Exam 4: Applications of the Derivative

The following is the graph of Where do the points of inflection of occur, and on which intervals is concave up?

The following table describes the signs of the first and second derivatives of a function : 0 1 2 + + + + + 0 - - 0 + - - - - Which of the following is a possible graph of

Find the maximum value and the minimum value of the function in the given interval. A) B)

Find the maximum value and the minimum value of the function in the given interval A) B)

Determine the greater between and without using a calculator, by considering the function for

A factory produces 2000 products each month. The expenses on each product are and the income from each product is . For each additional product beyond the first 2000 products, the income for each product reduces by (for example, for 2001 products the income from each product is ). A) Find the profit in a month for which additional product are produced. B) How many additional products should be produced to obtain maximum profit in a month?

Given such that and for all , is for all ? (Hint: Use the MVT.)

The following table describes the signs of the first and second derivatives of a function : 0 1 2 0 0 0 0 If , sketch a possible graph of

Given the function is

Find the dimensions and the perimeter of the rectangle with maximum perimeter, inscribed in the ellipse .

A right circular cylinder is to be constructed with a volume of 400 . The material used to build the top and base of the cylinder costs . The material for the remainder of the cylinder costs . Find the radius value which will minimize the material cost of constructing the cylinder.

Estimate the roots of the equation to three decimal places using the linear approximation for .

Show that the equation has a solution in the interval and use Newton's Method to approximate it to within an error of at most .

A ball produced on an assembly line is supposed to have a volume of . Use linearization to estimate the maximum allowable error in the radius if the volume of the sphere must have an error of less than .

Find the maximum value and the minimum value of the function in the given interval or in its natural domain A) B) C)

A producer can sell instruments per week for dollars each, where . His cost for producing instruments is dollars. Find the amount of instruments that should be produced in a week to obtain maximum profit.

A cylinder of radius and height has surface area and volume . Find the dimensions of a cylinder with volume and minimal surface area.

True or False: Let . By Rolle's Theorem, has at most 3 real roots.

The following function has a local extremum at a point in the interval :

By Rolle's Theorem the equation has at most two roots.