1. AP Calculus Warm up 9.3.14 What is the derivative of a function? The derivative of a function is also a function which gives us the instantaneous rate of change (or slope) of the curve at any given point.

2. And now its time for.. The Lame Joke of the day.. What did the turkey say to the computer? Google, Google, Google.

3. AP Calculus Warm up 9.3.14 What is the derivative of a function? The derivative of a function is also a function which gives us the instantaneous rate of change (or slope) of the curve at any given point.

4. Secant line = Average rate of change

5. The secant line becomes the tangent line

6. Tangent line = instantaneous rate of change.

7. Rates of change Average rate of change slope of secant line Use slope formula Instantaneous rate of change slope of Tangent line Use derivative

8. A beaker is being heated continuously over time (t) and is modeled by the twice differentiable function C (t). The table above gives selected values of the temperature in degrees Celsius. Estimate the instantaneous rate of change of the temperature at 14 minutes. Indicate units of measure. Time (t) minutes 0 2 9 20 35 Temperature C (t) in degrees Celsius 60 63 89 102 110

9. Find the Instantaneous Rate of change at (2, 12) f (x) 3x2

10. What is the equation of the tangent line of f(x) at ( 2, 12) Slope = 12 f (x) 3x2

11. Differentiation The process of finding the derivative of a function If a function has a derivative, it is called differentiable

12. What makes a function differentiable? 1. It MUST be continuous. 2. The derivative from the left must equal the derivative from the right.

13. Most common times when a derivative does not exist. 1. At a sharp corner 2. At a vertical tangent line or asymptote

14. KEY POINT Differentiability implies continuity, But continuity DOES NOT imply differentiability.

15. Notation

16. Review The derivative of a function is also a function which gives us the instantaneous rate of change (or slope) of the curve at any given point.

17. The graph of f is given below. Sketch out the graph of f