Exam 2: Functions, Limits, and the Derivative

Find the one-sided limit. ​

Find function f such that , , and

Find the limit, if it exists. ​ ​

Find for the function. Simplify your answer. ​ ​

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

Let , , and . Find the rule for the function .

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

Find the one-sided limit, if it exists. ​ ​

The volume of a spherical cancerous tumor is given by the function where r is the radius of the tumor in centimeters. By what factor is the volume of the tumor increased if its radius is quadruple?

An apple orchard has an average yield of 36 bushels of apples/tree if tree density is 21 trees/acre. For each unit increase in tree density, the yield decreases by 2 bushels. Letting x denote the number of trees beyond 21/acre, find a function in x that gives the yield of apples. ​

Find the limit. ​

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. ​ Round your answers to the nearest whole number. ​ How many homes had digital TVs at the beginning of 2000? ​ __________ ​ How many homes will have digital TVs at the beginning of 2005? ​ __________

Use the graph of the function f to find , and , if the limit exists. If it does not, answer none. ​ ​ = __________ ​ = __________ ​ = __________ ​

Let , . Find the rule for the function .

Find the domain of the function. ​ ​

Use the intermediate value theorem to find the value of c such that . ​ ​ c = __________

A rectangular box is to have a square base and a volume of 20 ft³. 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. ​

Use the method of bisection to find the root of the equation , accurate to two decimal places if necessary. ​ __________

Find the limit. ​

In the following figure, f(t) gives the population P​₁ of a certain bacteria culture at time t after a portion of bactericide A was introduced into the population at t = 0. The graph of g gives the population P​₂ of a similar bacteria culture at time t after a portion of bactericide B was introduced into the population at t = 0. ​ ​ Which population is decreasing faster at t​₁? ​ Population __________ ​ Which population is decreasing faster at ​t​₂? ​ Population __________ ​ Which bactericide is more effective in reducing the population of bacteria in the short run? ​ Bactericide __________ ​ Which bactericide is more effective in reducing the population of bacteria in the long run? ​ Bactericide __________