1. HOMEWORK Section 10-1 Section 10-1 (page 801) (evens) 2-18, 22-32, 34a (16 problems)

2. 10-1 Limits and Motion The tangent problem.

3. What you’ll learn about Average Velocity Instantaneous Velocity Limits Revisited The Connection to Tangent Lines The Derivative … and why The derivative is fundamental to all of calculus. It allows us to analyze rates of change for any function regardless of how complicated it is. The ability to find rates of change is essential in physics, economics, engineering, and even history.

4. Vocabulary 6543210 feet 0 5 10 15 20 25 30 35 seconds Average velocity: the change in position divided by the change in time 6 feet 30 seconds Ave Vel.= 6/30 = 0.2 ft/sec

5. Average Rate of Change ab f(b) f(a)

6. 6543210 feet 0 5 10 15 20 25 30 35 seconds Velocity: is it constant over the entire interval? Ave Velocity= 6/30 = 0.2 ft/sec Instantanious Velocity: the actual velocity at an instant in time. an instant in time. For many applications, instantanious rate of change instantanious rate of change is more important than the is more important than the average rate of change. average rate of change. We need infinitely small measurements that are still greater than zero.

7. Example: The distance on object travels is given by the relation: by the relation: For the interval: (2 sec --2.1 sec): For the interval: (2 sec --2.01 sec): For the interval: (2 sec --2.001 sec): For the interval: (2 sec --2.0001 sec): What is happening to the velocity as to the velocity as our time interval our time interval approaches zero? approaches zero?

8. Example: The distance on object travels is given by the relation: by the relation: The idea of the limit of a function is essential to calculating instantanious rates of change.

9. Limits at a (Informal) I would prefer to replace arbitrarily with the word infinitely.

10. Finding the slope of a curve at exactly one point. 43210 0 1 2 3 Finding the slope of a tangent line and the instantaneous velocity line and the instantaneous velocity are the exact same type of problem. are the exact same type of problem.

11. Your turn: 1. Find the instantaneous velocity at time = 3 for the following time—distance relation: for the following time—distance relation:

12. Derivative at a Point a f(a) Not as good a definition as the definition as the following slide. following slide.

13. Derivative at a Point (easier for computing) Remember the idea of instantaneous rate of change: (as f(a + h) gets infinitely close to f(a)). (as f(a + h) gets infinitely close to f(a)). Numerator: “rise” (with the rise being infinitely small) Denominator: “run” (with the fun being infinitely small) Derivative  think slope (of the tangent line at a point on the curve) (of the tangent line at a point on the curve)

14. Finding a Derivative (slope) at a Point This is a trivial problem. You know the slope of this line at any point on the line is 2. General form of a derivative Replace ‘x’ in the general form with the ‘x’ value of the point with the ‘x’ value of the point simplify Simplify  we already knew it would be 2 Rewrite f(x) using the input values above input values above

15. Derivative

16. Finding the Derivative of a Function (gives an equation that you can use to find the slope at any point in the original function) General form of a derivative Input values are ‘x+h’ and ‘x’ into the function. the function. Simplify Simplify Simplify

17. Your turn: 1. Find: f’(x) for Using:

18. Finding the derviative of a point (alternative method) 1. Find: f’(x) for the function. 2.Use ‘x’ value of the point as an input value to f’(x)

19. 2.Find the instaneous velocity at time t = 4 for the following time—distance relation for the following time—distance relation using: using: Your turn: Pretty silly really. You know the slope anywhere on the graph is 3 (y = mx + b) the graph is 3 (y = mx + b)

20. Newton vs. Liebnitz Both independently discovered calculus. They came up with different notation to describe the derivative. Liebnitz’s notation is much more useful later on in calculus.

21. This is too hard!! Figuring out these problems using following formula: is meant to help you figure out the relationship between the limit and the slope of the tangent line at a point. is meant to help you figure out the relationship between the limit and the slope of the tangent line at a point. (Infinitesimally small change in rise over an infinitesimally small change in run)

22. What you’ll learn in Caclulus. There are some very simple methods of finding the derviatives of various functions. Each method is different for different classes of functions. There are some very simple methods of finding the derviatives of various functions. Each method is different for different classes of functions. Polynomials –way easy Trig Functions – pretty easy for simple ones Composition of functions – another “cool” method Product of functions– really “slick” method Quotient of functions – not too hard Rational functions (ratio of functions)--harder

23. I’ll teach you the method for polynomials – way easy