On what interval is the function f(x)=5 x-e^x increasing? |...

Name: Problem 47, Chapter 3: Calculus
Uploaded: 2021-09-08T16:25:30-08:00
Duration: 2 min 24 s
Channel: Lauren Shelton
Description: On what interval is the function f(x)=5 x-e^x increasing?

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Key Concepts

Properties of Exponential Functions

Exponential functions possess unique characteristics, particularly that their derivative is the function itself. This property simplifies differentiation when an exponential term is involved. Understanding these properties is essential when exponential functions interact with algebraic expressions, as it allows for the efficient solving of equations and inequalities in calculus.

Solving Inequalities Involving Derivatives

After computing the derivative, setting it greater than zero typically leads to an inequality that must be solved to find where the function is increasing. This involves algebraic manipulation and may require the use of logarithms, especially when the inequality contains an exponential term. The solution of this inequality identifies the interval over which the derivative is positive.

Derivative as a Tool for Analyzing Function Behavior

The derivative of a function provides the rate at which the function is changing at any given point. By differentiating a function and analyzing where its derivative is positive, one can determine the intervals on which the function is increasing. This concept is foundational in calculus, enabling the understanding and graphing of function behavior.

Transcript

00:01 I suppose i'm given a function and i want to figure out if my function is increasing or decreasing.

00:05 Well, how do i determine just looking at the function without a graph if it's increasing or decreasing? well, to figure out if it's increasing, this occurs when the first derivative, if the original function is increasing, it's when the first derivative is positive or greater than zero.

00:22 On the contrary, if the original function is decreasing, this means that the first derivative is now less than zero or negative.

00:30 So in order to figure out if it's increasing, first i need to find the first derivative of f of x.

00:38 Derivative of 5x is just the constant 5.

00:40 Minus derivative of e to the x is just itself, and there's nothing special in the exponent, so don't need to do the chain rule.

00:47 Okay, so we need to figure out where it's greater than zero, or first i would need to figure out where it's equal to zero.

00:55 So i'm going to set the first derivative equal to zero.

00:58 Add e to the x to both sides.

01:01 This is going to give me 5 equals e to the x.

01:06 Now, how do we get x by itself? because i have e attached to it.

01:09 Well, what's the inverse of the exponential function? the logarithmic function.

01:14 Logbase e.

01:15 So i'm going to take the ln of both sides...

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