Consider the following function.

f(t) = t⁸

(a)Find the relative rate of change.

(b)Evaluate the relative rate of change at t = 1.Evaluate the relative rate of change at t = 8.

The demand function for beer is given below, where p is the retail price and

D(p)

is the demand in gallons per capita. For the demand function, find the elasticity of demand E(p) for any price p. [Note: You will find that demand is inelastic. This means that taxation, which acts like a price increase, is an ineffective way of discouraging liquor consumption, but is an effective way of raising revenue.]

D(p) = 5.945p^0.141

E(p) =

In 1998 one state raised cigarette taxes, with the result that the price of a pack of cigarettes rose from $2.60to $3.30. Demand then fell from55to 47.5 million packs. Use these data to estimate the elasticity of demand for cigarettes. (Hint: Find the percentage change in price and the percentage change in demand and use the formula

E= percent change in demand/percent change in price

Round your answer to the nearest percent.)

E= %

For the demand function

D(p),complete the following. D(p) = 90 3p

(a)Find the elasticity of demand

E(p) =

Because of the popularity and power of smartphones, demand for wireless communication is growing rapidly. Suppose monthly global data traffic is predicted to be 4.3e^0.583x petabytes, where x is the number of years since 2015. Find the relative growth rate in any year x. % per year

Consider the following function.

f(t) = e^t2

(a)Find the relative rate of change.

(b)Evaluate the relative rate of change at

t = 17.

The United States population (in millions) is predicted to be

P(t) = 317e^0.01t,

where t is the number of years after 2013. Find the instantaneous rate of change of the population in the year 2029. (Round your answer to two decimal places.) million people per year

Find the derivative of the function.

f(x) = x e^x

f'(x) =

According to a study, each additional year of education increases one's income by 15%. Therefore, with x extra years of education, your income will be multiplied by a factor of

1.15^x.

How many additional years of education are required to double your income? That is, find the x that satisfies

1.15^x = 2.

(Round your answer to one decimal place.) yr

After t days of advertisements for a new laundry detergent, the proportion of shoppers in a town who have seen the ads is

1 e^0.02t.

How long must the ads run to reach 50% of the shoppers? (Round your answer to the nearest whole number.) days

The Reed-Frost model for the spread of an epidemic predicts that the number I of newly infected people is

I = S(1 e^rx),

where S is the number of susceptible people, r is the effective contact rate, and x is the number of infectious people. Suppose that a school reports an outbreak of measles with

x = 8

cases, and that the effective contact rate is

r = 0.3.

If the number of susceptibles is

S = 550,

use the Reed-Frost model to estimate how many students will be newly infected during this stage of the epidemic. (Round your answer to the nearest whole number.) I =

A loan shark lends you $50 at 1% compound interest per week (this is a weekly, not an annual rate). (Round your answers to the nearest dollar.)

a)How much will you owe after 2 years? (Use 52 weeks per year.)$

(b)In "street" language, the profit on such a loan is known as the "vigorish" or the "vig." Find the shark's vig.$