IMP 3Name______

Small World Review Packet

The following problems are a review of the concepts discussed in our unit “Small World, Isn’t It?” You should work through the problems completely, showing all work.

Our unit started with the question of “How long will it take for everyone to be all scrunched together?” So, we started looking at different functions that could model growth. Our first function was the linear function. Use the idea of linearity to answer the following questions.

  1. My son Nathan wants to buy a Game Boy SP. He has $12 right now. He thinks that in 8 weeks he can have $60.
  2. Name the 2 points you can get from this information.
  1. Use those points to find the average rate of saving per week.
  1. Write the equation of this line in slope-intercept form.
  1. Write the equation of this line in point-slope form.
  1. Using your equation, find how much money Nate will have in 10 weeks.
  1. The Game Boy SP costs about $100. How many weeks will it take Nate to have enough money to buy it?
  1. Because I am such a wonderful mom, I decide to give Nate $8 to add to his original amount. It is still his job to save at the same rate. Write the equation of this new line in slope-intercept form.
  1. Graph the two equations and explain what you see.
  1. Using your new equation, find out how many weeks it will take him to have $100.
  1. We also discussed the slopes of horizontal and vertical lines. On the graph below, draw a horizontal line and a vertical line. Choose 2 points on the horizontal line and calculate the slope. Choose 2 points on the vertical line and calculate the slope.

Horizontal line points and slope

Vertical line points and slope

From here, our discussion turned to instantaneous rates of change. Use this idea to work through the following problems.

  1. On my summer vacation, I was bold or stupid enough to jump from a 36 foot cliff. The equation for that is h(t) = 36 – 16t2 .
  2. How high above the river was I one second after I jumped?
  1. How many seconds did it take me to hit the water?
  1. What was my average speed during the final half-second of my jump?
  1. What was my average speed during the final tenth of a second of my jump?
  1. What was my speed at the exact moment I hit the water?

We found that neither linear nor quadratic functions made very good models for population growth. So, we expanded our study to exponential and logarithmic functions. Use these ideas to answer the following questions.

  1. Simplify each of the following exponential expressions. Keep the exponent in your final answer.
  2. 73 78 712 =
  1. ( 5123)8 =
  1. 147  97 =
  1. Rewrite each of the following equations in either exponential form or logarithmic form. Also, solve each equation.

Alternate formSolution

a. 19x = 100______

b. 3x = 31______

c. ex = 50______

d. log5x = 6______

e. logx81 = 4______

  1. A Richter scale is a numerical way to describe the magnitude of an earthquake. This scale uses logarithms. The formula used to get the scale number is

R = log10a where R is the scale number and a is the amplitude or amount of ground motion.

  1. What is the scale number of an earthquake with an amplitude of 100,000?
  1. What is the amplitude of the great San Francisco quake of 1906 that measured 8.3 on the Richter scale?
  1. How many times as much ground motion does a 6 quake have compared to a 4 quake?
  1. Growth in nature fits fairly well into exponential functions. Use the following situation to answer the questions below.

An experiment began with 320 bacteria. The bacteria grew exponentially and 6 hours later there were 24,000.

  1. Find the exponential function (base e) that models this situation.
  1. How many bacteria will be present after 18 hours?
  1. Growth of money also fits into exponential functions use A = P(1+r)t.

Let’s take Nate’s original $12 and deposit it into an account earning 6% annual interest.

  1. How much money will be in the account in 10 years?
  1. How much money will there be if the account is compounded quarterly?
  1. How much money will there be if it is compounded daily? (Use 365.25 days)
  1. How much money will there be if it is compounded continuously?
  1. How many years will it take him to quadruple his money if he uses the annual interest rate of 6%?

We also used linear, quadratic and exponential functions to study the concept of a derivative.

  1. Use your calculator to find the value of the function and the derivative of the function f(x) = 5x3 + 2x2 – 4x at the given coordinates of x.

a.xf(x)f’(x)

-3

2

6

  1. Test the function for the “proportionality property”.
  1. Use your calculator to sketch the function. Set your window to the scale shown below. Answer the questions about the derivative of your sketched function. Then, use your answers to sketch the derivative function.

f’(x)  0 when ______

f’(x)  0 when ______

f’(x) = 0 when ______

  1. Use the shortcut to find the following derivatives.

a. f(x) = 71x3 - x54 + 99x +11 f’(x) = ______

b. f(x) = 1/5x10 + 13x8 – 1/6x – 1/8 f’(x) = ______