Aim: What do slope, tangent and the derivative have to do with each other?

1. Aim: What do slope, tangent and the derivative have to do with each other? Do Now: What is the equation of the line tangent to the circle at point (7, 8)?

2. A secant of a circle is a line that intersects the circle in two points. Tangents & Secants A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. B C

3. sin  cos  -1 Tan  y 1 radius = 1 center at (0,0) cos , sin  (x,y) 1  x -1 1

4. (x3, y3) (x2, y2) (x4, y4) (x1, y1) Tangents to a Graph slope is level: m = 0 slope is falling: m is (-) slope is steep! Unlike a tangent to a circle, tangent lines of curves can intersect the graph at more than one point.

5. (1, 1) 2 1 Finding the Slope (tangent) of a Graph at a Point This is an approximation. How can we be sure this line is really tangent to f(x) at (1, 1)?

6. f(x + h) – f(x) h x, f(x) h is the change in x f(x + h) – f(x)is the change in y Slope and the Limit Process A more precise method for finding the slope of the tangent through (x, f(x)) employs use of the secant line. (x + h, f(x + h)) This is a very rough approximation of the slope of the tangent at the point (x, f(x)).

7. (x + h, f(x + h)) f(x + h) – f(x) x, f(x) h Slope and the Limit Process h is the change in x f(x + h) – f(x)is the change in y As (x + h, f(x + h)) moves down the curve and gets closer to (x, f(x)), the slope of the secant more approximates the slope of the tangent at (x, f(x).

8. (x + h, f(x + h)) f(x + h) – f(x) h x, f(x) Slope and the Limit Process h is the change in x f(x + h) – f(x)is the change in y What is happening to h, the change in x? It’s approaching 0, or its limit at x as h approaches 0.

9. Slope and the Limit Process As h 0, the slope of the secant, which approximates the slope of the tangent at (x, f(x)) more closely as (x + h, f(x + h)) moved down the curve. At reaching its limit, the slope of the secant equaled the slope of the tangent at(x, f(x)).

10. The slope m of the graph of f at the point (x, f(x)) , is equal to the slope of its tangent line at (x, f(x)), and is given by provided this limit exists. difference quotient Definition of slope of a Graph

11. Model Problem Find the slope of the graph f(x) = x2 at the point (-2, 4). set up difference quotient Use f(x) = x2 Expand Simplify Factor and divide out Simplify Evaluate the limit

12. Slope at Specific Point vs. Formula What is the difference between the following two versions of the difference quotient? (1) Produces a formula for finding the slope of any point on the function. (2) Finds the slope of the graph for the specific coordinate (c, f(c)).

13. The derivative of f at x is provided this limit exists. The derivative f’(x) is a formula for the slope of the tangent line to the graph of f at the point (x,f(x)). Definition of the Derivative The function found by evaluating the limit of the difference quotient is called the derivative of f at x. It is denoted by f ’(x), which is read “f prime of x”.

14. Finding a Derivative Find the derivative of f(x) = 3x2 – 2x. factor out h

15. Aim: What is the connection between differentiability and continuity? Do Now: Find the equation of the line tangent to

16. f(x) is a continuous function (x, f(x)) f(x) – f(c) (c, f(c)) x – c x c alternative form of derivative Differentiability and Continuity What is the relationship, if any, between differentiability and continuity? Is there a limit as x approaches c? YES

17. Differentiability and Continuity Is this step function differentiable at x = 1? By definition the derivative is a limit. If there is no limit at x = c, then the function is not differentiable at x = c. Does this step function, the greatest integer function, have a limit at 1? NO: f(x) approaches a different number from the right side of 1 than it does from the left side.

18. Differentiability and Continuity If f is differentiable at x = c, then f is continuous at x = c. NO Is the Converse true? If f is continuous at x = c, then f is differentiable at x = c.

19. m = -1 m = 1 alternative form of derivative Graphs with Sharp Turns – Differentiable? f(x) = |x – 2| Is this function continuous at 2? YES Is this function differentiable at 2? One-sided limits are not equal, f is therefore not differentiable at 2. There is no tangent line at (2, 0)

20. Does a limit exist at 0? Graph with a Vertical Tangent Line f(x) = x1/3 Is f continuous at 0? YES NO f is not differentiable at 0; slope of vertical line is undefined.

21. Differentiability Implies Continuity corner vertical tangent a b c d f is not continuous at a therefore not differentiable f is continuous at b & c, but not differentiable f is continuous at d and differentiable

22. Summary 1. If a function is differentiable at x = c, then it is continuous at x = c. Thus, differentiability implies continuity. 2. It is possible for a function to be continuous at x = c and not be differentiable at x = c. Thus continuity does not imply differentiability.