1. DO NOW: Use Composite of Continuous Functions THM to show f(x) is continuous.

2. HW: Pg. 92-93 #1-3, 9-12, 16, 18, 23 2.4 – Rates of Change and Tangent Lines

3. Tangents DEFINITION The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope Provided that this limit exists.

4. Example 1 Find an equation of the tangent line to the parabola y = x 2 at the point P(1,1).

5. Tangent Line (2 equivalent statements) (SLOPE OF A CURVE AT A POINT (a,f(a)) )

6. NORMAL TO A CURVE The normal line to a curve at a point is the line perpendicular to the tangent at that point.

7. Example 1 - Extended Find an equation for the normal line to the parabola y = x 2 at the point P(1,1).

8. Example 2 Find an equation of the tangent line to the hyperbola y = 3/x at the point (3,1).

9. Example 2 (Solution)

10. Average Velocity Average velocity = The function f that describes the motion is called the position function of the object.

11. Instantaneous Velocity Now suppose we compute the average velocities over shorter and shorter intervals [a, a+h] That is -> we let h approach 0. The instantaneous velocity v(a) at time t = a to be the limit of the average velocities:

12. Average vs Instantaneous Velocity Average Velocity (secant line): Average velocity = Instantaneous Velocity (tangent line):

13. Example 3 Consider a ball dropped from a height of 450 m. Find: The velocity of the ball after 5 seconds How fast the ball is traveling when it hits the ground

15. Avg vs Instantaneous Rate of Change The average rate of change of y with respect to x over the interval [x 1, x 2 ]: The instantaneous rate of change of y with respect to x at x = x 1

16. Example 4 Temperature readings T (in C) were recorded every hour starting at midnight on a day in Whitefish, Montana. The time x is measured in hours from midnight. Find the average rate of change of temperature with respect to time – From noon to 3 PM – From noon to 2 PM – From noon to 1 PM Estimate the instantaneous rate of change at noon.

17. Example 4 (solution) Find the average rate of change of temperature with respect to time – From noon to 3 PM – From noon to 2 PM – From noon to 1 PM Estimate the instantaneous rate of change at noon.