Unformatted text preview: Add 9: h(x) = x + 9 Take cosine: g (x) = cos x Square: f (x) = x2 Section 1.5: Exponential Functions
Exponential functions are of the form f (x) = ax where a is a positive constant. Increasing, Decreasing or Constant Increasing, if 1 < a < ∞, Decreasing, if 0 < a < 1, Constant, if a = 1. The exponential function f (x) = ax will either be For the increasing exponential, the function will increase more rapidly (as x → ∞) the larger a gets. All exponential functions pass through the point (0, 1) (unless they are shifted, of course!). If a = 1 the domain is all reals and the range is (0, ∞). Laws of Exponents If a and b are positive numbers and x and y are any real numbers, then ax+y ax−y (ax )y (ab)x = = = = ax ay ax a−y axy ax bx Chapter 1 Review 6 Applications of Exponential Functions: Population Growth Suppose that we know a population doubles every hour. Say the population is p(t), and t is the time in hours. If our initial population is p0 , then we have: p(0) p(1) p(2) p(3) = = = = p0 2 p0 2p(1) = 22 p0 2p(2) = 23 p0 p(t) = 2t p0 And we see how the exponential function enters the ﬁeld of population growth. Q: What if the population tripled every hour, how would that change our ﬁnal answer? A: p(t) = 3t p0 . Applications of Exponential Functions: HalfLife The halflife is deﬁned for radioactive compounds to be the time it takes for the amount to decay into half that amount. The halflife for strontium90 (90 Sr) is 25 years. If we have 24mg of left after t years. m(0) = 24 1 m(25) = (24) 2 1 m(50) = m(25) = 2 1 m(50) = m(75) = 2 m(t) =
90 Sr, ﬁnd an expression for amount 1 m(0) 22 1 m(0) 23 1 (24) = 24 · 2−t/25 2t/25 There was some talk about the number e More on that later with Logarithms and natural Logarithms. Section 1.6: Inverse Function and Logarithms
Not all functions have inverses. Recall that a function is deﬁned as a rule f that assigns to each element x in a set A exactly one element, called f (x) in a set B . This requirement that a function have exactly one element imposes a condition on whether or not a function f has an inverse function. Deﬁnition A function f is called a onetoone function if it never takes on the same value twice, that is f (x1 ) = f (x2 ) whenever x1 = x2 . You can check whether or not a function is onetoone from the graph by using the horizontal line test. ...
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 Fall '08
 JUNGHENN
 Calculus, Exponential Function, Derivative, Exponential Functions, Exponentiation, Inverse function

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