1. 2.1 The Derivative and the Tangent Line Problem Objectives: -Students will find the slope of the tangent line to a curve at a point -Students will use the limit definition to find the derivative of a function -Students will understand the relationship between differentiability and continuity

2. Calculus grew out of the major problems that European mathematicians were working on during the 17 th century. One of these is the tangent line problem that we discussed in section 1.1.

3. We want to find the slope of the curve at point P. This is also the slope of the tangent line. The tangent line is the limit of the secant lines. P Q f(x) (x,f(x)) xx +∆x f (x +∆x)

4. So the slope of the tangent line is (the derivative) This equation gives the slope of the tangent line at point P. It also gives the slope of the graph of f at x = c

5. Ex 1) Find the general equation of the graph of at any point.

6. …So the derivative of is We use it to find the slope at a point. PointSlope of f at point (0,1) (-1,2) (2,5) compare with graph:

7. Derivative

8. Ex 2) Find the equation of the tangent line to at the point (8,2).

9. Ex 3) Find the slope of the graph of at (2,1). (how can we PROVE the slope is 2?)

10. Ex 4) Find y’ for

11. If we can take the derivative of a function, we say the function is differentiable. Reasons a function is NOT differentiable: 1) There is a discontinuity at x = c f ’(x) DNE at x = 0 since f is discontinuous at x = 0.

12. If we can take the derivative of a function, we say the function is differentiable. Reasons a function is NOT differentiable: 2) There is a sharp turn on the graph at x = c f ’(x) DNE at x = 0 since the derivative on the left does not equal the derivative on the right.

13. If we can take the derivative of a function, we say the function is differentiable. Reasons a function is NOT differentiable: 3) f (x) is vertical at x = c f ’(x) DNE at x = 0 since the tangent line is vertical at x = 0 (slope is undefined).

14. If f is differentiable at x = c, then f is continuous at c. BUT…. If f is continuous at x = c, then f is not necessarily differentiable at c.