Quiz # 5

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MATH 107 QUIZ 5 Nov 4, 2020 Instructor: Tim Elsner

NAME: _______________________________ Due: Tue Nov 10 by midnight I have completed this assignment myself, working independently and not consulting anyone except the instructor.

INSTRUCTIONS

• The quiz is worth 100 points, and there is an extra credit opportunity at the end. There are 10 problems. This quiz is open book and open notes. This means that you may refer to your textbook, notes, and online classroom materials, but you must work

independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish,

provided you turn in your quiz no later than Tuesday night midnight, Nov 10.

• Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also.

In your document, be sure to include your name and the assertion of independence of work.

• General quiz tips and instructions for submitting work are posted in the Quizzes module.

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1. (4 pts) Which of these graphs represent a one-to-one function? Answer(s): ____________ (no explanation required.) (There may be more than one graph that qualifies.) (A)

(B) (C)

(D)

2. (6 pts) Based on data about the growth of a variety of ornamental cherry trees, the following logarithmic

model about these trees was determined:

h(t) = 7 ln(t) + 10, where t = age of tree in years and h(t) = height of tree, in feet. (Note that "ln" refers to the natural log function) (explanation optional) Using the model,

(a) At age 4 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

(b) At age 20 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

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3. (4 pts) Convert to a logarithmic equation: 2x = 64. (no explanation required) 3. ______

A. log𝑥 64 = 2

B. log2 64 = 𝑥

C. log2 𝑥 = 64

D. log8 64 = 2

4. (8 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, your solution

√2x + 13 = x + 5

5. (8 pts)

(a) log7 1 =_______ (fill in the blank)

(b) Let 𝑥 = log4 64 Write the exponential form of the equation.

(c) Determine the numerical value of x = log4 64, in simplest form. Show work.

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6. (10 pts) Let f (x) = x2 + 5x + 8 and g(x) = x + 2

(a) Write the composite function (𝒇 𝒐 𝒈)(𝒙) and simplify the results. (don’t solve) Show work.

(b) Now find (𝒇 𝒐 𝒈)(−𝟐) . Show work.

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7. (16 pts) Let 𝑓(𝑥) = 5𝑥+3

𝑥− 2

(a) Find f − 1(x) , the inverse function of f. Show all work.

(b) What is the domain of f(x) in interval notation ?

What is the domain of the f -1(x) in interval notation

(c) What is f (1) ? ______ show work

(d) What is f − 1 (-8) ________ show work

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8. (18 pts) Let f (x) = e x - 1 + 3. (hint: use a graphing utility, such as desmos.com)

Answers can be stated without additional work/explanation.

(a) Which describes how the graph of f can be obtained from the graph of y = ex ? Choice: ________

A. Shift the graph of y = ex to the left by 1 unit and up by 3 units.

B. Shift the graph of y = ex to the right by 1 unit and up by 3 units.

C. Shift the graph of y = ex to the right by 3 units and up by 1 unit.

D. Shift the graph of y = ex to the left by 3 units and down by 1 unit.

(b) What is the domain of f in interval notation ?

(c) What is the range of f in interval notation ?

(d) What is the horizontal asymptote expressed as an equation?

(e) What is the y-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).

(f) Which is the graph of f ?

GRAPH A GRAPH B GRAPH C GRAPH D

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NONLINEAR MODELS - For the latter part of the quiz, we will explore some nonlinear models.

9. (16 pts) QUADRATIC REGRESSION

Data: On a particular summer day, the outdoor temperature was recorded at 8 times of the day, and the following table

was compiled. A scatterplot was produced and the quadratic (parabola) of best fit was determined.

t = Time of day (hour)

y = Outdoor Temperature (degrees F.)

7 52

9 67

11 73

13 76

14 78

17 79

20 76

23 61

Quadratic Polynomial of Best Fit:

y = − 0.35t2 + 11t − 6 where t = Time of day (hour) and y = Temperature (in degrees F)

REMARKS: The times are the hours since midnight. For example, t=7 means 7 am, and 13 means 1 pm.

(a) Using algebraic techniques we have learned, find the maximum temperature predicted by the quadratic model and

find the time when it occurred. Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45).

(For example, a time of 18.25 hours is reported as 6:15 pm.) Report the maximum temperature to the nearest tenth of a

degree. Show algebraic work. Be careful ! in translating the decimal hr to the hr and nearest 15 min. (see table)

decimal min

0.00 :00

0.25 :15

0.50 :30

0.75 :45

y = -0.35t2 + 11t - 6 R² = 0.97

0

10

20

30

40

50

60

70

80

90

0 4 8 12 16 20 24

T e

m p

e ra

tu re

( d

e g

re e

s)

Time of Day (hour)

Temperature on a Summer Day

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(b) Use the quadratic polynomial to estimate the outdoor temperature at 9:30 am, to the nearest tenth of a

degree. Show work.

(c) Use the quadratic polynomial y = − 0.35t2 + 11t − 6 together with algebra to estimate the time(s) of

day when the outdoor temperature y was 70 degrees. That is, solve the quadratic equation

y = 70 = − 0.35t2 + 11t − 6 .

Show algebraic work in solving. State your results clearly; report the time(s) to the nearest quarter hour.

10. (10 pts) + (extra credit at the end) EXPONENTIAL REGRESSION

Data: A cup of hot coffee was placed in a room maintained at a constant temperature of 69 degrees, and the

coffee temperature was recorded periodically, in Table 1.

TABLE 1

t = Time Elapsed (minutes)

C = Coffee Temperature (degrees F.)

0 166.0

10 140.5

20 125.2

30 110.3

40 104.5

50 98.4

60 93.9

REMARKS:

Common sense tells us that the coffee will be cooling off

and its temperature will decrease and approach the ambient

temperature of the room, 69 degrees.

So, the temperature difference between the coffee

temperature and the room temperature will decrease to 0.

We will fit the temperature difference data (Table 2) to an

exponential curve of the form y = A e−bt.

Notice that as t gets large, y will get closer and closer to 0,

which is what the temperature difference will do.

So, we want to analyze the data where t = time elapsed

and y = C − 69, the temperature difference between the

coffee temperature and the room temperature.

TABLE 2

t = Time Elapsed (minutes)

y = C − 69 Temperature Difference (degrees F.)

0 97.0

10 71.5

20 56.2

30 41.3

40 35.5

50 29.4

60 24.9

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Exponential Function of Best Fit (using the data in Table 2):

y = 90 e − 0.023

t where t = Time Elapsed (minutes) and y = Temperature Difference (in degrees)

(a) Use the exponential function to estimate the temperature difference y when 30 minutes have elapsed. Report

your estimated temperature difference to the nearest tenth of a degree. (explanation/work optional)

(b) Since y = C − 69, we have coffee temperature C = y + 69. Take your difference estimate from part (a) and

add 69 degrees. Interpret the result by filling in the blank: When 25 minutes have elapsed, the estimated coffee temperature is ________ degrees.

(c) Suppose the coffee temperature C is 100 degrees. Then y = C − 69 = ____ degrees is the temperature

difference between the coffee and room temperatures.

EXTRA CREDIT (6 pts):

Show algebraic work to solve this part (d) equation for t, to the nearest tenth. Interpret your results clearly in the

context of the coffee application. [Use additional paper if needed]

y = 90 e-0.023t

R² = 0.985

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70

T e

m p

e ra

tu re

D if

fe re

n ce

( d

e g

re e

s)

Time Elapsed (minutes)

Temperature Difference between Coffee and Room