Exam 2: Functions, Limits and the Derivative

Find the limit.

Find function f such that and

Let . Find the derivative of and the point on the graph of where the tangent line to the curve is horizontal.

The function P, whose graph follows, gives the prime rate (the interest rate banks charge their best corporate customers) as a function of time for the first 32 wk during a year.Determine the values of t for which P is discontinuous.

Evaluate h (2), where

Sketch the graph of the function and evaluate , if it exists, for the given value of .

A manufacturer has a monthly fixed cost of $45,000 and a production cost of $9 for each unit produced.The product sells for $51/unit. What is the cost function? __________ What is the revenue function? __________ What is the profit function? __________

The number of homes with digital TVs is expected to grow according to the function where t is measured in years, with t = 0 corresponding to the beginning of 2000, and f(t) is measured in millions of homes.How many homes had digital TVs at the beginning of 2000?

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. __________

Find the one-sided limit, if it exists.

An apple orchard has an average yield of 35 bushels of apples/tree if tree density is 25 trees/acre.For each unit increase in tree density, the yield decreases by 3 bushels.Letting x denote the number of trees beyond 25/acre, find a function in x that gives the yield of apples.

Find the limit, if it exists.

Let , .Find the rule for the function f + g.

Find function f such that and

Find for the function.Simplify your answer.

Use the vertical-line test to determine whether the graph represents y as a function of x.

Economists define the disposable annual income for an individual by the equation D = (1 - r)T, where T is the individual's total income and r is the net rate at which he or she is taxed.What is the disposable income for an individual whose income is $80,000 and whose net tax rate is 21%? $__________

A rectangular box is to have a square base and a volume of 20 .The material for the base costs , the material for the sides costs , and the material for the top costs .Letting x denote the length of one side of the base, find a function in the variable x giving the cost of constructing the box.

The 1980s saw a trend toward old-fashioned punitive deterrence as opposed to the more liberal penal policies and community-based corrections popular in the 1960s and early 1970s.As a result, prisons became more crowded, and the gap between the number of people in prison and the prison capacity widened.The number of prisoners (in thousands) in federal and state prisons is approximated by the function where t is measured in years, with t = 0 corresponding to 1983.The number of inmates for which prisons were designed is given by where C(t) is measured in thousands and t has the same meaning as before.Find an expression that shows the gap between the number of prisoners and the number of inmates for which the prisons were designed at any time t.

Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time t (in minutes) is described by the function Find the rate of population growth at min. rate = __________