M 112 Short Course in Calculus

1
M 112 Short Course in Calculus

• Chapter 1 Functions and Change
• Sections 1.5 Exponential Functions
• V. J. Motto

2
1.4 Exponential Functions

• An exponential function is a function of the form
• Where a ? 0, b gt 0, and b ? 1. The exponent
  must be a variable.

3
Illustration 1
4
Illustration 2 Different bs, b gt 0
What conclusions can we make looking at these
graphs? Use your calculator to sketch these
graphs!
5
Illustration 3 Different bs, 0 lt b lt1
Graph these on your calculator. What can we
conclude? Are there other ways to write these
equations?
6
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

7
Illustration 4 (page 39)

• Population of Nevada 2000-2006

• Dividing each years population by the previous
  years population gives us
• We find a common ratio!

8
Illustration 4 (continued)
Thus, the modeling equation is P(t)
2.020(1.036)t

• These functions are called exponential growth
  functions. As t increases P rapidly increasing.

9
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

10
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.

11
Comments

• The largest possible domain for the exponential
  function is all real numbers, provided a gt0.

12
Linear vs Exponential

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

13
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

14
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

15
Example 3 (page 42)

• Which of the following table sof values could
  correspond to an exponential function or linear
  function? Find the function.

16
Example 3 (continued)

1. f(x) 15(1.5)x, (common ratio)
2. g is not linear and g is not exponential
3. h(x) 5.3 1.2x

17
Research Homework

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