Quiz 1 practice, umich, math 115, fall 2020 PDF

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Math 115 — Quiz 1 Practice

Fall 2020

page 1

1. [4 points] Suppose that as an airplane gains altitude, the air temperature outside the plane decreases linearly as a function of the plane’s height above sea level. At a height of 8,000 feet, the temperature outside the plane is 30 ◦ F, and at a height of 14,000 feet, the temperature outside the plane is 9 ◦ F. Let T (h) be the temperature outside the plane (in ◦ F) at a height of h feet. Find a formula for T (h). 2. [8 points] The mass of a sample of a radioactive substance is decaying exponentially, and decreased by 12.6% between 2010 and 2015. a. [2 points] By what percent will the mass of the sample decrease between 2015 and 2020? Give your answer as a percentage accurate to two decimal places (like 1.23% or 34.56%). b. [3 points] By what percent does the mass of the sample decrease every 10 years? Give your answer as a percentage accurate to two decimal places (like 1.23% or 34.56%). c. [3 points] By what percent does the mass of the sample decrease each year? Give your answer as a percentage accurate to two decimal places (like 1.23% or 34.56%). 3. [4 points] After being mentioned on a popular news show, the number of views received by a video on the internet grows quickly. The number V of millions of views received by the video h hours after being mentioned on the show is modeled by the formula V = 0.65 · 1.013h−24 . a. [2 points] Which one of the following statements correctly explains the significance of the number 0.65 in the above formula? Write the letter of your answer in the blank below. A. B. C. D. E. F.

The video had 650,000 views when it was mentioned on the news show. The video had 650,000 views 24 hours after it was mentioned on the news show. The video had 650,000 views 24 hours before it was mentioned on the news show. The number of views is growing by 65% per hour. The number of views is growing by 0.65% per hour. The number of views is growing by 65% per day.

b. [2 points] Which one of the following statements correctly explains the significance of the number 1.013 in the above formula? Write the letter of your answer in the blank below. A. B. C. D. E. F.

The video had 1.013 million views when it was mentioned on the news show. The video had 1.013 million views 24 hours after it was mentioned on the news show. The video had 1.013 million views 24 hours before it was mentioned on the news show. The number of views is growing by 1.3% per hour. The number of views is growing by 1.013% per hour. The number of views is growing by 1.3% per day.

4. [8 points] The population of a country is growing at a constant percent rate, with a population of 42 million in 2002 and a population of 48 million in 2010. Write a formula for the population P (t) of the country, in millions of people, t years after the year 1990. 5. [14 points] Luna knows how much Link loves catnip mouse toys, so she has a small catnip garden outside to grow catnip for him. She wants to expand the size of her garden by planting catnip seeds. Let G be the new total size of the garden in square meters if she plants c grams of catnip seeds. Then G = h(c), where h is the function   15 + c h(c) = 3 . 10 Luna does not have space to make a garden larger than 40 square meters. a. [3 points] The function h is a linear function. What is the slope of h? Give a practical interpretation of the slope in the context of this problem. b. [3 points] What is the vertical intercept (G-intercept) of the graph of G = h(c)? Give a practical interpretation of the vertical intercept in the context of this problem. c. [4 points] Find a formula for h−1 (G). d. [4 points] In the context of this problem, what are the domain and range of the function h−1 (G)?

Math 115 — Quiz 1 Practice

Fall 2020

page 2

6. [12 points] Luna plants the new catnip seeds in her garden on August 6, and catnip begins to grow. She will use the catnip to make catnip mouse toys for her friend Link. Consider the following two functions. • Let a(t) be the number of ounces of catnip in the garden t weeks after August 6. • Let b(c) be the number of mouse toys Luna can make with c ounces of catnip. Assume that the functions a and b are invertible. a. [2 points] Give a practical interpretation of the equation b(a(6)) = 40. b. [2 points] Write an expression which equals the number of ounces of catnip needed to make 30 mouse toys. c. [3 points] Write an equation which expresses the following fact: On August 27 (three weeks after August 6), Luna only needs 4 more ounces of catnip to be able to make 20 mouse toys for Link. d. [2 points] Let p(t) be the number of pounds of catnip in the garden t weeks after August 6. Write a formula for p(t) in terms of the functions a, b, a−1 and/or b−1 . Note that there are 16 ounces in 1 pound. e. [3 points] Let f (x) be the number of ounces of catnip in the garden x days after August 20. Write a formula for f (x) in terms of the functions a, b, a−1 and/or b−1 . Note that there are seven days in a week and that August 20 is two weeks after August 6. 7. [10 points] Below is a graph of y = T (x) for some function T . The domain of T (x) is x ≥ −4, and the graph of T (x) has a horizontal asymptote at y = 1. y 4

y = T (x)

3 2 1

x −4 −3 −2 −1

1

2

3

4

In parts a–b below, the graph of the function is a transformation (a combination of stretches or compressions, shifts, and/or reflections) of the graph of y = T (x). Write a formula for each function in terms of the function T . b. [3 points] a. [3 points] y y 5

60

4

30

y = B(x) x

3 −4

2

y = A(x)

1

−30

2

4

−60 x

−2 −1

−2

1

2

3

4

A(x) =

5

−90

6

B(x) =

c. [4 points] Sketch a graph of y = T (2x − 6). Be sure the coordinates of any important points on your graph are clearly labeled.

