MAT 213 Brief Calculus Section 2.1 Change, 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 change Units: output units Percent Change = Units: percent We calculated percent change back in 1.3 with exponential functions 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

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) Height (ft)

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) Height (ft)

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

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 line Use 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 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

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