MATH 131-503 Fall 2015
c
Wen
Liu
1.5
1.5 Exponential Functions
An exponential function is a function of the form
f (x) = ax
where a is a positive constant.
Graphs of y = ax , 0 < a < 1 (called “Exponential Decay”), y = 1x , and y = ax , a > 1 (called
“Exponential Growth”) are shown below.
Laws of Exponents: If a, b are positive numbers and x, y are any real numbers, then
ax ay = ax+y
ax
= ax−y
y
a
(ax )y = axy
(ab)x = ax bx
Examples:
1. Simplify the expression:
48x−5 (y 2 )−3
√
9x2 3 y
−3
Page 1 of 4
MATH 131-503 Fall 2015
1.5
c
Wen
Liu
2. (p. 60) A bacterial culture starts with 500 bacteria and doubles in size every half hour.
(a) How many bacteria are there after t hours?
(b) Graph the population function and estimate the time for the population to reach 100, 000.
3. (p. 56) The half-life of strontium-90, 90 Sr, is 25 years. This means that half of any given quantity
of 90 Sr will disintegrate in 25 years.
(a) If a sample of 90 Sr has a mass of 24 mg, find an expression for the mass m(t) that remains after
t years.
(b) Find the mass remaining after 40 years, correct to the nearest milligram.
(c) Use a graphing device to graph m(t) and use the graph to estimate the time required for the mass
to be reduced to 5mg.
Page 2 of 4
MATH 131-503 Fall 2015
c
Wen
Liu
1.5
Of all possible bases for an exponential function, there is one that is most convenient for the purposes
of calculus. We call the function f (x) = ex the natural exponential function.
Examples:
4. Starting with the graph of y = ex , write the equation of the graph that results from the following
changes: reflecting about the x-axis and then about the y-axis.
5. Match the exponential function with its graph.
(a) g(x) = 2x
(b) g(x) = 2x + 1
(c) g(x) = 2x−2
(d) g(x) = 2−x
Page 3 of 4
MATH 131-503 Fall 2015
1.5
c
Wen
Liu
6. Immediately following an injection, the concentration of a drug in the bloodstream is 500 milligrams per milliliter. After t hours, the concentration is 85% of the level of the previous hour. Find
a model for C(t), the concentration of the drug after t hours.
Page 4 of 4