Math 115 — Quiz 1 Practice

Fall 2020

page 3

8. [4 points] Let P (t) be the population (in thousands) of Pikipek on Akala Island t years after the start of the year 2000. Suppose that a formula for P (t) is P (t) = 5 · (1.04)2t−15 . a. [1 point] Approximately how many Pikipek were there on Akala Island at the start of the year 2000? b. [3 points] When will the Pikipek population reach 25,000? Leave your answer in exact form. 9. [9 points] Show all of your work on these problems, and leave your answers in exact form. a. [4 points] Solve the equation 2 ln(x) + 4 =3 ln(5x) for the variable x. b. [5 points] Solve the equation

50 · 1.05t = 200 · e−0.2t

for the variable t. 10. Below are the graphs of seven equations. Match the graphs to the equations by writing the correct letter in the blank after each formula. Note that a, b, c, and d are positive constants. The two plots below are not necessarily drawn to the same scale, and the x- and y-axes in each plot may have different scales. y

C

y A

D E F G

B x

• y = b + dx • y = b + cx • y = a + dx • y = a + cx • y=d·

 a x

• y=c·

 a x

• y=c·

 b x

b

b

a

x

Math 115 — Quiz 1 Practice

Fall 2020

page 4

11. Suppose that the function G(x) has domain (0, 10) and range [5, ∞). a. Determine the domain and range of the function H (x) defined by H (x) = G(x + 5) + 8. b. Determine the domain and range of the function J (x) defined by J (x) = 2G(−x). c. Determine the domain and range of the function K (x) defined by K (x) = −G(2(x − 1)). 12. Suppose that f (x) is an even function and that the point (20, 30) is on the graph of y = f (x). Which two points must be on the graph of y = 4f (x − 8)? 13. A graph of y = m(x) is shown on the right. Note that the domain of m(x) is x ≥ −4 and that the graph of y = m(x) contains the three points marked on the graph.

y y = m(x)

(0, 2) x

(−4, 0) (6, −2)

Each of the functions graphed below is a transformation of the function m. Find a formula for each function in terms of the function m. Note that the scale of each axis in the graphs below might be different from the scale of the axes in the graph above. a.

b. y

y y = p(x)

(1, 8)

(−9, 6) (4, 6)

(−5, 4) x y = q(x)

x (10, −6)

c.

d. y

y y = r(x)

y = s(x) (4, 2) (0, 2)

x

(−4, 0) (2, 0) (−3, −2)

x

(16, −2)

Math 115 — Quiz 1 Practice

Fall 2020

page 5

14. [9 points] Consider the function Q defined by Q(x) =

10 . 6 + ln x

a. [3 points] Determine the domain of the function Q(x). Give your answer using one or more inequalities or using interval notation. b. [4 points] The function Q is invertible. Find a formula for Q−1 (y). c. [2 points] Consider the function R defined by R(x) =

50 . 8 + ln x

Which of the following formulas correctly expresses the function R in terms of the function Q? Circle the one correct answer, or circle none of these. R(x) = 5Q(x + 2)

R(x) = 5Q(x) + 2

R(x) = 5Q(x + e2 )

R(x) = 5Q(e2 x)

R(x) =

5 2 + Q(x)

R(x) = 25 + 5Q(x)

R(x) = 5Q(x + ln 2)

R(x) =

15 Q(x) 4

none of these 15. [12 points] Some mischievous fish decide to stop the filter in their tank so that algae will grow. After a few attempts, they manage to lodge a small rock in the filter. 20 hours after the filter stops, there are 120 grams of algae in the tank. 5 hours later, there are 138 grams of algae. If you use decimal approximations on this page, be sure that your answers are accurate to at least 3 decimal places. a. [3 points] Let L(t) be the number of grams of algae in the tank t hours after the filter stops, assuming the amount of algae is growing linearly. Find a formula for L(t). b. [4 points] Let E(t) be the number of grams of algae in the tank t hours after the filter stops, assuming the amount of algae is growing exponentially. Find a formula for E(t). c. [5 points] A smart fish who has lived in the tank for a long time believes that the amount of algae will grow according to a logistic model, with the amount of algae in the tank approaching a maximum of 600 grams. Suppose that the amount of algae in the tank is given by F (t) =

600 1 + be−c(t−20)

for some constants b and c, where t is the number of hours since the filter stopped. Find the value of the constants b and c.

Math 115 — Quiz 1 Practice

Fall 2020

page 6

16. [15 points] Luna owns a food truck that sells tuna tacos for cats. Luna is the only cat working in the truck today, so she can barely keep up with the orders! Luna starts working at 11:00 AM. Let f (t) be the number of tacos that Luna has made after working for t minutes, and let p(x) be Luna’s profit (in dollars) from selling x tacos. Assume for this problem that the functions f and p are invertible. Your answers on this page might involve the functions f , p, f −1 , and/or p−1 . a. [2 points] How many tacos has Luna made after 35 minutes? b. [2 points] How many minutes does Luna have to work to make a profit of $250? c. [2 points] Write an inequality which expresses the following fact: At 11:45, Luna has made more than twice as many tacos as she had made by 11:30. d. [2 points] How many tacos does Luna need to sell to make twice as much profit as she makes from selling 100 tacos?   e. [3 points] In the equation f −1 p−1 (400) + 50 = 220, what are the units on the numbers 400, 50, and 220? f. [2 points] Let g(h) be the number of tacos Luna has made after working for h hours. Find a formula for g(h) in terms of the function f . Recall that there are 60 minutes in 1 hour. g. [2 points] Let j(t) be the total number of tacos Luna has made t minutes after 11:30 AM. Find a formula for j(t) in terms of the function f . 17. [5 points] Below is the graph of w = h(z) for some piecewise-linear function h. The domain of h(z) is −7 ≤ z ≤ 4. Evaluate each quantity. Your answers should be exact. w a. [2 points] h(0) = 4 b. [1 point] h−1 (3) = 2 z −6

−4

2

−2

4

−2 −4

h(z)

c. [2 points] h(2h−1 (−1)) =...