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Graph several members of the family of functions $$ f(x) = \frac{1}{1 + ae^{bx}} $$where $ a > 0 $. How does the graph change when $ b $ changes? How does it change when $ a $ changes?

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01:38

Jeffrey Payo

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 4

Exponential Functions

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Johns Hopkins University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

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A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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or it in this problem, we want to change the values of A and B in the equation on what I've done first is I've let in. Why want? I've let the A value B one and then for why, to a value is to and for why. Three. The A value is three, but I left the B value as one each time, just so that we can see what happens when we change a. And so let's look at those three graphs. They don't look terribly different. It looks like that changing the value of A to a larger number is making the graph a little lower. And if I go back and if I change the A value to nine instead of three just to be a little bit more extreme about it and look at those graphs, you can see that the graph with nine for a is has decreased quite a bit more quickly. Okay, let's see what happens if we change the value of beat. So now I'm going to turn off my second graf on my third graf in turn on my fourth graf and my fifth graf and so all three of these graphs have the same value of a A equals one, but they have different values of B 12 and three. Let's compare those graphs so it looks like having a greater value of B is making it a bit steeper. You can see the green one seems to be the spread as a decline seems to have shortened in with and gotten more steeper. So the next thing I decided to do was change my three to a nine to make it a little bit more extreme. So now be is nine, and let's take a look at how that changed things. So looking at the green one, we got an even narrower interval of decline and in much steeper decline.

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