1. The Derivative Calculus. At last. (c. 5)

2. POD Review each other’s answers for c. 4: 23, 25, and 27

3. The Basic Idea The derivative is simply the slope of the tangent at a point on a curve. It models instantaneous rate of change– say, how fast you travel at a given moment. A diagram…

4. The Basic Idea

5. Want to see it animated? Sometimes a moving picture helps.moving picture Pay attention to how the orange secant line travels as the second point approaches the first point. The derivative is all about finding the slope and equation for a tangent at a given x, or in general. But finding a slope implies having two points. How do we find the slope at one point?

6. How to do it We’ll start to find the derivative by finding the slope of the secant. We’ll do this by using the difference quotient between x and x+h. (What is the difference quotient again?) Slope is difference in y over difference in x, so the difference quotient is the ideal tool. Use the function f(x) = x 2 + 3.

7. How to do it Slope is difference in y over difference in x, so the difference quotient is the ideal tool. Use the function f(x) = x 2 + 1. What two points will we use to find the slope?

8. Use it Find the slope of a secant between two points on the function f(x) = x 2 + 1. The set up:

9. Use it Find the slope of a secant between two points on the function f(x) = x 2 + 1. The finish:

10. Use it The slope of a secant between two points on the function f(x) = x 2 + 1 is This is the equation to find the slope between any two points on the curve. But we want the slope of the tangent. How do we find it based on this slope?

11. Use it The slope of the secant: To find the slope of the tangent, think about the h value. What happens to it as the points come closer?

12. Use it The slope of the secant: To find the slope of the tangent, think about the h value. It approaches 0 as the two points get closer and the secant becomes a tangent. So…

13. Use it … the slope of the tangent: At any value of x, the slope of the tangent at that point on the curve f(x) = x 2 + 1 will be 2x. The slope at the point (1, 2) is 2·1 = 2. The slope at the point (3, 10) is 6. What are the equations for those tangents?

14. Use it The slope at the point (1, 2) is 2·1 = 2. The equation for the tangent at the point (1,2) is y – 2 = 2(x – 1). The slope at the point (3, 10) is 6. The equation for the tangent at the point (3,10) is y – 10 = 6(x – 3).

15. Definition of the derivative The limit of as h approaches 0 is called the derivative of f(x) with respect to x. The notation is usually f’ (x). So. In general,

16. Now Read pp. 26-35. Do pp. 35 – 6: 3,5, 9, 13, 15, 19, 23. You will have shortcuts soon to find the derivative, but you will also need to be able to find it using the difference quotient.