Tangent Lines and Derivatives

1. Tangent Lines and Derivatives 2.7

2. The First of the TWO GRANDE Sized Questions • What are the two primary questions that Calculus seeks to answer?

3. Tangent Lines and Derivatives • In this section, we see how limits arise when we attempt to find: • The tangent line to a curve. • The instantaneous rate of change of a function.

4. The Tangent Problem

5. What is a Tangent Line, Anyways? • A tangent lineis a line that justtouches a curve. • The figure shows the parabola y =x2and the tangent line t that touches the parabola at P(1, 1).

6. Equation of Tangent Line • We will be able to find an equation of t as soon as we know its slope m. • The difficulty is that we know only one point, P, on t. • We need two points to compute the slope.

7. A Road Trip - Average Rate of Change • Suppose we went on a road trip to Las Vegas. Suppose further that we drove 210 miles in a 3 hour period of time. • What was our average speed? • If you left at 7PM and arrived at 10PM, how fast were you driving at 7:35?

8. Average Rate of Change • Earlier in the course, we defined the average rate of change of a function f between the numbers a and x as:

9. A Road Trip Continued • Consider the following diagram:

10. Derivatives – Tangent Lines • We have seen that the slope of the tangent line to the curve y = f(x) at the point (a, f(a)) can be written as: • It turns out that this expression arises in many other contexts as well—such as finding velocities and other rates of change. • Since this type of limit occurs so widely, it is given a special name and notation.

11. Derivative—Definition • The derivative of a function f at a number x, denoted by f'(x), is:if this limit exists.

12. Instantaneous Rate of Change—Definition • If y = f(x), the instantaneous rate of change of y with respect to xat x = a is the limit of the average rates of change as x approaches a:

13. Derivative- Instantaneous Rate of Change • Suppose we consider the average rate of change over smaller and smaller intervals by letting x approach a. • The limit of these average rates of change is called the instantaneous rate of change.

14. What can we conclude about the slope of the tangent line and the instantaneous rate of change at the same point? That’s right! They’re the same!!!!

15. Interpreting the Derivative • Notice that we now have two ways of interpreting the derivative: • f'(a) is the slope of the tangent line to y = f(x) at x =a. • f'(a) is the instantaneous rate of change of y with respect to x at x =a.

16. E.g. 3—Finding a Derivative at a Point • Find the derivative of f(x) = 5x + 3at the number 2.

17. More Examples

18. E.g. 4—Finding a Derivative • Let . • Find f'(a). • Find f'(1), f'(4), and f'(9).

19. Example (a) E.g. 4—Finding a Derivative • We use the definition of the derivative at a:

20. Example (a) E.g. 4—Finding a Derivative

21. Example (b) E.g. 4—Finding a Derivative • Substituting a = 1, a = 4, and a = 9 into the result of (a), we get:

22. Example (b) E.g. 4—Finding a Derivative • These values of the derivative are the slopes of the tangent lines shown here.