Chapter 1 | Functions and Graphs
101
Figure 1.45 The graph of f ( x ) = e x has a tangent line with slope 1 at x =0.
Example 1.35 Compounding Interest
Suppose $500 is invested in an account at an annual interest rate of r =5.5 % , compounded continuously. a. Let t denote the number of years after the initial investment and A ( t ) denote the amount of money in the account at time t . Find a formula for A ( t ). b. Find the amount of money in the account after 10 years and after 20 years. Solution a. If P dollars are invested in an account at an annual interest rate r , compounded continuously, then A ( t ) = Pe rt . Here P =$500 and r =0.055. Therefore, A ( t ) =500 e 0.055 t . b. After 10 years, the amount of money in the account is A (10) =500 e 0.055·10 =500 e 0.55 ≈$866.63.
After 20 years, the amount of money in the account is
A (20) =500 e 0.055·20 =500 e 1.1 ≈ $1, 502.08.
1.29 If $750 is invested in an account at an annual interest rate of 4 % , compounded continuously, find a formula for the amount of money in the account after t years. Find the amount of money after 30 years.
Logarithmic Functions Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels. The exponential function f ( x ) = b x is one-to-one, with domain (−∞, ∞) and range (0, ∞). Therefore, it has an inverse function, called the logarithmic function with base b . For any b >0, b ≠1, the logarithmic function with base b , denoted log b , has domain (0, ∞) and range (−∞, ∞), and satisfies log b ( x ) = y if and only if b y = x . For example,
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