Exploration for introduction to FTC Name: Suppose you are given the graph of a continuous function such as f(x) = x(x -3)(x -5) over the interval [a,b] . Let a>0 and b>5 (choose x randomly and do not choose an integer) 1. Given the graph of f . Choose any x greater than a in the interval [a,b] and mark it on the x-axis. AF 2. Using only vertical line segments, shade in the region between the graph of f and the x-axis from a to x. (Some shading might be below the x-axis.) 3. Your shaded region represents a definite integral. Explain why this integral can be written as J f(1)di . (Why is it not written as . f(x)dx ?) * is one ofyour bounds 4a. Compare your picture with your classmate(s). How does your shaded area compare to your classmate? What caused them to be different? They have different a point less or mere shaded 4b. Your shaded area is unique and depends on the x you chose. Therefore set up an integral defining your unique area. Call it F(x) . X S F ( x ) = f ( t > at 5. Recall the F'(x) is the limit of AF as Ax gets smaller and smaller. Represent AF in your picture by drawing AX one more vertical shading segment to the right of your last one. Shade your AF darkly (so its looks different than the other shaded part). Describe what does AF represents. 6. Represent Ax in your picture. What does Ax represent? 7. What is the height of your vertical segment?8. Set up 2 equations for the additional area and then use substitution to show why Newton and Leibniz concluded that F'(x) = f(x). 9. Copy the part of the Fundamental Theorem of Calculus that represents what you discovered. 10. Give an example of using this part of the theorem. (hint: start with a rate of change)