Subject: Calculus

Exponential Function

Any function of the form f(x)=a^x where a is any real-valued number is called an exponential function(a is also called as base). This is often used to characterize any constant change in the independent variable which in turn corresponds to a proportional change in the dependent variable. So say y=3^x obviously an exponential function. So when x increases or decreases, the value of y also decreases or increases noticeably.

Exponential functions are very useful in Science, Engineering and Mathematics. It is used to model growth and decay in values such as population, radioactive decay, biological multiplication and etc. It is even used to describe and predict values with exponential growth and decay.

Exponential function with base e

One of the most-distinguished and most important exponential function is the one with base e( a=e). So, the exponential would become

f(x) = e^x

e^x is very popular in growth and decay modeling. In Chemistry, given an initial amount of a substance S_o, we can easily determine the remaining amount of the substance S when it is subject to some chemical reaction. In this case, the independent variable is usually expressed as time t. So,

S=S_oe^{kt}

where k is just any constant. k determines the nature of the phenomena whether it is growth or decay. When k is negative, then it is decay and when it is positive, then obviously its growth.

Graph of Exponential Function

Plots of the exponential function f(x)=a^{x} with different values of a is shown in the figure below.

Notice in the graph that there is a sudden increase in the value of f(x) upon small increments in the value of x.

Rules regarding exponential functions

1. a^{x+y}=a^x a^y

2. a^{xy}=\left (a^x\right )^y=\left (a^y\right )^x

3. a^{0} = 1

4. a^{-x} = \frac{1}{a^x}

Derivative and integral of exponential functions

Exponential functions invokes special interest in mathematics especially in calculus due to its special properties regarding integration and differentiation.

Derivative of exponential functions

1. \frac{d}{dx}e^x = e^x
2. \frac{d}{dx}a^x = a^x\cdot \ln{a}

Integral of exponential functions

1. \int e^x dx = e^x +C
2. \int a^x dx = \frac{1}{\ln{a}}a^x + C

Example #1

Give reasons why the area under the exponential function e^x from negative infinity to a certain point x is just equal to the value of the function at point x.

Reason: To get the area, we need to integrate e^x from negative infinity to x. Now, the integral of exponential function is just equal to the function itself. The lower limit of negative infinity becomes zero. Thus, area of the exponential function e^x from negative infinity to a certain point x is just equal to the value of the function at point x.

Example #2

Without actually performing integration, can you give a good guess of the area under the function y=e^x in the 2nd quadrant?

Reason: This question is in connection to the previous example. In example #1, we showed that the area of the exponential function e^x from negative infinity to a certain point x is just equal to the value of the function at point x. Since the area under the curve in the first quadrant ranges from negative infinity to x=0, then the area would be equal to the value of the function at x=0.

Substituting x=0 to y=e^x,

y=e^0 = 1

Thus, the area is equal to 1. See how convenient it is to solve for the area without actually doing integration.

Example #3

In rules regarding exponential functions, simplify the following functions.

e^{\cos^2{x}}\cdot e^{\sin^2{x}}

Solution: From a^{x+y}=a^x a^y,

e^{\cos^2{x}}\cdot e^{\sin^2{x}} = e^{\cos^2{x}+\sin^2{x}}

Note that \cos^2{x}+\sin^2{x}=1, thus

e^{\cos^2{x}+\sin^2{x}} = e^1

Thus,

e^{\cos^2{x}}\cdot e^{\sin^2{x}} = e

Example #4

Tell which of the following are exponential functions.

  • f(x) = 6e^{x}
  • f(x) = e^{6x}
  • f(x) = x^{e}
  • f(x) = 32^{e}
  • f(x) = 32^{x}

ANSWER:

  • f(x) = 6e^{x}
  • f(x) = e^{6x}
  • f(x) = 32^{x}

Example #5

Find the derivative of the following exponential function.

f(x) = 1^{x}

Solution: From the property of derivative of exponential functions, \frac{d}{dx}a^x = a^x\cdot \ln{a}, where a=1,

D[f(x)] = \frac{d}{dx}1^{x}
\frac{d}{dx}1^{x}=1^{x} \cdot \ln{1} = 0

This answer makes sense because the derivative of a constant is zero and the function itself is a constant. The number one raise to any number is always one - a constant.

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