Graphs of Functions & Their Derivatives (College Board AP® Calculus AB)

The function is defined on the closed interval [-5, 5]. The graph of , the derivative of , consists of two line segments and a semicircle, as shown in the figure.

Does have a relative minimum, a relative maximum, or neither at ? Give a reason for your answer.

On what open intervals, if any, is the graph of concave down? Give a reason for your answer.

It is known for , where is a positive non-zero constant, that the rate of change is . For a particular value of , the maximum value of is 40. Find the value of .

Let and be the functions defined by and . The graphs of and , shown in the figure above, inersect at and , where and .

For let be the vertical distance between the graphs of and . Is increasing or decreasing at ? Give a reason for your answer.

Selected values of the differentiable function are shown in the table above.

Justify why there must be at least one value of for , at which is equal to zero.

Let be the continuous function defined on whose graph, consisting of three straight line segments and a semicircle, is given above. Let be the function .

Find the -coordinate of each point at which the graph of has a horizontal tangent line. For each of these points, determine whether has a relative minimum, relative maximum or neither a minimum or maximum at the point. Justify your answers.

For , find all the values of for which the graph of has a point of inflection. Explain your reasoning.

Let be a differentiable function. On the interval , the graph of , the derivative of , consists of a semicircle and two line segments, as shown in the figure above.

Find the -coordinates of all points of inflection of the graph of for . Justify your answer.

Let be the function defined by . On what intervals, if any, is decreasing for ? Show the analysis that leads to your answer.

Given that , find the absolute maximum value of the function defined in part (b) on the interval . Justify your answer.

Let be a continuous function defined on the closed interval . The graph of , consisting of four line segments, is shown above. Let be the function defined by

Find the average rate of change of on the interval [-6, 6]. Does the Mean Value Theorem guarantee a value , for which is equal to this average rate of change? Justify your answer.

Birds enter an area of woodland at a rate modeled by the function . Birds leave the area at a rate modeled by the function . Both and are measured in birds per hour, and is measured in hours since midnight .

At what value of , for , is the greatest number of birds in the area of woodland? Justify your answer.

Is the rate of change in the number of birds in the area of woodland increasing or decreasing at noon ? Explain your reasoning.

The continuous function is defined on the closed interval . The figure above shows the graph of , consisting of two line segments and a quarter of a circle centered at the point .

The function is given by . Find the absolute maximum value of on the interval . Justify your answer.

The rate at which a hot object cools down is proportional to the difference between its current temperature and the ambient temperature of the surrounding environment. If is the temperature of the object in degrees Farenheit at time hours, and the surrounding temperature is constant at °F, the rate of change of the temperature is given by:

Initially, at , the temperature of the object is °F.

Find in terms of . Use to explain why the graph of cannot resemble the following graph.