MATH107 One to One Function and Exponential Form of The Equation Assignment

Please see problems on attachment (10 problems to solve)

MATH 107 QUIZ 5
April-May, 2019
NAME: _______________________________
Instructor: S. Sands
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 Sunday, May 5.
• 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.
• If you have any questions, please contact me by e-mail.
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 U.S. households, the following logarithmic model was determined:
f (t) = 4645.3 ln(t) − 35240, where t = year and f (t) = number of U.S. households with cable
television, in millions of households.
(Note that “ln” refers to the natural log function) (explanation optional)
Using the model,
(a) How many U.S. households had cable television in the year 1999, to the nearest million?
(b) How many U.S. households had cable television in the year 2006, to the nearest million?
3. (4 pts) Convert to a logarithmic equation: 9x = 27.
(no explanation required)
3. ______
A. log 𝑥 27 = 9
B. log 𝑥 9 = 27
C. log 9 𝑥 = 27
D. log 9 27 = 𝑥
4. (8 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your
final conclusion.
√3𝑥 + 22 = 𝑥 + 6
5. (8 pts)
(a)
log 6 1 =_______ (fill in the blank)
(b) Let 𝑥 = log 6
(c)
1
√6
State the exponential form of the equation.
Determine the numerical value of log 6
1
√6
in simplest form. Work optional.
6. (10 pts) Let f (x) = 3×2 – x + 7 and g(x) = 2x – 1
(a) Find the composite function (𝑓 𝑜 𝑔)(𝑥) and simplify the results. Show work.
(b) Find (𝑓 𝑜 𝑔)(3) . Show work.
𝑓(𝑥 ) =
7. (16 pts) Let
(a) Find f
−1
6𝑥 + 5
4𝑥 + 7
, the inverse function of f. Show work.
(b) What is the domain of f ? What is the domain of the inverse function f
(c) What is f (−2) ?
(d) What is f
−1
f (−2) = ______
−1
?
work/explanation optional
( ____ ), where the number in the blank is your answer from part (c)?
work/explanation optional
8. (18 pts) Let f (x) = 3ex − 2.
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. Stretch the graph of y = ex vertically by a factor of 3, and then shift to the right by 2 units.
B. Shift the graph of y = ex downward by 2 units, and then stretch vertically by a factor of 3.
C. Stretch the graph of y = ex vertically by a factor of 3, and then shift downward by 2 units.
D. Shift the graph of y = ex to the right by 3 units, and then shift downward by 2 units.
(b) What is the domain of f ?
(c) What is the range of f ?
(d) What is the y-intercept?
(e) What is the horizontal asymptote?
(f) Which is the graph of f ?
GRAPH A (below)
GRAPH C (below)
GRAPH B
(below)
GRAPH D (below)
NONLINEAR MODELS – For the latter part of the quiz, we will explore some nonlinear models.
9. (18 pts) QUADRATIC REGRESSION
Data: On a particular spring day, the outdoor temperature was recorded at 8 times of the day. The parabola of best fit
was determined using the data.
Quadratic Polynomial of Best Fit:
y = −0.24t2 + 6.84t + 27.6
for 0  t  24 where t = time of day (in hours)
and y = temperature (in degrees)
REMARKS: The times are the hours since midnight.
For instance, t = 6 means 6 am. t = 20 means 8 pm.
t = 18.25 hours means 6:15 pm
(a) Use the quadratic polynomial to estimate the outdoor temperature at 4:00 pm, to the nearest tenth of a
degree. (work optional)
(b) 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 instance, 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.
(c) Use the quadratic polynomial y = −0.24t2 + 6.84t + 27.6 together with algebra to estimate the time(s) of day
when the outdoor temperature y was exactly 60 degrees.
That is, solve the quadratic equation 60 = −0.24t2 + 6.84t + 27.6 .
Show algebraic work in solving. Round the results to the nearest tenth. Write a concluding sentence to
report the time(s) to the nearest quarter-hour, in the usual time notation. (Use more paper if needed)
10. (8 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 65 degrees, and the
coffee temperature was recorded periodically, in Table 1.
TABLE 1
t = Time
Elapsed
(minutes)
0
10
20
30
40
50
60
C = Coffee
Temperature
(degrees F.)
162.0
136.5
121.2
106.3
100.5
94.4
89.9
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, 65 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 − 65, the temperature difference between the
coffee temperature and the room temperature.
TABLE 2
t = Time
Elapsed
(minutes)
y = C − 65
Temperature
Difference
(degrees F.)
0
10
20
30
40
50
60
97
71.5
56.2
41.3
35.5
29.4
24.9
Temperature Difference (degrees)
120
Temperature Difference between Coffee and Room
100
80
y = 89.976e-0.023t
R² = 0.9848
60
40
20
0
0
10
20
30
40
50
60
70
Time Elapsed (minutes)
Exponential Function of Best Fit (using the data in Table 2):
y = 89.976 e − 0.023 t where t = Time Elapsed (minutes) and y = Temperature Difference (in degrees)
(a) Use the exponential equation to estimate the temperature difference y when 5 minutes have elapsed.
Report your estimated temperature difference to the nearest tenth of a degree. (explanation/work optional)
(b) Since y = C − 65, we have coffee temperature C = y + 65. Take your difference estimate from part (a)
and add 65 degrees. Interpret the result by filling in the blank:
When 5 minutes have elapsed, the estimated coffee temperature is ________ degrees.
(c) Suppose the coffee temperature C is 90 degrees. Then y = C − 65 = ____ degrees is the temperature
difference between the coffee and room temperatures.
(d) Consider the equation _____ = 89.976 e − 0.023t where the ____ is filled in with your answer from part
(c).
EXTRA CREDIT (5 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]