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. 6/1/2016 2

Illustration 1 6/1/20163

Illustration 2: Different b’s, b > 0 6/1/20164 What conclusions can we make looking at these graphs? Use your calculator to sketch these graphs!

Illustration 3: Different b’s, 0 < b <1 6/1/20165 Graph these on your calculator. What can we conclude? Are there other ways to write these equations?

Commentson y = b x 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 = 6/1/20166

Illustration 4: (page 39) Population of Nevada Dividing each year’s population by the previous year’s population gives us We find a common ratio! 6/1/20167

Illustration 4 (continued) These functions are called exponential growth functions. As t increases P rapidly increasing. 6/1/20168 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 6/1/20169

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. 6/1/201610

Comments The largest possible domain for the exponential function is all real numbers, provided a >0. 6/1/201611

Linear vs Exponential Linear function has a constant rate of change An exponential function has a constant percent, or relative, rate of change. 6/1/201612

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 a) Increasing by 0.4 mg per minute b) Decreasing by 0.4 mg per minute c) Increasing by 3% per minute d) Decreasing by 3% per minute 6/1/201613

Example 1: (continued) Solution a) A = t - linear increase b) A = 25 – 0.4t - linear decrease c) A = 25(1.03) t - exponential growth d) A = 25(0.97) t - exponential decay 6/1/201614

Example 3: (page 42) Which of the following table sof values could correspond to an exponential function or linear function? Find the function. 6/1/201615

Example 3: (continued) a) f(x) = 15(1.5) x, (common ratio) b) g is not linear and g is not exponential c) h(x) = x 6/1/201616

Research Homework Search the internet (or mathematics books you own) and find a demonstration that discovers the value of e. 6/1/201617