1. 1 The Derivative and the Tangent Line Problem Section 2.1

2. 2 After this lesson, you should be able to: find the slope of the tangent line to a curve at a point use the limit definition of a derivative to find the derivative of a function understand the relationship between differentiability and continuity

3. 3 Tangent Line Remember our tangent line problem from section 1.1? Just in case you’ve forgotten, here are the cliff notes: Finding the slope of a curve at x = c is equivalent to finding the slope of the tangent line to the curve at x = c. To approximate the slope of the tangent line at c we can use the secant line through (c, f(c)) and another point on the curve really close to c. If we keep moving that second point closer and closer to the point of tangency, c, we get this very slick definition of the slope of the tangent line (See next slide).

4. 4 If f is defined on an open interval containing c, and if the limit exists, then the line passing through with slope m is the tangent line to the graph of f at We can also say the m is the slope of the graph of f at x = c. Definition of Tangent Line with Slope m

5. 5 Let's see how this applies to some basic functions. Example involving a linear function: Find the slope of the graph of f(x) = 5x - 7 at the point (-1, -12). So we need to find the slope of the graph of f at c = -1.

6. 6 For non-linear functions the slope is not constant. Example involving a quadratic function. Find the slope of the tangent line to the graph of f(x) = x 2 + 6 at any point on the graph.

7. 7 Quadratic Example continued From this, find the slope of f at: a) (0, 6) b) (-3, 15) Now pay attention…here comes the important part 

8. 8 There are many notations for indicating the derivative of f at x: "f prime of x" "the derivative of y with respect to x" "y prime" "the derivative of f with respect to x" ___________________________ is the process of finding the derivative of a function. A function is ______________________________ at x if its derivative exists at x. Definition of the derivative of f The derivative of f at x is given by for all x for which the limit exists.

9. 9 Using the definition of the derivative 1) Given f(x) = x 3 - 12x, a) find f '(x) using the definition of the derivative.

10. 10 Using the definition of the derivative (continued) b) What is the slope of f at (-2, 16)? Check this out on the calculator.

11. 11 Using the definition of the derivative 2) Given, a) find f '(x) using the definition of the derivative.

12. 12 Using the definition of the derivative (continued) b) What can you say about the slope of the graph of f at (1, 0)?

13. 13 Using the definition of the derivative 3) Find the equation of the tangent line to the graph of at x = -2. a) First find the _________ of the graph of f at any point x in the domain.

14. 14 Using the definition of the derivative (continued) 3 b) Next, find the slope of f when x = _____. 3 c) Write the equation of the line with m = _______ that contains the point (-2, ____).

15. 15 Using your calculator Graph the function on your calculator. Now, hit  DRAW Select 5: Tangent( Type the x value, which in this case is -2, and then hit  1 2 3 4 Here’s the equation of the tangent line…notice the slope…it’s approximately what we found 3 d) Using your calculator, graph f and its tangent line at x = -2.

16. 16 Differentiability Implies Continuity If f is differentiable at x, then f is continuous at x. Some things which destroy differentiability: 1.A discontinuity (a hole or break or asymptote) e.g. For does not exist because f is not continuous at x = 3. 2.A sharp corner (continuous but not differentiable) e.g. For does not exist because there is a sharp corner at x = 0. 3.A vertical tangent line (continuous but not differentiable) e.g. For does not exist because there is a vertical tangent at x = 0. The converse is not true; i.e. continuity does not imply differentiability!

17. 17 Homework Section 2.1 page 103 #1, 5-27 odd, 33, 37-40 all, 81-85 odd, 93