Question

Find the find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point. $$f^{\prime}(x)=5(2 x-1)^{4} ;(1,3)$$.

Name: Problem 51, Chapter 6: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-06T21:25:08-08:00
Duration: 2 min 41 s
Channel: Priyanka Sadarangani
Description: Find the find the function f given that the slope of the tangent line to the graph of f at any point (x, f(x)) is f^'(x) and that the graph of f passes through the given point. f^'(x)=5(2 x-1)^4 ;(1,3).

   Find the find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=5(2 x-1)^{4} ;(1,3)$$.

Applied Calculus for the Managerial, Life, and Social Sciences

Soo T. Tan 10th Edition

Chapter 6, Problem 51

Instant Answer

Solved by Expert Priyanka Sadarangani

05/06/2021

Step 1

So, we have \[f(x) = \int f^{\prime}(x) \, dx = \int 5(2 x-1)^{4} \, dx.\] Show more…

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Find the find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point. $$f^{\prime}(x)=5(2 x-1)^{4} ;(1,3)$$.

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Key Concepts

Initial Condition

An initial condition provides a specific value of the function at a particular point, allowing for the determination of the constant of integration. Here, the point (1, 3) gives the necessary information to uniquely identify the antiderivative among all possible functions.

Antiderivative

The antiderivative is the process of finding a function whose derivative is the given function. In this problem, determining f(x) from f'(x) involves integrating f'(x) and then adding an arbitrary constant, usually denoted as C.

Substitution Method

The substitution method (or u-substitution) is used to simplify integrals involving composite functions. By letting u equal an inner function—in this case, 2x - 1—the integration transforms into a simpler integral in terms of u, which facilitates the computation of the antiderivative.

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