1. The Derivative

2. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine the slope of the tangent line to a given curve when shown a graph. Find the equation of the secant line through two points on a function. Find the equation of the tangent line to a given function through a specified point. Solve problems involving the derivative.

3. Newton’s Quotient and limits process for finding the derivative: Add h (where ) to a and compute f(a + h). Compute the change in the function value. Calculate and simplify Newton’s quotient for. Calculate the limit

4. Example 1 Estimate the slope of the tangent line to the curve in the graph to the right.

5. Example 2 Estimate the slope of the tangent line to the curve in the graph to the right.

6. Example 3 Estimate the slope of the tangent line to the curve in the graph to the right.

7. Example 4 Find each of the following for the function below:

8. Example 5 Find each of the following for the function below:

9. Example 6 Find each of the following for the function below:

10. Example 7 For the function below, find the equation of the secant line through the points x = 3 and x = 5.

11. Example 8 For the function below, find the equation of the tangent line through the point x = 3.

12. Example 9 For the function below, find the equation of the secant line through the points x = 25 and x = 36.

13. Example 10 For the function below, find the equation of the tangent line through the point x = 25.

14. Example 11 Find the x- values where the function to the right is not differentiable.

15. Example 12 For the function shown in the sketch to the right, give the intervals (or points) on the x-axis where the rate of change of the function f(x) with respect to x is positive, negative, and zero.

16. Example 13 Identify which graph below represent velocity and which represents distance. Graph 1 Graph 2

17. Example 14 The cost in dollars of producing x tacos is, for. a.Find the exact cost of producing the 101st taco. b.Use the marginal cost to approximate the cost of producing the 101st taco.