MAT 213Brief Calculus

Section 2.1Change, Percent Change, and

Average Rates of Change

Change• A main function of calculus is describing change

• While it is useful to know what something is, it can be more useful to know how it is changing

• Examples– Change in home sales– Change in profit/revenue/costs per unit produced– Change in number of students attending SCC per

semester

Change = New Value – Old Value, sometime referred to as absolute or total changeUnits: output units

Percent Change =

Units: percent

We calculated percent change back in 1.3 with exponential functions

valueold

change

Two ways of measuring change

Example

After researching several companies, you and your cousin decided to purchase your first shares of stock.

You bought $200 worth of stock in one company;

Your cousin invested $400 in another.

At the end of a year, your stock was worth $280, whereas your cousin’s stock was valued at $500.

Who earned more money?

Who did better?

• We have another way of expressing change which involves spreading the change over an interval

• This is called the average rate of change

• We divide the change (or total change) by the length of the interval

• Example– If you drove 100 miles in 2 hours, how fast were

you going?– Were you going the same speed the whole time?

Change = New Value – Old Value

Units: output units

Percent Change =

Units: percent

Average Rate of Change =

Units: units of output per unit input

valueold

change

interval oflength

change

Average Rate of Change

Suppose a large grapefruit is thrown straight up in the air at t = 0 seconds

The grapefruit leaves the thrower’s hand at high speed, slows down until it reaches its maximum height, then

speeds up in the downward direction and finally, “SPLAT!”

During the first second, the grapefruit moves 22 feet During the next second, the grapefruit moves 12 feet

…

Time (sec) 0 1 2 3 4 5

Height (ft) 4 26 38 40 32 14

Average Rate of Change

Use the table to compute the average rate of change of the grapefruit over the following intervals

4 ≤ t ≤ 5

1 ≤ t ≤ 3

0 ≤ t ≤ 5

Time (sec) 0 1 2 3 4 5

Height (ft) 4 26 38 40 32 14

Average Rate of Change

The grapefruit’s journey can be modeled by this quadratic function:

Use this function to compute the average rate of change over the following intervals

4 ≤ t ≤ 5

1 ≤ t ≤ 3

0 ≤ t ≤ 5

4275)( 2 tttH

Average Rate of Change

Graphically, the average rate of change over the interval a ≤ x ≤ b is the slope of line connecting f(a) with f(b)

We refer to this as the secant lineUse the graph of H(t) to compute the average rate of change of the grapefruit over the following intervals

4 ≤ t ≤ 5

1 ≤ t ≤ 3

0 ≤ t ≤ 5•Notice that secant lines are linear functions that have the average rate of change between a and b as a slope

• If an annual interest r is compounded n times per year, then the balance, B, on an initial deposit P after t years is

• If money is invested at a nominal rate of r and compounded continuously, we have the formula

nt

n

rPB

1

rtPeB

A couple of formulas for compounding interest

• The nominal rate or annual percentage rate (APR) is the percentage 100r%

• The percentage change of the amount accumulated over one compounding period is

• The effective rate or annual percentage yield (APY) is the percentage change in the amount accumulated over one year

%100 nr

• Suppose you have money to place into an interest bearing account. You have the following options

1. 5.9% compounded quarterly

2. 5.8% compounded continuously

3. 6% compounded annually

Which option would you choose and why?

In groups let’s try the following from the book

• 3, 13, 23, 27