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

Instructor: Thomas Jay Instructor Contact Information: email: thomas.jay@hccs.edu Phone: (713) 718-5845 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: 978-1337604741 Hardbound Textbook + WebAssign Multi-Term Printed Access Card: Edwards ISBN-13: 978-1337604758 Hardbound Textbook: ISBN-13: 978-1337275347 WebAssign Multi-Term Printed Access Card: ISBN-13: 978-1285858265 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

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.

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.

Learning Web

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

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

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

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.

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

Graphical Method

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

Limits That Fail to Exist

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

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

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.

Limits That Fail to Exist

A Formal Definition of Limit

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

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

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

A Formal Definition of Limit

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

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

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.

Simplifying the expression when