2.2c - The Derivative Function (f 0 ! f ) (Going Backwards) Miss. Gomero Calculus Introduction OBJECTIVE: To do the reverse of what we did in the previous lesson: the graph of the derivative will be given and we will sketch a possible graph of the original function. To do this, we must pay attention to four behaviors of the graph of y = f 0 (x). f 0 (x) = 0. f 0 (x) > 0 or f 0 (x) < 0. f 0 (x) is increasing or f 0 (x) is decreasing. f 0 (x) has a maximum or minimum. Our goal is to translate these four behaviors of the f 0 (x) graph to the corresponding behaviors of f (x) graph. Gomero 2.2c - The Derivative Function (f 0 ! f ) (Going Backwards) Important Properties to Note Properties: Relationship b/w f’ and f graphs If f 0 (a) = 0, then the graph of y = f (x) has a horizontal tangent line at x = a. i.e., y = f (x) has a max or min at x = a. If f 0 (x) > 0 over an interval, then y = f (x) is increasing over that interval. If f 0 (x) < 0 over an interval, then y = f (x) is decreasing over that interval. If f 0 (x) is increasing over an interval, then y = f (x) is concave up over that interval. If f 0 (x) is decreasing over an interval, then y = f (x) is concave down over that interval. If the f 0 -graph has a max/min at x = a, then the f -graph has an inflection point at x = a. Gomero 2.2c - The Derivative Function (f 0 ! f ) (Going Backwards) Example 1 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = 3x + 3 y x Example 2 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x 2 + 4x 3 Example 3 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x3 + 7x 3 20 3 y x Practice 1 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x2 2x + 4 Practice 2 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x 2 + 2x + 3 Practice 3 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x 3 + 2x 2 + 3x + 4 Practice 4 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x 3 + 2x 2 + 3x + 4 Practice 5 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). f 0 (x) = x 3 + 5x Example/Practice 6 The graph of y = f 0 (x) is given. In the same coordinates system, sketch a possible graph of y = f (x). Assume that f (0) = 1 and f (1) = 1.5. HW: Complete hmwk#25 handout. Gomero 2.2c - The Derivative Function (f 0 ! f ) (Going Backwards)