Exam 4: Applications of Differentiation

Find the minimum value of the function $f ( x ) = x \ln x$ .

A particle starts at the origin and moves along the parabola $y = x ^ { 2 }$ such that its distance from the origin increases at 4 units per second. How fast is its x-coordinate changing as it passes through the point $( 1,1 )$ ?

A company estimates that the initial cost of producing a commodity would be $680: The additional cost of manufacturing each item can be modeled by the function $m ( x ) = 3 x + 0.01 x ^ { 2 }$ , where x is the number of items produced and m is the manufacturing cost, in dollars. A market survey indicates the demand function can be estimated by $p ( x ) = 12 - 0.002 x$ .(a) Find the production level that maximizes profit. What is the maximum profit? (b) What should the selling price be to maximize the profit?

A certain company has cost function $C ( x ) = 1000 + 2 x + 0.01 x ^ { 2 }$ and demand function $P ( x ) = 74 - 0.02 x$ . Find the production level x that will yield a maximum profit.

A direction field for a function f is given below. Use this to sketch the antiderivative F that satisfies $F ( 0 ) = 1$ .

Give an example of a function that is defined on a closed interval but which does not have an absolute maximum on the interval. Does the existence of this function contradict the Extreme Value Theorem? Why or why not?

A student 5 feet tall is 10 feet away from a lamppost 15 feet tall. She is walking away from the lamppost at 2 feet per second. How fast is the tip of her shadow moving away from the foot of the lamppost?

Use the graph to state the absolute and local maximum and minimum values of the function.

If a diagonal of a square increases at the rate of $1 \mathrm { inch } / \mathrm { s }$ , how fast is the area changing when the side of the square is 13 inches?

The graph of the derivative f(x) is given. Find the critical numbers of f and determine whether f has a local maximum and minimum at each critical number.

Farmer Brown wants to fence in a rectangular plot in a large field, using a rock wall which is already there as the north boundary. The fencing for the east and west sides of the plot will cost $3/yard, but she needs to use special fencing which will cost $5 / yard on the south side of the plot. If the area of the plot is to be 600 square yards, find the dimensions for the plot which will minimize the cost of the fencing. Dimensions below are listed east by south.

A car is traveling at 50 miles per hour when the brakes are fully applied, producing a constant deceleration of 22 feet per second $2$ . What is the distance covered before the car comes to a stop?

Given $f ( x ) = x ^ { 2 } - 3$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

At what value of x does the function $f ( x ) = 3 x - \sqrt [ 3 ] { x }$ change from increasing to decreasing?

The pressure P (in kilopascals), volume V (in liters), and temperature T (in degrees Kelvin), of a mole of an ideal gas are related by the equation $P V = 8.31 T$ . Find the rate at which the pressure in a container of gas is changing when the temperature is $300 ^ { \circ } \mathrm { K }$ and increasing at a rate of $0.1 ^ { \circ } \mathrm { K } / \mathrm { s }$ and the volume is 100 liters and increasing at a rate of $0.2 \mathrm {~L} / \mathrm { s }$ .

A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3 feet per second. Does the area also increase at a constant rate? Explain your answer.

A company has a cost function $C ( x ) = 1000 - 10 x + x ^ { 2 }$ . Find the marginal cost of producing 100 units.

Is it possible for the velocity of an object to be zero at the same time that its acceleration is not zero? Explain.

Given $f ( x ) = x ^ { 5 } - 3$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

The cost of operating a bus between Moose Jaw and Saskatoon is $100 + 5 x$ , where x is the number of passengers. If a ticket costs $20, there will be 10 passengers, while if a ticket costs $15, there will be 20 passengers. Assuming the demand function is linear, what is the maximum profit?