2. Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications

3. Objectives:  To be able to use the derivative to analyze function  Draw the graph of the function based on the analysis  Apply the principles learned to problem situations

4. Example 1: Find the equation of the tangent line to the parabola, y = x 1/2 at a. (0,0) b. (1,1).

5. Example 2: Find the point on the parabola y = x 2 – 2x + 1 where the tangent line is horizontal.

6. Example 3: Locate the point where the tangent line is a. horizontal b. Vertical. x 3 + y 3 = 6xy

7. Example 4: Find the equation of the tangent line to the ellipse at the end of the latus rectum found in the first quadrant. Equation of ellipse is x 2 /16 + y 2 /25 = 1

8. Example 5: Find the equation of the tangent line at t = 0.x = t 2 + ty = t 2 - t

9. Example 6: Find the equation of the tangent line to y = 1 + x – 2x 3 at x = 1.

10. Example 7: Find the equation of the tangent line to y = x / (2x – 1) at x = 1.

11. Example 8: Find the equation of the tangent line to y = 2 /(3-x) 1/2 at x = - 1.

12. Example 9: Find the equation of the tangent line to y = x / (x 2 -3) at x = 2.

13. Example 10: Find the equation of the horizontal tangent line of x = t(t 2 – 3) y = 3(t 2 – 3)

14. Example 11: 2(x 2 +y 2 ) 2 = 25(x 2 –y 2 ) is an equation of a curve called the lemniscate. (a) find the equation of its tangent line at (3, 1). (b) Locate the points where the tangent line is horizontal.

15. Example 12: Find the equation of the tangent line at the given point. x 2 /16 - y 2 /9 = 1 at ( -5, 9/4 )

16. Example 13: Find the equation of the tangent line at the given point. y 2 = 5x 4 – x 2 at ( 1, 2 )

17. Example 14: Find the equation of both lines through ( 2, - 3) that are both tangent to the parabola y = x 2 + x.

18. Example 15: Where does the normal line to the parabola y = x – x 2 at ( 1, 0 ) intersect the curve a second time?

19. Example 16: Find the cubic function y = ax 3 + bx 2 + cx + d where its graph has horizontal lines ( -2, 6) and ( 2, 0).

20. Example 17: The vertex of a parabola is the point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of y 2 = -2x + 8

21. Example 18: The vertex of a parabola is the point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of y = x 2 + 8x – 5.

22. Example 19: The vertex of a parabola is the point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of x 2 – 4x + y = 0.

23. Example 20: Locate the points where the tangent line is horizontal. y = x 3 – x 2 – x + 1

24. Example 20: Locate the points where the tangent line is horizontal. y = 2x 3 – 3x 2 – 6x + 37