Question

Let $f(x)=x^{3}$ a. Find the point on the graph of $f$ where the tangent line is horizontal. b. Sketch the graph of $f$ and draw the horizontal tangent line.

Name: Problem 45, Chapter 3: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-05-05T17:42:47-08:00
Duration: 2 min 3 s
Channel: Wendi Zhao
Description: Let f(x)=x^3 a. Find the point on the graph of f where the tangent line is horizontal. b. Sketch the graph of f and draw the horizontal tangent line.

     Let $f(x)=x^{3}$
a. Find the point on the graph of $f$ where the tangent line is horizontal.
b. Sketch the graph of $f$ and draw the horizontal tangent line.

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan 10th Edition

Chapter 3, Problem 45

Instant Answer

Solved by Expert Wendi Zhao

05/05/2021

Step 1

The derivative of a function gives the slope of the tangent line at any point on the graph. Show more…

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Let $f(x)=x^{3}$ a. Find the point on the graph of $f$ where the tangent line is horizontal. b. Sketch the graph of $f$ and draw the horizontal tangent line.

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Key Concepts

Horizontal Tangent Line

A horizontal tangent line occurs at points where the derivative is zero since a zero slope implies that the line is flat. Setting the derivative equal to zero allows one to solve for the x-values at which the function's graph has a horizontal tangent.

Sketching Graphs of Functions

Understanding the overall shape of a function, particularly in the case of cubic functions, is critical when sketching its graph. This involves identifying key features such as intercepts, symmetry, and points of inflection, which are essential for accurately representing the function, including drawing specific tangent lines.

Derivative

The derivative of a function describes the instantaneous rate of change at any point and is used to determine the slope of the tangent line to the function's graph. In this context, differentiating the function helps in finding where the tangent line has specific properties, such as being horizontal.

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