1. 2.7/2.8 Tangent Lines & Derivatives

3. P = (a, f(a)) with a slope of,
Tangent Line
The line through
P = (a, f(a)) with a slope of,

4. The derivative of f at a is f(a):

5. Find the slope of the tangent line at x = 4.
Ex 1:
Find the slope of the tangent line at x = 4.
Find the slope of the tangent line at x = a.

6. Ex 2: Find f (x) and use it to find the equation of the tangent line to f (x) at x = 9.

7. Average Velocity: the slope of the secant line between two points, P & Q, on a position curve.
Don’t Forget!

8. Instantaneous Velocity: the slope of the tangent line to a point P on a position curve.
Don’t Forget!

9. 2.7 pg # 1, 2, 3, 5, 11, 13, 15, 25

10. NOTE: f(x) = the slope of the tangent line at x
s(a) = the velocity at t = a
| s(a)| = the speed at t = a

11. Ex 3: The displacement (in meters) of a particle moving in a straight line is given where t is seconds.

12. a) Find the average velocity on the interval [4, 4
a) Find the average velocity on the interval [4, 4.5] b) Find the instantaneous velocity at t = 4 c) Find the speed at t = 5

13. Let P(t) be the population of the United States at time t, in years.
Ex 4:
Let P(t) be the population of the United States at time t, in years.
What are the units of P(t)?
What does P(1992) = 2.7 million mean?

14. Ex 5:
Sketch the graph of the function g for which g(0) = 0, g(0) = 3, g(1) = 0, & g(2) = 1.

15. Ex 6:
The limit represents the derivative of f at some number a. State f and a.

16. The limit represents the derivative of f at some
Ex 7:
The limit represents the derivative of f at some
number a. State f and a.

17. 2.8 pg # 1, 3, 4, 5, 13, 15, 19 – 24 all, 25, 27, 28, 31