Exam 12: Differentiation

The volume V of a spherical cell is given by V = π ,where r is the radius.Find the rate of change of volume with respect to the radius when r = 3 × cm.Give your answer in terms of π.

Find the slope of the tangent line to the curve y = at the point (0, ).

If y = ,find .

If the concentration of salt in a particular saline solution is given by the function S(t)= 1.565,find .

Two children are selling cups of lemonade for $0.25 each.If they sell q cups of lemonade,the total revenue r is given by r = 0.25q.Find the marginal revenue.

Find all points x for which the curve y = 2 + 3x + 5 has a tangent line that is perpendicular to the line

Find an equation of the tangent line to the curve y = (x - when x = 0.

The concentration of salt in a solution is given by C(t)= .Find the rate of change of the concentration,

Suppose that the total cost function for the production of x units of a product is given by Find the derivative of this function.

If a manufacturer's average cost equation is = + 400 - 25q + ,find the total cost function c,and the marginal cost function .What is the marginal cost when 8 units are produced?

By using the definition of a derivative,find f (x)where f(x)= .

The weekly output of a certain product is Q(x)= 200x + where x is the number of workers.Find the equation of the tangent line to this equation when there are 60 workers.

A smoothie stand usually sells 150 smoothies per day at $4 each.A business student's research tells her that for every $0.10 decrease in the price,the stand will sell 5 more smoothies per day.The revenue function for the smoothie stand is given by: R(x)= (4 - 0.1x)(150 + 5x)where x is the number of $0.10 reductions in price.Find ,the marginal revenue.

The revenue R from the sale of x units of a product is R = + 20x.The number of units sold after t weeks of advertising is x = 9 + .Find when t = 2.

If f(x)= ,then f'(x)=

Find if y = - + .

Let p = 20 + be the demand function for a manufacturer's product.Find the rate of change of price p per unit with respect to quantity q.How fast is the price changing with respect to q when q = 2? Assume that p is in dollars.

Differentiate: f(x)=

If y = - 2 ,the relative range of change y with respect to x when x = -1 is

If the profit for a shoe company (in millions of dollars)is given by the equation find the slope of the tangent to this curve at (1,4).