1. MATH 2413
Summer 2017
2 – 4:30 M - F Room 216
Professor Thomas Jay

2. Instructor: Thomas Jay
Instructor Contact Information: Phone: (713)
Office location and hours: Office: 359 I; Office Hours: 9 – 10 M – F
Course Description Math 2413: Calculus I. Topics includes: Limits and continuity; the Fundamental Theorem of Calculus; definition of the derivative of a function and techniques of differentiation; applications of the derivative to maximizing or minimizing a function; the chain rule, mean value theorem, and rate of change problems; curve sketching; definite and indefinite integration of algebraic, trigonometric, and transcendental functions, with an application to calculation of areas.
Prerequisites
MATH 2412 or consent of the Department Chair
Textbook Options for: Calculus, 11th Edition, by Ron Larson & Bruce H. Edwards
Loose-leaf Textbook + WebAssign Multi-Term Printed Access Card: Edwards ISBN-13: 
Hardbound Textbook + WebAssign Multi-Term Printed Access Card: Edwards ISBN-13: 
Hardbound Textbook: ISBN-13: 
WebAssign Multi-Term Printed Access Card: ISBN-13: 
Calculator: A scientific calculator is required for this course. A graphing calculator is recommended. The use of a TI – 89 or TI nSpire must be approved by the instructor.
Course Goal
This course provides the background in mathematics for sciences or further study in mathematics and its applications

4. Resources
CalcChat.com provides complete solutions to all the odd-numbered exercises in the text. This is a free website maintained by the author. LarsonCalculus.com is a new website provided by the author that offers multiple tools and resources.

5. WebAssign
WebAssign is required. WebAssign allows you to work your assignments online. You can have a sample problem worked for you, get help with your assigned problem, and even look at the text book online. You have two free weeks to try it out. After two weeks, all your work is lost until you purchase a code. You will be given a username and password and already be registered for the WebAssign class.

6. Learning Web

14. Limits There are three ways to determine a limit. Numerically
Graphically
Analytically

15. Numerical method X -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x) .2911 .2889
.2887
.2884
.2863

16. Example 1 – Estimating a Limit Numerically
Evaluate the function at several
x-values near 0 and use the results to estimate the limit

17. Example 1 – Solution
The table lists the values of f(x) for several x-values near 0.
From the results shown in the table, you can estimate the limit to be 2.

18. x
1.9
1.99
1.999
2.001
2.01
2.1
f(x) x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
2.95
2.995
2.9995
3.001
3.01 x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
2.95
2.995
2.9995
3.0001
3.001
3.01 x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
2.95
2.995
2.9995
3.01 x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
2.95
2.995
2.9995 x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
2.95 x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
2.95
2.995

19. Graphical Method

20. Example 1 – Solution
cont’d
This limit is reinforced by the graph of f shown in Figure 1.6.
Figure 1.6

23. Limits That Fail to Exist

24. Example 3 – Different Right and Left Behavior
Show that the limit does not exist.
Solution:
Consider the graph of the function
In Figure 1.8 and from the definition of absolute value,
you can see that
Figure 1.8

25. Example 3 – Solution
cont’d
So, no matter how close x gets to 0, there will be both positive and negative x-values that yield f(x) = 1 or f(x) = –1.
Specifically, if (the lowercase Greek letter delta) is a positive number, then for x-values satisfying the inequality 0 < | x | < , you can classify the values of | x | / x as –1 or 1 on the intervals

26. Example 3 – Solution
cont’d
Because | x | / x approaches a different number from the right side of 0 than it approaches from the left side, the limit does not exist.

27. Limits That Fail to Exist

28. A Formal Definition of Limit

29. A Formal Definition of Limit
The first person to assign mathematically rigorous meanings to these two phrases was Augustin-Louis Cauchy. His definition of limit is the standard used today.
In Figure 1.12, let (the lower case Greek letter epsilon) represent a (small) positive number.
Then the phrase “f(x) becomes arbitrarily close to L” means that f(x) lies in the interval (L – , L + ).
Figure 1.12

30. A Formal Definition of Limit
Using absolute value, you can write this as
Similarly, the phrase “x approaches c” means that there exists a positive number such that x lies in either the interval or the interval
This fact can be concisely expressed by the double inequality

31. A Formal Definition of Limit
The first inequality
expresses the fact that The second inequality
says that x is within a distance of c.

32. A Formal Definition of Limit

33. Example 6 – Finding a  for a Given 
Given the limit
find  such that whenever
Solution:
In this problem, you are working with a given value of
 –namely,  = 0.01.
To find an appropriate  , try to establish a connection
between the absolute values

34. Example 6 – Solution Notice that
cont’d
Notice that
Because the inequality is equivalent to
you can choose
This choice works because
implies that

35. Analytical method
Usually, the analytical method, or algebraic method, involves substituting the value of the variable into the expression and evaluating. Sometimes that does not yield a limit. In this example if we substitute 0 for x we end up with 0/0. Since division by 0 is not allowed, this does not give us the desired limit. We will discuss later how to do this specific example analytically.

36. Simplifying the expression when