Exam 4: The Derivative in Graphing and Applications

Find the relative extrema for f(x) = 12x^2/3 - 3x.

Answer true or false. f(x) = ln 4x has a point of inflection.

Answer true or false. f(x) = e^5x has a point of inflection.

Answer true or false. A point that has an x-coordinate where f '(x) = 0 is a point of inflection.

Sketch the graph of y = (x + 3)^2/3. Find any stationary points, inflections points, and cusps which may or may not exist.

Let s = 3t² - 6t- 6 be the position function of a particle. Find the maximum speed of the particle during the time interval 1 $\le$ t $\le$ 4.

The position function of a particle is given by s(t) = 4t³ -4t + 7. Find the velocity function.

Find the relative extreme values for on the interval [-10, 10] and determine where those values occur.

A slice of pizza, in the form of a sector of a circle, is to have a perimeter of 14 inches. What should be the radius of the pan to make the slice of pizza largest. (The area of a sector of a circle, where $\theta$ is the central angle in radians and the arc length along a circle is S = r $\theta$ with $\theta$ in radians.)

An open cylindrical trash can is to hold 11 cubic feet of material. What should be its dimensions if the cost of material used is to be a minimum? [Surface Area, S = $\pi$ r² + 2 $\pi$ rh where r = radius and h = height.]

Does satisfy the hypothesis of the Mean-Value Theorem over the interval [-5, 5]? If so, find all values of C that satisfy the conclusion.

Answer true or false. A graphing utility can be used to show that Rolle's Theorem can be applied to show that f(x) = (x - 8)² has a point where f '(x) = 0.

Sketch the graph of y = 4 - 2x - x². Find the relative maximum.

Sketch the graph of y = x⁴ -14x² + 50. Find any stationary points and any points of inflection.

Find the extreme values for on the interval [-3, 3] and determine where those values occur.

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 70?

Determine the x-coordinate of each critical point of .

Answer true or false. on [-3, 3] satisfies the hypotheses of Rolle's Theorem.

Find the extreme values for on the interval (0, 4) and determine where those values occur.

A cyclist starts from rest and travels 25 miles along a straight road in 30 minutes. Use the Mean-Value Theorem to show that at some instant during the trip his velocity was exactly 50 miles per hour.