The Exponential Function
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Lesson Summary:
In this lesson, we learn about the exponential function and its inverse, the natural logarithm. We find that the number e is defined as the inverse of the natural logarithm of x, and that these two functions are inverses of each other. We also learn about the derivative and integral of the exponential function, as well as its limit as x approaches infinity. Finally, we see how to differentiate and integrate exponential functions with bases other than e.
Lesson Description:
The inverse of the natural logarithm.
Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at http://online.math.uh.edu/HoustonACT/videocalculus/index.html.
Additional Resources:
- Table of Derivatives - Table of derivatives of several different functions.
- Derivatives and Integrals of exponential function - A list of derivatives and integrals of the exponential function.
- Derivatives and Integrals of exponential and natural logarithm functions - A list of derivatives and integrals of the exponential and natural logarithm function.
Questions answered by this video:
- What is the exponential function?
- What is the definition of e?
- What is the inverse of ln x?
- What is the derivative of e^x?
- What is the derivative of e^u?
- What is the integral of e^u?
- What is the integral of e^kx?
- What is the derivative and integral of b^x?
- What is the change of base formula for logarithms?
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Staff Review
The number e and the exponential e^x are defined and explained. Also, the inverse of the natural logarithm is discovered and explained with graphs and animations. Properties of the natural logarithm and the exponential function are explained. Several examples of finding the derivative of e^u for various u functions are shown. We also see how to sketch the graph of f(x) = x^2e^-x. A really great, comprehensive video on logarithms.
