Riemann Sums & Definite Integrals (College Board AP® Calculus AB)

is a differentiable function. The table shown gives values of the function and its first derivative at selected values of .

Let be the function defined by . Find . Show the work that leads to your answer.

Is the function defined in part (a) increasing, decreasing, or neither at ? 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 .

On what open intervals is the graph of concave up? Give a reason for your answer.

The temperature of coffee in a mug at time is modeled by a strictly decreasing, twice-differentiable function , where is measured in degrees Fahrenheit and is measured in minutes. At time , the temperature of the coffee is . The mug of coffee is then left to cool, beginning at time . Values of at selected times for the first 15 minutes are given in the table above.

Use the data in the table to evaluate . Using correct units, interpret the meaning of in the context of this problem.

For , the average temperature of the water in the mug is . Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate . Does this approximation overestimate or underestimate the average temperature of the coffee over these 15 minutes? Explain your reasoning.

For , the function that models the coffee temperature has first derivative given by . Based on the model, what is the temperature of the coffee at time ?

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

Find the absolute minimum value of on the closed interval . Justify your answer.

The density of algae on the surface of a circular garden pond at a distance of meters from the center of the pond is given by an increasing, differentiable function , where is measured in kilograms per square meter. Values of for selected values of are given in the table above.

The total mass, in kilograms, of algae in the pond is given by the integral expression . Approximate the value of using a right Riemann sum with the four subintervals indicated by the data in the table.

Is the approximation found in part (a) an overestimate or underestimate of the total mass of algae in the pond? Explain your reasoning.