Exam 4: Applications of Differentiation

One model of a company's costs is given by the short-run Cobb-Douglas cost curve $C ( x ) = K x ^ { \frac { 1 } { a } } + F$ , where a is a positive constant, F is the fixed cost, and K measures the available technology.The average cost of manufacturing a quantity x of a good is defined to be $A ( x ) = \frac { C ( x ) } { x }$ .(a) What is the average cost function for the Cobb-Douglas model? (b) If average cost is minimized when it is equal to the marginal cost, find the quantity x which yields the minimum average cost.?

Find, correct to six decimal places, the root of $\sin x + x - 1 = 0$ .

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow \infty } \frac { 2 x + \cos x } { x }$ , then find the limit.

Find all critical numbers for the function $f ( x ) = \sin x + \cos x \text { for } 0 < x < \pi$ .

A kite is flying 100 feet above the ground at the end of a string 125 feet long. The girl flying the kite lets out the string at a rate of 1 foot per second. If the kite remains 100 feet above the ground, how fast is its horizontal distance from the girl increasing?

Find the minimum and maximum values of $y = x ^ { 3 } - 9 x + 8$ on the interval $[ - 3,1 ]$ .

If the cost of manufacturing x units per day of a certain commodity is $c = 600 + 0.04 x + 0.002 x ^ { 2 }$ and if each unit sells for $10.00, what daily production will maximize the profit?

Sketch the graph of $y = x ^ { \frac { 1 } { 3 } }$ . Clearly the only x-intercept is zero. However, Newton's method fails to converge here. Explain this failure.

The length of a rectangle is increasing at the rate of 2 feet per second, while the width is increasing at the rate of 1 foot per second. When the length is 4 feet and the width is 3 feet, how fast is the length of the diagonal of the rectangle increasing?

Use Newton's method to find the root of $6 x ^ { 3 } - x ^ { 2 } - 19 x + 6 = 0$ that lies between 0 and 1.

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

Use Newton's method with the initial approximation $x _ { 1 } = 2$ to find $x _ { 2 }$ , the second approximation to a root of the equation $x ^ { 5 } - 34 = 0$ .

Find all critical numbers for the function $f ( x ) = \sqrt [ 3 ] { 9 - x ^ { 2 } }$ .

Let $a > 1$ and $b > 1$ . Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { a ^ { x } - b ^ { x } } { x }$ .

A ball is thrown directly upward at a speed of 64 feet per second from a cliff 80 feet above the ground.(a) Find expressions for the velocity and height of the ball t seconds after it was released.(b) At what time does the ball reach its highest point? How high above the ground at the base of the cliff does it reach? (c) When does the ball strike the ground at the base of the cliff? What is its velocity at that instant?

Given that $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ has critical numbers at $x = 1 \text { and } x = 3 \text {, }$ find a and b.

The length of a rectangle is decreasing at the rate of 1 foot per second, but the area remains constant. How fast is the rectangle's width increasing when its length is 10 feet and its width is 5 feet?

Two straight roads intersect at right angles in Newtonville. Car A is on one road moving toward the intersection at a speed of 50 miles/h. Car B is on the other road moving away from the intersection at a speed of 30 miles/h. When car A is 2 miles from the intersection and car B is 4 miles from the intersection, how fast is the distance between the cars changing?

How many points of inflection does $f ( x ) = x ^ { 3 } e ^ { - x }$ have?

Use Newton's method to approximate the root of $f ( x ) = x ^ { 4 } - 4 x - 2$ that lies between $x = 1$ and $x = 2$ .