The derivative
M119
M119
The derivative
1 / 15
2.1 Instantaneous rate of change
Recall that the average rate of change between x = a and x = b is:
f (b) f (a)
y
=
.
x
ba
M119
The derivative
2 / 15
2.1 Instantaneous rate of change
Recall that the average r
1.7. Exponential Growth and Decay
In the function P = P0 e kt , P0 is the initial quantity
and k is the continuous growth or decay
The population of a country in 2010 was 120 million and the population
was growing at a continuous rate of 3.5% per year. Wh
Examples 1 and 2 are about specific taxes. Examples 3 and 4 are about percentage taxes.
Answers to questions 34, 36, and 38 are at the end.
Example 1)
Let = () = 15 + 4000 and = () = 8000 10 be supply and demand equations when no
tax is present. To keep t
7.3 Definite Integrals
If F is an antiderivative of continuous function G , then
Fundamental Theorem of Calculus:
Z b
G (x) dx = F (b) F (a)
a
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Antiderivatives
1/6
7.3 Definite Integrals
Evaluate the following definite integrals:
R2 2
1
0 9x x + 3 dx
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M119
TEST 3
18th Nov 2015
READ THIS: For each of the problems 1-12 that begin on the next page, find the one correct answer choice from the
options listed. Indicate your answer
7.5 Antiderivatives: Graphically
If F is an antiderivative of continuous function G , then
Fundamental Theorem of Calculus:
Z b
F (b) = F (a) +
G (x) dx
a
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Antiderivatives
1/3
7.5 Antiderivatives: Graphically
In particular,
Fundamental Theorem of Calc
1.6. Natural Logarithm
The natural logarithm, written as ln x is the inverse of exponential function with base e
In other words, for example,
e 1.6094 = 5 means that ln 5 = 1.6094.
More generally, e c = x ln x = c
M119
Functions and Change
1/8
1.6. Natura
1.9. Power functions
y = 3 6x is an exponential function
y = 3x 6 is a power function
If y = k x p , then it is a power function where k is
called the constant of proportionality and p is the
power
M119
Functions and Change
1/9
1.9. Power functions
Which
1.5. Exponential Function
Consider the table:
0
6
1
18
2
54
3
162
4
486
Is it an increasing function or decreasing function?
M119
Functions and Change
1/9
1.5. Exponential Function
Consider the table:
0
6
1
18
2
54
3
162
4
486
Is it an increasing function
1.4. Applications to Economics
A company buys a $ 6000 machine to produce T-shirts. Using this
machine, the company produces T-shirts at the cost of $ 2 per shirt.
Write the cost function in terms of the quantity q and graph.
M119
Functions and Change
1 /
1.8. New Functions from Old
The function f (g (x) is a composite of two
functins f (x) and g (x) in which there is an inside
function g (x) and an outside function f (x)
Warning: In genearl f (g (x) 6= g (f (x)
M119
Functions and Change
1/8
1.8. New Funct
7.1 Constructing Antiderivatives
If the derivative of F (x) is f (x), then an antiderivative of f (x) is F (x)
M119
Antiderivatives
1/8
7.1 Constructing Antiderivatives
If the derivative of F (x) is f (x), then an antiderivative of f (x) is F (x)
For exam
M119
Exam 2 - Spring 2017
Exam Question 1: The population of a town was 350 at the start of 2017
1) A
2) E
3) D
4) C
5) D
6) B
7) D
8) B or C
B is the intended answer.
Nevertheless, while grading these exam papers, it was felt that the distinction between
Appendix B. Compound Interest
Effective annual rate:
Formula to calculate effective annual rate is:
r
R = (1 + )n
n
where
r =
n =
(nominal) annual interest rate
number of times interest compounded in a year
Now effective annual rate is R 1 written in perc
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EXAM 3
14th April 2015
READ THIS: For each of the problems that begin on the next page, find the one correct answer choice from the
options listed. Indicate yo
7.2 Integration by substitution
Integration by substitution is the reverse process of chain rule.
Use integration by substitution to find the following integrals:
Hint: Look for an inside function whose derivative is a factor.
R
2
1
2xe x dx
M119
Antideri
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M119
TEST 3
18th Nov 2015
READ THIS: For each of the problems 1-12 that begin on the next page, find the one correct answer choice from the
options listed. Indicate your answer
1.3 Average rate of change
Even if the rate of change is not constant, the average rate of change can
be computed by the same formula:
Average Rate of change from t = a to t = b is equal to
M119
Functions and Change
f (b) f (a)
ba
1/5
1.3 Average rate of
7.4 Integration by parts
Integration by parts:
Z
Z
uv 0 dx = uv u 0 v dx
M119
Antiderivatives
1/3
7.4 Integration by parts
Evaluate the following integrals:
R 4x
1
xe dx
M119
Antiderivatives
2/3
7.4 Integration by parts
Evaluate the following integrals:
R
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Seat number:
EXAM 3
14th April 2015
READ THIS: For each of the problems that begin on the next page, find the one correct answer choice from the
options listed. Indicate yo
I201 SUMMER 2017: HW 9
This HW assignment is due no later than the night before your lab section (6/11), at 11:59PM.
You must submit an electronic copy on Canvas.
Requirements :
Start the HW early. Some of the problems might be challenging, and you want
2.11 Write Python expressions corresponding to these statements:
(a) The sum of negative integers -7 through -1
(b) The average age of a group of kids at a summer camp given that 17 are 9 years old, 24 are 10
years old, 21 are 11 years old, and 27 are 12
I201: HW 1
This HW assignment is due at the beginning of your Lab 2 (January 18, 19 or
20.)
Comments :
Make sure to clearly write your name, HW #, lab #, and due date.
Please staple the pages together. DO NOT just fold the pages together.
If you use pa
I201 SUMMER 2017: HW 4
This HW assignment is due no later than the night before your lab section (5/21), at
11:59PM. You must submit an electronic copy on Canvas.
Requirements :
Start the HW early. Some of the problems might be challenging, and you want
I201 SUMMER 2017: HW 3
This HW assignment is due no later than the night before your lab section (5/16), at
11:59PM. You must submit an electronic copy on Canvas.
Requirements :
Start the HW early. Some of the problems might be challenging, and you want
I201 SUMMER 2017: HW 5
This HW assignment is due no later than the night before your lab section (5/28), at
11:59PM. You must submit an electronic copy on Canvas.
Requirements :
Start the HW early. Some of the problems might be challenging, and you want
cfw_1) (10 901nm) Construct a. truth table for each of these compound propositions:
(ahMew)
P a OWE Pam?)
q
TTT'r T
(151 1: p
FTr
171:1: F I:
1: r
*7 T "1 ~- T 1:
T 1' T: T 1: F
T F T T F F
T 'F F T 1:
'F T T T T:
1: T F 1: F
7:3: i 5E 3: T: F
' (d)(y>:r)
I201 SUMMER 2017: HW 7
This HW assignment is due no later than the night before your lab section (6/4), at
11:59PM. You must submit an electronic copy on Canvas.
Requirements :
Start the HW early. Some of the problems might be challenging, and you want t
I201: HW 2
This HW assignment is due at the beginning of your Lab 3 (January 25, 26 or
27.)
Comments :
Make sure to clearly write your name, HW #, lab #, and due date.
Please staple the pages together. DO NOT just fold the pages together.
If you use pa
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