10. ss 4 2 - Notes PDF

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Section 4.2 Exponential Functions An exponential function, f , is a function of the form • f (x) = ax with a > 0, a 6= 1 Examples. The graphs of f (x) = 2x and f (x) = ( 12 )x are shown below.

f (x) = ( 12 )x

f (x) = 2x Note the following: • The graph of y = ax , a > 1, is similar to the graph of y = 2 x .

• The graph of y = ax , 0 < a < 1, is similar to the graph of y = ( 12 )x . Note the difference between exponential functions and power functions: • y = ax is an exponential function • y = xa is a power function Properties of exponentials (a, b > 0): • as+t = as at • (as )t = ast • (ab)s = as bs • a−s =

1 as

= ( 1a )s

Properties of the graph of y = ax , a > 1:

y = ax , a > 1 • The y−intercept is at 1. • y → ∞ as x → ∞ • y → 0 as x → −∞ • The graph is strictly increasing (x1 < x 2 implies ax1 < ax 2 ) 1

• the graph is continuous and smooth (no breaks and no sharp corners) Properties of the graph of y = ax , 0 < a < 1:

y = ax , 0 < a < 1 • The y−intercept is at 1 • y → 0 as x → ∞ • y → ∞ as x → −∞ • The graph is decreasing (x1 < x2 implies ax1 > a x2 ) • The graph is continuous and smooth (no breaks and no sharp corners) Example. Compare the graphs of y = 2 x and y = 2x−1 . Solution. There are two ways to compare the graphs: (1) The graph of y = 2x−1 is a horizontal translation (by one unit) of the graph of y = 2 x , i.e., x if 2 = f (x), then 2x−1 = f (x − 1). (2) Since 2x = 2 · 2 x−1, the y-values of 2x are 2 times the y values of 2x−1 y = 2x

y = 2x

y = 2 x−1

In mathematics, the number e is usually reserved for the base of the natural logarithm. We will discuss, in a later section, the origins and use of the number e. For now, it suffices to know that e ≈ 2.718. 4.2 EXERCISES In Problems 1-3, approximate each number using a calculator. Express your answer rounded to three decimal places. √

1. (d) 3 5 2. (d) 2π 3. (d) π e 4. In Figures 11-18, the graph of an exponential function is given. Match each graph to one of the following functions: A. y = 3 x 2

B. y = 3−x C. y = −3x D. y = −3−x E. y = 3x − 1 F. y = 3 x−1 G. y = 31−x H. y = 1 − 3 x

In Problem 5, use a transformation to graph the function. Determine the domain, range, and horizontal of each function. Verify your results using a graphing utility. 5. f (x) = e −x 6. Exponential Probability Between 12:00PM and 1:00PM, cars arrive at Citibank’s drive-thru at the rate of 6 cars per hour(0.1 car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within t minutes of 12:00PM. F (t) = 1 − e −0.1t (a) Determine the probability that a car will arrive within 10 minutes of 12:00PM(that is before 12:00PM). (b) Determine the probability that a car will arrive within 40 minutes of 12:00PM(that is before 12:00PM). (c) Graph f using your graphing utility. (d) Determine how many minutes are needed for the probability to reach 50% (e) What value does F approach as t becomes unbounded in the positive direction?

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