l' l Prior Knowledge Questions (Do these BEFORE using the Gizmo.) IIIIE! 1. The slope of a line tells you the rate of change of y, relative to x. What is the slope of the line shown here? 2. The graph of y = x2 3 is shown to the right. In general, how is the \"slope\" of a parabola different from the slope of a line? Gizmo Warm-up The derivative of a function f(x), denoted f'(x), is the rate of change of the function at a point. If the graph is a curve, the derivative is the slope of the tangent line. In the Graphs of Derivative Functions Gizmo, you will nd the derivatives of several functions, and explore the graphs of derivative functions. At the top left of the Gizmo, select Linear function. Set a to 3 and b to 5 to graph y = 3x 5. (To quickly set a specic value, type the value in the text box, and hit Enter.) 1. What do you think the derivative of fix) = 3x- 5 is? Why? Select Show derivative to check your answer. 2. Vary b. What do you notice about how the vertical shift effects the derivative? Activity A (continued from previous page) 3. With Show tangent line still selected, turn off Show derivative. Select Cubic function, and graph f(x) = x3 + 5x2 + 3x - 4. A. The graph should look like the one shown. Where is the "slope" of f(x) = x3 + 5x2 + 3x - 4: *Use the terms local max and local min. in your answer positive? negative? - 6 -2 2 1-21 B. Drag the red point along the curve, and watch how the dark blue tangent line changes. *Use the terms local max and local min in your answer - 6 Where is the slope of the tangent line zero? Explain why this makes sense using your knowledge of tangent slopes. 4. Graph f(x) = 0.4x3 + 2. Select Show derivative, and drag the red point along the curve. A. What does the light blue curve tell you about the derivative of f(x) = 0.4x3 + 2? B. Vary a, b, c, and d. What type of function is the derivative of a cubic? C. Vary a, b, c, and d, and look for a pattern in the derivative. In general, what is the derivative of f(x) = ax3 + bx2 + cx + d? f(x) =