Exam 4: Applications of the Derivative

Choose the one alternative that best completes the statement or answers the question. -The graph below shows the first derivative of a function y = f(x). Select a possible graph f that passes through the point P.

Use differentiation to determine whether the integral formula is correct. -

Use l'Hopital's Rule to evaluate the limit. -

Find the limit. -

Find the absolute extreme values of the function on the interval. -f(x) = , -1 $\le$ x $\le$ 8

Find the extreme values of the function and where they occur. -y = - 3 + 6x - 8

Find the linearization L(x) of f(x) at x = a. -f(x) = x + , a = 3

Provide an appropriate response. -Consider the quartic function f(x) = a + b + c + dx + e, a ≠ 0. Must this function have at least one critical point? Give reasons for your answer. (Hint: Must for some x?) How many local extreme values can f have?

Solve the problem. -A = $\pi$ , where r is the radius, in centimeters. By approximately how much does the area of a circle decrease when the radius is decreased from 5.0 cm to 4.8 cm? (Use 3.14 for $\pi$ .)

Use a calculator to compute the first 10 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places. -f(x) = 1 - ln(x + 8); = -6

Use Newton's method to find an approximate answer to the question. Round to six decimal places. -Where is the first local maximum of f(x) = 3x sin x on the interval (0, $\infty$ ) located?

Find the extreme values of the function and where they occur. -y =

Use Newton's method to approximate all the intersection points of the pair of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. Round to six decimal places. -y = cos x and y = 3x

Find the linearization L(x) of f(x) at x = a. -f(x) = cos x, a = 0

Find the extreme values of the function and where they occur. -y = + 2x

Find the absolute extreme values of the function on the interval. -f(x) = - x, -4 $\le$ x $\le$ 2

Graph the equation. Include the coordinates of any local extreme points and inflection points. -y = 3x² + 24x

Use l'Hopital's Rule to evaluate the limit. -

Provide an appropriate response. -It took 20 seconds for the temperature to rise from 4° F to 166° F when a thermometer was taken from a freezer and placed in boiling water. Although we do not have detailed knowledge about the rate of temperature increase, we can know for certain that, at some time, the temperature was increasing at a rate of ° F/sec. Explain.

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. -f(x) = ,