^ { 2 } x$
D) $\sin ^ { 2 } x = 1 + \tan ^ { 2 } x$
E) $\sec ^ { 2 } x = 1 + \csc ^ { 2 } x$

Question 44

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

A)
B)
C)
D)
E)

Question 45

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

A)
B)
C)
D)
E)

Question 46

Sales of computers are subject to seasonal fluctuations. Computer City's sales of computers in 1995 and 1996 can be approximated by the function $s ( t ) = 0.468 \sin ( 1.25 t + 1.73 ) + 0.79 ( 1 \leq t \leq 6 )$
Where t is time in quarters ( $t = 1$ represents the end of the first quarter of 1995) and $s ( t )$ is computer sales (quarterly revenue) in billions of dollars. Estimate Computer City's maximum and minimum quarterly revenue from computer sales.

A) $s _ { \max } = 1.526$ , $s _ { \min } = 0.074$
B) $s _ { \max } = 0.79$ , $s _ { \min } = 0.468$
C) $s _ { \max } = 2.048$ , $s _ { \min } = 1.112$
D) $s _ { \max } = 2.194$ , $s _ { \min } = 0.146$
E) $s _ { \max } = 1.258$ , $s _ { \min } = 0.322$

Question 47

Model the curve with a sine function.
Note that the period of the curve is $P = 0.4$ and its range is $[ - 3 , - 1 ]$ .

A) $f ( x ) = 2 - \sin x$
B) $f ( x ) = - 2 + 5 \sin x$
C) $f ( x ) = - 2 + \sin ( 5 \pi x )$
D) $f ( x ) = 2 - \sin ( 5 \pi x )$
E) $f ( x ) = - 2 + \sin ( \pi x )$

Question 48

Use the addition formulas: $$\begin{array} { l } \sin ( x + y ) = \sin x \cdot \cos y + \cos x \cdot \sin y \\\sin ( x - y ) = \sin x \cdot \cos y - \cos x \cdot \sin y \\\cos ( x + y ) = \cos x \cdot \cos y - \sin x \cdot \sin y \\\cos ( x - y ) = \cos x \cdot \cos y + \sin x \cdot \sin y\end{array}$$
To express $\tan ( x + 23 \pi )$ in terms of $\tan ( x )$ .

A) $\frac { \tan ( x ) } { 23 \pi }$
B) $\tan ( x ) + 23 \pi$
C) $23 \pi \tan ( x )$
D) $\tan ( x )$
E) $\tan ( x ) - 23 \pi$

Question 49

Use the formula for $\sin ( x + y )$ to simplify the expression $\sin \left( t + \frac { 17 \pi } { 2 } \right)$ .

A) $\cos t - 17 \pi$
B) $\cos t + 17 \pi$
C) $\frac { \cos t } { 17 \pi }$
D) $17 \pi \cos t$
E) $\cos t$

Question 50

Use the addition formulas: $$\begin{array} { l } \sin ( x + y ) = \sin x \cdot \cos y + \cos x \cdot \sin y \\\sin ( x - y ) = \sin x \cdot \cos y - \cos x \cdot \sin y \\\cos ( x + y ) = \cos x \cdot \cos y - \sin x \cdot \sin y \\\cos ( x - y ) = \cos x \cdot \cos y + \sin x \cdot \sin y\end{array}$$
To calculate $\cos \left( \frac { \pi } { 3 } \right)$ , given that $\sin \left( \frac { \pi } { 6 } \right) = \frac { 1 } { 2 }$ and $\cos \left( \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 }$ .

A) $\cos \left( \frac { \pi } { 3 } \right) = 0$
B) $\cos \left( \frac { \pi } { 3 } \right) = \frac { \sqrt { 3 } } { 2 }$
C) $\cos \left( \frac { \pi } { 3 } \right) = \frac { 1 } { 2 }$
D) $\cos \left( \frac { \pi } { 3 } \right) = - \frac { \sqrt { 3 } } { 2 }$
E) $\cos \left( \frac { \pi } { 3 } \right) = - \frac { 1 } { 2 }$

Question 51

Use the conversion formula $\cos x = \sin \left( \frac { \pi } { 2 } - x \right)$ to replace the expression $g ( x ) = 25 \cos [ 2 \pi ( 3 x - 1 ) ] + 9$
By a sine function.

A) $g ( x ) = 25 \sin \left( 6 \pi x + \frac { 5 \pi } { 2 } \right) - 9 + \frac { \pi } { 2 }$
B) $g ( x ) = 25 \sin \left( 6 \pi x - \frac { 5 \pi } { 2 } \right) + 9$
C) $g ( x ) = 25 \sin \left( - 6 \pi x - \frac { 5 \pi } { 2 } \right) + 9$
D) $g ( x ) = 25 \sin \left( - 6 \pi x + \frac { 5 \pi } { 2 } \right) + 9$
E) $g ( x ) = 25 \sin \left( 12 \pi x + \frac { 3 \pi } { 2 } \right) + 9$

Question 52

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.

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 53

Model the curve with a cosine function.
Note that the period of the curve is $P = 14$ , its range is $[ 0,120 ]$ the graph of the cosine function is shifted upward 60 units and shifted to the right 7 units.

A) $f ( x ) = 120 \cos \left( \frac { \pi ( x - 60 ) } { 60 } \right) + 7$
B) $f ( x ) = 120 \cos \left( \frac { \pi ( x - 60 ) } { 60 } \right) - 7$
C) $f ( x ) = 60 \cos \left( \frac { \pi ( x + 7 ) } { 7 } \right) + 60$
D) $f ( x ) = 7 \cos \left( \frac { \pi ( x - 60 ) } { 60 } \right) + 7$
E) $f ( x ) = 60 \cos \left( \frac { \pi ( x - 7 ) } { 7 } \right) + 60$

Question 54

Use the conversion formula $\cos x = \sin \left( \frac { \pi } { 2 } - x \right)$ to replace the expression $f ( t ) = 5.2 \cos ( 6 \pi t ) + 10$
By a sine function.

