Calculus Assignment
1. In 1997 the cost of a steak dinner at Your Local Beef Bar was $13.49. In 2006 that same steak dinner cost $ 19.95. Assuming an exponential growth model, what would be the cost of that steak dinner in 2020?
2. If bacteria grows exponentially, and it takes 12 hours for bacteria to double in number, what is the growth rate?
3. Iodine-131 decays at a rate of 8.6 % per day. What is its half-life (in days)?
4. A company introduces a new product in a certain city. The product is advertised on local television and it is found that the percent of people [P(t)%] who bought the product after t ads had been shown on TV is given by P(t) = 100/1 + 19e-0.15t.
a) What percentage of the population buys the product after the ad has been shown 5 times? After 20 times?
b) Find the rate of change of P(t)?
c) How fast is the percentage of buyers changing when t = 5? When t = 20?
d) How many times does the ad need to run so that 80% of the people buy the product?
5. An investment grows at a rate of 0.85 % compounded continuously. How long (in years) will it take for the investment to triple in value?
6. Find the derivative of
y = (e5x - e-5x)/(e5x + e-5x)
7. Find the derivative of yex - x e2y = x y.
8. A warm object is placed in air of temperature 35 o and the object cools from 120o to 60o in 40 minutes. Find the temperature of the object after 100 minutes.
9. Consider the demand function q = D(x) = √(300-x).
a) Write the elasticity function.
b) Find the elasticity at x = $ 250, and state whether the demand is elastic or inelastic.
c) Find the price for which the total revenue is a maximum.
d) What is the maximum revenue?
10. Find the exact (no decimals) equation of the tangent line to the function
f (x) = ex + xe + x + e at x = 1.
11. Find the minimum and maximum values of f (x) = x3e-3 x on [0, 10].
12. Find the derivative of y = (e3 x2) ln (x6)
13. Find the equation of the line perpendicular to y = x ln x and parallel to x + 2 y - 1 = 0.