Exam 4: Applications of Differentiation

Determine whether Rolle's Theorem can be applied to on the closed interval . If Rolle's Theorem can be applied, find all values of c in the open interval such that . ​

Analyze and sketch a graph of the function . ​

A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded for the first two minutes is given by . Find . ​

Find the differential dy of the function . ​

The graph of f is shown below. For which value of x is minimum? ​ ​

Locate any relative extrema and inflection points of the function . Use a graphing utility to confirm your results. ​

Find the differential of the function . ​

Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes. ​

The height of an object t seconds after it is dropped from a height of 500 meters is . Find the time during the first 4 seconds of fall at which the instantaneous velocity equals the average velocity. ​

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval . ​

The ordering and transportation cost C for components used in a manufacturing process is approximated by , where C is measured in thousands of dollars and x is the order size in hundreds. According to Rolle's Theorem, the rate of change of the cost must be 0 for some order size in the interval . Find this order size. Round your answer to three decimal places. ​

Find the point on the graph of the function that is closest to the point . Round all numerical values in your answer to four decimal places. ​

Find the relative minima of on the interval by applying the First Derivative Test. Round numerical values in your answer to three decimal places. ​

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 38 feet. ​ ​

Use differentials to approximate the value of . Round your answer to four decimal places. ​

For the function : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ Use a graphing utility to confirm your results. ​

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity. ​

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . . Determine the speed of the ball bearing after t seconds. ​

Compare dy and for at with . Give your answers to four decimal places. ​

The sum of the perimeters of an equilateral triangle and a square is 8. Find the dimensions of the triangle and the square that produce a minimum total area. ​