A) $f ( t ) = 5.2 \sin \left( \frac { \pi } { 2 } - 6 \pi t \right) + 10$
B) $f ( t ) = 5.2 \sin \left( \frac { \pi - 6 \pi t } { 2 } \right) + 10$
C) $f ( t ) = 5.2 \sin \left( \frac { \pi } { 2 } - 6 t \right) + 10$
D) $f ( t ) = 6 \sin \left( \frac { \pi } { 2 } - 5.2 \pi t \right) + 10$
E) $f ( t ) = 10 \sin \left( \frac { \pi } { 2 } - 6 \pi t \right) + 5.2$

Question 55

Model the curve with a sine function.
Note that the period of the curve is $P = 32$ and its range is $[ - 40,0 ]$ , the graph of the sine function is shifted to the right 7 units.

A) $f ( x ) = 20 \sin \left( \frac { \pi ( x + 7 ) } { 16 } \right) + 20$
B) $f ( x ) = - 20 \sin \left( \frac { \pi ( x + 7 ) } { 16 } \right) + 20$
C) $f ( x ) = 40 \sin \left( \frac { \pi ( x - 7 ) } { 16 } \right) - 40$
D) $f ( x ) = 40 \sin \left( \frac { \pi ( x + 7 ) } { 16 } \right) - 40$
E) $f ( x ) = 20 \sin \left( \frac { \pi ( x - 7 ) } { 16 } \right) - 20$

Question 56

Model the curve with a sine function.

Note that the period of the curve is $P = \frac { 1 } { 5 }$ and its range is $[ - 2.2,2.2 ]$ and the graph of the sine function is shifted to the left 0.55 units.

A) $f ( x ) = 2.2 \sin ( 10 \pi ( x + 0.55 ) )$
B) $f ( x ) = 2.2 \sin ( 10 \pi ( x - 0.55 ) )$
C) $f ( x ) = 2.2 \sin ( 10 \pi x + 0.55 )$
D) $f ( x ) = 2.2 \sin ( 10 \pi ( 2 x + 0.55 ) )$
E) $f ( x ) = 4.4 \sin ( 5 \pi ( x + 0.55 ) )$

Question 57

Calculate the derivative.
​

Question 58

The depth of water $d ( t )$ at my favorite surfing spot varies from 8 to 20 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 the depth of water as a function of time t in hours since midnight on Sunday morning.

A) $d ( t ) = 10 \sin \left( \frac { 2 \pi ( t - 1.625 ) } { 13.5 } \right) + 4$
B) $d ( t ) = 6 \sin \left( \frac { 2 \pi ( t + 1.625 ) } { 11.5 } \right) - 14$
C) $d ( t ) = - 6 \sin \left( \frac { - 2 \pi ( t + 1.625 ) } { 13.5 } \right) + 14$
D) $d ( t ) = 6 \sin \left( \frac { 2 \pi ( t - 1.625 ) } { 13.5 } \right) + 14$
E) $d ( t ) = 14 \sin \left( \frac { 2 \pi ( t - 1.625 ) } { 13.5 } \right) + 6$

Question 59

Model the curve with a cosine function.
Note that the period of the curve is $P = \frac { 1 } { 5 }$ , its range is $[ - 3.3,3.3 ]$ and the graph of the cosine function is shifted to the right 0.35 units.

A) $f ( x ) = 6.6 \cos ( 20 \pi ( 2 x - 0.35 ) )$
B) $f ( x ) = 3.3 \cos ( 10 ( x - 0.35 ) )$
C) $f ( x ) = 6.6 \cos ( 20 \pi ( 2 x + 0.35 ) )$
D) $f ( x ) = 3.3 \cos ( 10 \pi ( x - 0.35 ) )$
E) $f ( x ) = 3.3 \cos ( 10 \pi ( x + 0.35 ) )$

Question 60

Model the curve with a cosine function.
Note that the period of the curve is $P = \frac { 1 } { 3 }$ and its range is $[ - 1,1 ]$ .

A) $f ( x ) = \cos ( 6 x )$
B) $f ( x ) = \cos ( 6 \pi x )$
C) $f ( x ) = \cos \left( \frac { x } { 6 } \right)$
D) $f ( x ) = \cos \left( \frac { \pi x } { 6 } \right)$
E) $f ( x ) = 6 \cos ( \pi x )$

Question 61

Starting with the identity and then dividing both sides of the equation by a suitable trigonometric function, derive the trigonometric identity.
​

Question 62

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.)

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 63

The depth of water 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.

Question 64

Model the curve with a sine function.
​ ​
Note that the period of the curve is , its range is and the graph of the sine function is shifted to the left 0.9 units. Write the model function as a function of x and π.

Question 65

Model the curve with a cosine function.
​ ​
Note that the period of the curve is , its range is 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.

Question 66

Model the curve with a sine function.
​
​ ​
Note that the period of the curve is and its range is . Write the model function as a function of x and π.

Question 67

Sales of computers are subject to seasonal fluctuations. Computer City's sales of computers in 1995 and 1996 can be approximated by the function
​ ​
where ​t is time in quarters ( represents the end of the first quarter of 1995) and is computer sales (quarterly revenue) in billions of dollars. Estimate Computer City's maximum and minimum quarterly revenue from computer sales.
​
Maximum sales __________ billions of dollars
​
Minimum sales __________ billions of dollars