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02_23_activity_7_exponential_functions__dt3_.pdf
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Differentiation Techniques 3: Derivatives of Exponential Functions
113
Pre-Assignment for Derivatives of Exponential
Functions
Instructions: Print out the activity. Complete questions 1 through 3 of the activity.
Once you have completed this pre-assignment, go to the Canvas page for your class
and complete the pre-assignment quiz that you will find there. You have as many tries
as you like for each question. This must be completed BEFORE your peer led session
on Friday.
Bring your printed activity with your completed pre-assignment to your peer led session
in order to be eligible to take the quiz that will occur at the beginning of your peer led
session. Your quiz will be based on last week’s activity.
If you need help completing the pre-assignment, feel free to drop in at SMART lab (at
the library tutoring services).
SMART Lab Hours are:
M – Th:
F:
Sa:
Su:
9am – 9pm
9am – 4pm
closed
1 – 5pm
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Differentiation Techniques 3: Derivatives of Exponential Functions
DT3: Derivatives of Exponential Functions
Model 1: Graph of y = ex
The symbol e is used to represent the number whose nonrepeating decimal expansion begins:
2.718281828459045235360287471352662497757247093699959574966….
s
=
y f=
( x) e x
10
5
x
1
2
3
Construct Your Understanding Questions (to do in class)
1. According to Model 1, what do the numbers e and π have in common?
2. Use the graph of f ( x ) = e x in Model 1 to estimate the value of f at x = 1.1.
3. In Model 1, estimate the slope of the tangent line to the graph of y = f ( x ) at x = 1.1 (shown).
4. Which question above is asking you to find f (1.1) , and which is asking you to find f ′(1.1) ?
Explain your reasoning.
Differentiation Techniques 3: Derivatives of Exponential Functions
Model 2:
115
f ( x) = c x for Various Values of c
Note that decimals have been rounded to the thousandths position.
c
x
f ( x) = c x
f ′( x)
2
2
2
2
1
2
3
4
2
4
8
16
1.386
2.773
5.545
11.090
e
e
e
e
e
1
1.1
2
3
4
2.718
3.004
7.389
20.086
54.598
2.718
3.004
7.389
20.086
54.598
3
3
3
3
1
2
3
4
3
9
27
81
3.296
9.888
29.663
88.988
4
4
4
4
1
2
3
4
4
16
64
256
5.545
22.181
88.723
354.891
f ′( x)
f ( x)
Construct Your Understanding Questions (to do in class)
5. Why does it make sense to list c = e on the table after c = 2 and before c = 3 ?
6. (Check your work) Are your answers to Questions 2 and 3 consistent with the corresponding
table entries?
7. Identify the simple pattern in the gray boxes on the table in Model 2. What can you say about the
function f ( x ) = e x based on this pattern?
116
Differentiation Techniques 3: Derivatives of Exponential Functions
8. (Check your work) For the generalized function f ( x ) = c x …
a. According to the table in Model 2, it appears that f ( x ) = f ′( x ) when c =
b. Does f ( x ) = f ′( x ) for any of the other value of c listed on the table?
9. Fill in the last column in Model 2 (use a calculator if you wish). Round your entries to the
thousandths position.
a. (Check your work) Confirm that when c = 2 ,
f ′( x)
= 0.6931 (after rounding).
f ( x)
(2 x )′
= ln 2 .
2x
Do the other entries in the table (i.e., when c = e, 3, and 4 ) follow this same pattern?
If so, describe this pattern in your own words.
b. Rounded to the nearest thousandth, ln 2 = 0.6931 . In other words,
10. Based on your answer to the previous question, find each derivative.
a. for f ( x ) = 2 x , what is f ′( x ) =
b. for f ( x ) = e x , what is f ′( x ) =
c. for f ( x ) = 3x , what is f ′( x ) =
d. for f ( x ) = 4 x , what is f ′( x ) =
11. Write a formula for f ′( x ) , if f ( x ) = c x for some positive constant c.
12. Is your answer to the previous question consistent with your answer to part b of question 11 and
with the fact that ln e = 1 ? Explain.
Differentiation Techniques 3: Derivatives of Exponential Functions
117
13. (Check your work) Are your answers to the previous two questions consistent with Summary Box
DT3.1? If not, go back and revisit your work.
Summary Box DT3.1: Derivative of an Exponential Function
If f ( x) = c x where c is a positive, real number, then f ‘( x) = (ln c)c x .
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Differentiation Techniques 3: Derivatives of Exponential Functions
Activity Report
Derivatives of Exponential Functions
We verify that we all understand and agree with the solutions to these questions.
Group Number: _______
Manager: _____________________________________________
Recorder: _____________________________________________
Spokesperson: _________________________________________
Strategy Analyst: _______________________________________
Critical Thinking Question: to be agreed upon by the group, and written below by the recorder.
A student from another group says that the derivative of f ( x) = 2 x is f ‘( x) = x 2 x −1 . What
would your group say to this student? Explain (using complete sentences) and fix any errors in
the student’s thinking.
For instructor’s use only
All questions on activity filled out
Names and U-Numbers PRINTED on activity report
Critical thinking correct, fully justified, and written in complete sentences
Satisfactory/Unsatisfactory
…
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