Derivatives of Sine and Cosine Using the Creating the Derivative mathlet, select the (default) function f (x) = sin(x) from the pull-down menu in the lower left corner of the screen. Do not check any of the boxes. Move the slider or use the >> button to display the graph of the sine func tion. a) For approximately what values of x is the slope of f (x) = sin(x) equal to 0? b) At approximately what values of x is the slope of f (x) = sin(x) largest? c) For each of the values you listed in (b), is the slope positive or negative? d) Use the information you have collected to sketch the graph of f � (x), the derivative of the sine function. e) Check the box next to the red f � (x) to check your work. Solution a) For approximately what values of x is the slope of f (x) = sin(x) equal to 0? The slope of the graph is 0 wherever the tangent line is horizontal. For the sine function, the tangent line is horizontal at the peaks and troughs of its 3π π π 3π graph: x = ..., − , − , , , ... 2 2 2 2 b) At approximately what values of x is the slope of f (x) = sin(x) largest? The slope is the largest when the graph is the steepest. For the sine function, the graph is steepest when crossing the x-axis, at x = ... − 2π, −π, 0, π, 2π... c) For each of the values you listed in (b), is the slope positive or negative? The slope of the graph of the sine function at the x-intercepts alternates between positive and negative as the graph goes up and down across the axis. The slope is positive at x = ... − 2π, 0, 2π... and negative when x = ... − π, π, 3π, ... d) Use the information you have collected to sketch the graph of f � (x), the derivative of the sine function. The graph you drew should have peaks at −2π, 0 and 2π and troughs at 3π π π 3π −π and π. Its x-intercepts should occur at x = ..., − , − , , , ... Be 2 2 2 2 cause the graph of sin(x) is smooth and continuous, your graph should be connected. It turns out that the graph of the derivative of sin(x) is also smooth, but it’s hard to deduce this just by looking at the graph. e) Check the box next to the red f � (x) to check your work. Perhaps you already knew that the derivative of sin(x) is cos(x). Remember this exercise next time you have trouble remembering whether the derivative of cos(x) is positive or negative sin(x), whether the graph of sin(x) goes up ﬁrst or down, or what the graph of cos(x) looks like.