Calculus I (MAT 145) Dr. Day Monday Feb 4, 2013

1. Calculus I (MAT 145)Dr. Day Monday Feb 4, 2013 • Everything You Always Wanted to Know About Limits (2.2 through 2.6) • Assignments MAT 145

2. Two-Point Slope to One-Point Slope (2.1) • Determining Slope for a Function: Average Rate of Change • Applying the Idea of a Limit to the Secant Line Slope • What Do We Get? • Instantaneous Rate of Change • Slope of the Tangent Line • Slope of the curve at a Point • Derivative of the Function at a Point MAT 145

3. What About Limits?(2.1-2.3; 2.6; PIP pp 14-17; 20) • Concept of a Limit: Approachment • Symbolizing Limits • Determining Limits • Properties of Limits • What About Limits Involving Infinity? MAT 145

4. What About Limits? Concept: Approachment (PIP p 14) • What patterns of approachment in outputs are apparent when the inputs approach some value? • Reminder! We’re not concerned with behavior AT the specific point, only the behavior that’s apparent as we approach it! • Numerical, Graphical, and Symbolic Perspectives • One-Sided and Two-Sided Limits MAT 145

5. Determining Limits (PIP p 16) • Look at the graph of the function: Can you SEE the pattern? • Investigate numerical inputs and outputs for approachment patterns. • Use the symbolic representation for the function. • Sub in the limiting input value • Manipulate the Expression • Additional Symbolic Techniques MAT 145

6. Here are inputs (x) and outputs (sin(x))/(x) for a functional relationship. • What value of x is being approached based on the entries in the left column? • What value of (sin(x))/(x)is being approached based on the entries in the right column? • Based on this numerical evidence, we can say that: • As x approaches _?_, (sin(x))/(x) approaches _?_. MAT 145

7. Here is the graph of a function y = f(t). Use it to answer these questions. (1) What is the value of the function when t = 3? (2) As t gets closer and closer to 4, the value of the function is getting closer and closer to _?_. (3) What is the value of the function when t = 2? (4) As tapproaches -1, the outputs approach _?_. (5) What is the value of the function when t = 0? (6) As t approaches 2, the function values approach _?_. (7) As t approaches 0 from the left side, the function values approach _?_. (8) As t approaches 0 from the right side, the function values approach _?_. MAT 145

8. To explore a limit problem, Henry created a table of values for the function y=Q(t). His table is shown here. Why is Henry’s table not useful in helping to determine the following limit? MAT 145

9. Use limit symbolism to represent the limit situation shown in this figure. MAT 145

10. MAT 145

11. Use limit symbolism to describe both the one-sided and the two-sided limits that seem apparent as x approaches 0 in the top graph and as x approaches 3 in the other graph. If a particular limit does not exist, signify that with DNE. MAT 145

12. Properties of Limits(PIP p 15) • the limit of a sum is the sum of the limits • the limit of a difference is the difference of the limits • the limit of a constant times a function is that constant times the limit of the function • the limit of a product is the product of the limits • the limit of a quotient is the quotient of the limits (as long as …) • the limit of a function raised to a power is the limit of the function raised to that power • the limit of a constant is that constant MAT 145

13. LimitsInvolving Infinity Limits Involving Infinity (PIP p 20) • Horizontal and Vertical Asymptotes • Special Cases: Rational Functions MAT 145

14. What Makes a Function Continuous? (PIP pp 18-19) Informal Perspective Formal Definition One-Sided Continuity Continuity on an Interval Connections: Limits and Continuity MAT 145

15. Assignments • WebAssign • 2.2 due tonight • 2.3 due Wednesday night MAT 145