2. Problem of the Day If f ''(x) = x(x + 1)(x - 2) 2, then the graph of f has inflection points when x = a) -1 onlyb) 2 only c) -1 and 0 onlyd) -1 and 2 only e) -1, 0, and 2 only

3. Problem of the Day If f ''(x) = x(x + 1)(x - 2) 2, then the graph of f has inflection points when x = a) -1 onlyb) 2 only c) -1 and 0 onlyd) -1 and 2 only e) -1, 0, and 2 only

4. Chap. 4: Integral Calculus Introduction ©2002 Roy L. Gover (www.mrgover.com) Connection between differential and integral calculus Example of an integral calculus application

5. Important Idea Differential Calculus solves the tangent line problem Instantaneous rate of change Related rates Maxima & minima Optimization

6. Important Idea Integral Calculus solves the area problem Area of a region Volumes of solids Work, force & fluid pressure Accumulated rates of change

7. The tangent line problem and the area problem are related… one is the inverse of the other Important Idea

8. “This discovery (called the Fundamental Theorem of Calculus) brought differential and integral calculus together to become the single most powerful insight mathematicians had for understanding how the universe works.” -Dan Kennedy, textbook author

9. Example Velocity is the tangent line at any point on the position curve. But...

10. Area is Distance d=vt Given a constant velocity of 40 mph, find distance.

11. Distance is area even when velocity is not constant.

12. Distance is area even if the velocity function is complex

13. For the remainder of this course, we will focus on ways to solve problems involving area under a curve. The process of finding area is called antidifferentiation or integration.