5.5 Base Other Than e and Application

After this lesson, you should be able to: Define exponential functions that have bases other than e. Differentiate and integrate exponential functions that have bases other than e. Use exponential functions to model compound interest and exponential growth.

Definition of Exponential Function to Base a Exponential Functions Definition of Exponential Function to Base a The exponential function obey the usual laws of exponents.

Definition of Logarithmic Function to Base a Properties of Inverse Functions

Differentiation and Integration After we developed the derivatives for Natural Exponential Functions, it is quite easy to derived exponential functions with bases other than e. Notice that when a ≠1 and a > 0 and Taking the derivatives for both functions, we have When f(x) = x, we also can conclude

Theorem 5.13 Derivatives for Bases Other Than e

Example 1 Find the derivative of f (x) = 10x f (x) = 53x f (x) = log2cos x Solution

Example 2 Find the derivative of Solution Use the product rule

When an integrand involves an exponential function to a base other than e, there are two options: Convert to base e using the formula integrate directly by using the integration formula Example 3 Find Solution

Theorem 5.14 The Power Rule for Real Exponents

Example 4 Find the derivative of Solution Note that we can not simply write the derivative as Indeed, there is no simple differentiation rule to find its derivative. We have to take the logarithm first.

If the interest rate is compounded annually, at the end of 3rd Applications of Exponential Functions Question You deposit $1500 in an account that pays 7.5% interest compounded annually. What is the balance after 3 year? How does this compare to the same investment compounded monthly and daily? If the interest rate is compounded annually, at the end of 3rd year, the asset value is: A3= P(1 + r )3 = 1500 (1 + 0.075)3 = $1863.45 If the interest rate is compounded monthly, at the end of 3rd year, the asset value is: A3= P (1 + r / 12)12·3 = 1500 (1 + 0.075 / 12)36 = $1877.17

If the interest rate is compounded daily, at the end of 3rd year, the asset value is: A3= P (1 + r / 365)365·3 = 1500 (1 + 0.075 / 365)1095 = $1878.44 From the above question, we see that the total asset is affected by the factor (1 + r / 12)12·3 = [(1 + r / 12)12]3 (1 + r / 365)365·3 = [(1 + r / 365)365]3 Or in general, the factor of (1 + r / n) n or simply, (1 + 1 / n) n Mathematician Euler found that the (1 + 1 / n) n is approaching a fixed decimal number, 2.7182818284590…, as n  

The number was named after Euler as Euler number or natural base. The numbers e and possesses some properties: The number en increases when the n increases with an upper bound less than 3. 2) The number en is approaching to 2.7182818284590… or approximate to 2.718281828459 as n increases. The number e is an irrational number (non-terminating and non-repeating), like . 4)

Theorem 5.15 A Limit Involving e

You deposit P dollars in an bank account that pays interest Applications of Exponential Functions You deposit P dollars in an bank account that pays interest rate of r per year. If the rate is compounded n times every year, then the asset value at the end of t years is If the interest rate is compounded continuously, then at the end of t years, the asset value is:

Example 5 You deposit $1500 dollars in an bank account that pays 7 Example 5 You deposit $1500 dollars in an bank account that pays 7.5% per year interest compounded a) semi-annually b) monthly c) continously for 3 years. Find the balance. a) b) c)

Homework 5.5 P. 366 Q xxxxxxxxxxxx