Derivatives (229 problems)

Find the domain of the following function.

Explain the Chain Rule in your own words. Find an example of a function in which the use of the rule is necessary.

Differentiate by Logarithmic Differentiation

Farmer Simmons wants to fence in rectangular plot of area 1800 ft². He wants also to use additional fencing to build two internal divider fences, both parallel to the same two outer boundary sections. What is the minimum total length of fencing that this project will require?

A rectangular box, with a top, is to have volume 800 in³, and its base is to be exactly four times as long as it is wide. What is the minimum possible surface area of such box?

A right circular cone of radius r and height h has a slant height . What is the maximum possible volume of a cone with a slant height 10?

Two towns are located near the straight shore of a lake. Their nearest distances to points on the shore are 1 mile and 2 miles respectively, and these points on the shore are 6 miles apart. Where should a fishing pier be located to minimize the total amount of paving necessary to build a straight road from each town to the pier?

A Norman window has the shape of a semi-circle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 50 feet?

A cylinder is inscribed in a right circular cone of height 30 and radius (at the base) equal to 10. What are the dimensions of such a cylinder which has maximum volume?

We need to design a cylindrical can with radius r and height h. The top and bottom must be made of cooper, which will cost 2 cents/in². The curved side is to be made of aluminum, which will cost 1 cent/in². We seek the dimensions that will maximize the volume of the can. The only constraint is that the total cost of the can is to becents.

For speeds between 35 and 65 miles per hour, a diesel truck gets 480/x miles per gallon when driven at a constant speed of x miles per hour. Diesel gasoline costs $2.80 per gallon and the driver is paid $14 per hour. What is the most economical constant speed between 35 and 65 miles per hour at which the truck should be driven?

A manufacturer has been selling lamps at $16 apiece, and at this price, consumers have been buying 5000 lamps per month. The manufacturer wishes to raise the price and estimates that for each $1 increase in the price 400 fewer lamps will be sold each month. The manufacturer can produce the lamps at a cost of $9 per lamp. At what price should the manufacturer sell the lamps to generate the greatest possible profit and what is the maximum profit that would be attained under these conditions?

A colony of bacteria is estimated to have a population of million t hours after the introduction of a toxin designed to kill the bacteria.

a. What is the population at the time the toxin is introduced?

b. At what time is the population of the bacteria colony the greatest and what is that population?

c. At what time is the bacteria population decreasing at a rate of approximately 250,000 bacteria per hour?

You are standing on the west bank of a 200 meter wide frozen river that runs north and south. You plan to walk to a point that is 700 meters south of your position on the opposite side of the river. You know that you can walk across the frozen river at 10 meters per minute and that you can walk on solid land at 5 meters per minute. To what point on the far bank of the frozen river should you walk to minimize the amount of walking time and how long will you be walking along the shoreline?

Minimizing Average Cost. Suppose the total cost function for manufacturing a certain product is C(x) = 0.2(0.01x² + 120) dollars, where x represents the number of units produced. Find the level of production that will minimize the average cost

Apply the first derivative test to classify each of the critical points of the function (maximum or minimum or not an extremum).

Farmer John wants to fence in a rectangular plot of area 2400 ft². He wants also to use additional fencing to build an internal divider fence parallel to two of the boundary sections. What is the minimum total length of fencing that this project would require? Apply the first derivative test to verify that your answer yields the global minimum

An oil can is to have volume 1000 in³ and is to be shaped like a cylinder with a flat bottom but capped by a hemisphere. Neglect the thickness of the material of the can, and find the dimensions that will minimize the total amount of material needed to construct it.

Find the absolute maximum and absolute minimum ofon the given closed interval.

(a)

(b)

Show that the given equation has exactly one solution in the indicated interval

(a)

(b)