M 112 Short Course in Calculus

M 112 Short Course in Calculus Chapter 1 – Functions and Change Sections 1.5 Exponential Functions V. J. Motto

1.4 Exponential Functions An exponential function is a function of the form Where a ≠ 0, b > 0, and b ≠ 1. The exponent must be a variable.

Illustration 1

Illustration 2: Different b’s, b > 0 What conclusions can we make looking at these graphs? Use your calculator to sketch these graphs!

Illustration 3: Different b’s, 0 < b <1 Graph these on your calculator. What can we conclude? Are there other ways to write these equations?

Comments on y = bx All exponential graphs • Go through the point (0, 1) • Go through the point (1, b) • Are asymptotic to x-axis. • The graph f(x) = b-x =

Illustration 4: (page 39) Population of Nevada 2000-2006 Dividing each year’s population by the previous year’s population gives us We find a common ratio!

Illustration 4 (continued) These functions are called exponential growth functions. As t increases P rapidly increasing. Thus, the modeling equation is P(t) = 2.020(1.036)t

Illustration 5: Drugs in the Body Suppose Q = f(t), where Q is the quantity of ampicillin, in mg, in the bloodstream at time t hours since the drug was given. At t = 0, we have Q = 250. Since the quantity remaining at the end of each hour is 60% of the quantity remaining the hour before we have

Illustration 5: (continued) You should observe that the values are decreasing! The function Q = f(t) = 250(0.6)t Is an exponential decay function. As t increases, the function values get arbitrarily close to zero.

Comments The largest possible domain for the exponential function is all real numbers, provided a >0.

Linear vs Exponential • Linear function has a constant rate of change • An exponential function has a constant percent, or relative, rate of change.

Example 1: (page 41) The amount of adrenaline in the body can change rapidly. Suppose the initial amount is 25 mg. Find a formula for A, the amount in mg, at time t minutes later if A is • Increasing by 0.4 mg per minute • Decreasing by 0.4 mg per minute • Increasing by 3% per minute • Decreasing by 3% per minute

Example 1: (continued) Solution • A = 25 + 0.4 t - linear increase • A = 25 – 0.4t - linear decrease • A = 25(1.03)t - exponential growth • A = 25(0.97)t - exponential decay

Example 3: (page 42) Which of the following table sof values could correspond to an exponential function or linear function? Find the function.

Example 3: (continued) • f(x) = 15(1.5)x, (common ratio) • g is not linear and g is not exponential • h(x) = 5.3 + 1.2x

Research Homework Search the internet (or mathematics books you own) and find a demonstration that discovers the value of e.

M 112 Short Course in Calculus