consider the function f(x) shown in the graph below. (note that you can click on the graph to get a larger…

consider the function f(x) shown in the graph below. (note that you can click on the graph to get a larger version of it, and that it may be useful to print that larger version to be able to work with it by hand.) carefully sketch the derivative function of the given function (you will want to estimate values on the derivative function at different x values as you do this). use your derivative function graph to estimate the following values on the derivative function. at x = -3 -1 1 3 the derivative is

consider the function f(x) shown in the graph below. (note that you can click on the graph to get a larger version of it, and that it may be useful to print that larger version to be able to work with it by hand.) carefully sketch the derivative function of the given function (you will want to estimate values on the derivative function at different x values as you do this). use your derivative function graph to estimate the following values on the derivative function. at x = -3 -1 1 3 the derivative is

Answer

Explanation:

Step1: Recall derivative - slope relationship

The derivative of a function at a point is the slope of the tangent line to the function at that point.

Step2: Analyze (x = - 3)

At (x=-3), the function (y = f(x)) is increasing. The tangent - line has a positive slope. Estimating from the graph, the slope of the tangent line at (x = - 3) is approximately (2).

Step3: Analyze (x=-1)

At (x=-1), the function (y = f(x)) is still increasing, but less steeply than at (x=-3). The slope of the tangent line (derivative) is positive and smaller than at (x=-3). Estimating, the derivative is approximately (1).

Step4: Analyze (x = 1)

At (x = 1), the function (y=f(x)) has a horizontal tangent line. So the slope of the tangent line (derivative) is (0).

Step5: Analyze (x = 3)

At (x = 3), the function (y = f(x)) is decreasing. The slope of the tangent line (derivative) is negative. Estimating from the graph, the derivative is approximately (-1).

Answer:

(x) Derivative
(-3) (2)
(-1) (1)
(1) (0)
(3) (-1)