Question

Give an example of: A function whose derivative is $g^{\prime}(x)=2 x$ and whose graph has no $x$ -intercepts.

Name: Problem 106, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-16T18:58:39-08:00
Duration: 1 min 42 s
Channel: Gregory Higby
Description: Give an example of: A function whose derivative is g^'(x)=2 x and whose graph has no x -intercepts.

   Give an example of:
A function whose derivative is $g^{\prime}(x)=2 x$ and whose graph has no $x$ -intercepts.

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 106

Instant Answer

Solved by Expert Gregory Higby

04/16/2021

Step 1

To find the function $g(x)$, we need to integrate the derivative. The integral of $2x$ is $x^{2}$. So, we have $g(x)=x^{2}+C$, where $C$ is the constant of integration. Show more…

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Give an example of: A function whose derivative is $g^{\prime}(x)=2 x$ and whose graph has no $x$ -intercepts.

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Key Concepts

Function Behavior and Graph Constraints

Understanding how the graph of a function behaves is crucial, especially when specific characteristics are desired, such as the graph having no x-intercepts. This involves ensuring that the function maintains a consistent sign over its entire domain, which can often be achieved by an appropriate choice of the constant of integration.

Antiderivative

In calculus, an antiderivative is a function whose derivative equals a given function. Finding an antiderivative requires integrating the function and introducing a constant of integration, as multiple functions can differ by a constant and still have the same derivative.

Constant of Integration

When finding an antiderivative, the constant of integration represents the family of all antiderivatives. Its value is determined by additional conditions or constraints, enabling the selection of a specific function from the general family.

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