Deck 9: Trigonometric Models

Question 1

Use the formula for to simplify the expression .

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Answer 1

Question 2

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 ( 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)
B)
C)
D)
E)

Answer 2

Question 3

The depth of water at my favorite surfing spot varies from 9 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.

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E)

Answer 3

Question 4

Use the addition formulas: sin ( x + y )= sin x cos y + cos x sin y sin ( x - y )= sin x cos y - cos x sin y cos ( x + y )= cos x cos y - sin x sin y cos ( x - y )= cos x cos y + sin x sin y to express in terms of .

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Question 5

Model the curve with a cosine function. Note that the period of the curve is , its range is the graph of the cosine function is shifted upward 45 units and shifted to the right 19 units.

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Question 6

Starting with the identity , choose the right trigonometric identity.

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Question 7

Use the conversion formula to replace the expression by a sine function.

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Question 8

Sketch the curves without any technological help. ;

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Question 9

Model the curve with a sine function. Note that the period of the curve is and its range is , the graph of the sine function is shifted to the right 3 units.

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Question 10

The uninflated cost of Dugout brand snow shovels currently varies from a high of $22 on January 1 ( t = 0 )to a low of $8 on July 1 ( t = 0.5). Assuming this trend were to continue indefinitely, calculate the uninflated cost of Dugout snow shovels as a function of time t in years. (Use a sine function.)

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E)

Question 11

Sketch the curves without any technological help. ;

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E)

Question 12

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 to the right 0.7 units.

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Question 13

Model the curve with a sine function. Note that the period of the curve is its range is 2.4, 2.4 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 .

Question 14

Use the conversion formula to replace the expression by a sine function.

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E)

Question 15

Model the curve with a sine function. Note that the period of the curve is and its range is .

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Question 16

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

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Question 17

Model the curve with a sine function. Note that the period of the curve is and its range is 2.2, 2.2 and the graph of the sine function is shifted to the left 0.9 units.

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E)

Question 18

Use the conversion formula to replace the expression by a sine function.

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B)
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D)
E)

Question 19

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

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D)
E)

Question 20

Use the addition formulas : to calculate , given that and .

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Question 21

Find the derivative of the function.

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Question 22

Find the derivative of the function.

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Question 23

Find the derivative of the function.

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Question 24

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

Question 25

Find the derivative of the function.

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E)

Question 26

Find the derivative of the function.

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E)

Question 27

Find the derivative of the function.

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D)
E)

Question 28

Calculate the derivative.

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E)

Question 29

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 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

Question 30

Find the derivative of the function.

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B)
C)
D)
E)

Question 31

Find the derivative of the function.

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B)
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D)
E)

Question 32

Find the derivative of the function.

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B)
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D)
E)

Question 33

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 65 units and shifted to the right 9 units. Write the model function as a function of (x)and .

Question 34

Find the derivative of the function.

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D)
E)

Question 35

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 .

Question 36

Find the derivative of the function.

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B)
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D)
E)

Question 37

Find the derivative of the function.

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B)
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D)
E)

Question 38

Find the derivative of the function.

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B)
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D)
E)

Question 39

Find the derivative of the function.

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D)
E)

Question 40

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 41

Evaluate the integral.

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E)

Question 42

Evaluate the integral.

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Question 43

Evaluate the integral.

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E)none of these

Question 44

Recall that the average of a function on an interval is Calculate the 2-unit moving average of the function.

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Question 45

The cost of Dig-It brand snow shovels is given by 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)$21.99 per year
B)$14 per year
C)$45.98 per year
D)$46.98 per year
E)$43.98 per year

Question 46

Recall that the average of a function on an interval is Find the average of the given function. over .

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Question 47

Use geometry to compute the given integral.

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E)none of these

Question 48

Use geometry to compute the given integral.

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E)none of these

Question 49

Evaluate the integral.

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Question 50

Calculate the derivative.

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Question 51

Evaluate the integral

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Question 52

Evaluate the integral.

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Question 53

Calculate the derivative.

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Question 54

Evaluate the integral.

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Question 55

Calculate the derivative.

Question 56

Calculate the derivative.

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Question 57

Evaluate the integral.

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Question 58

Evaluate the integral.

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Question 59

Evaluate the integral.

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E)none of these

Question 60

Evaluate the integral.

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Question 62

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

Question 63

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 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.

A)TV = $1,400,000, FV = $745,106.09
B)TV = $0, FV = $207,020.55
C)TV = $0, FV = $277,680.69
D)TV = $0, FV = $54,779.56
E)none of these

Question 64

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

Question 65

Evaluate the integral.

Question 66

Decide whether the integral converges. If the integral converges, compute its value.

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