Question

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. $$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Name: Problem 60, Chapter 6: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-10T15:08:37-08:00
Duration: 3 min 9 s
Channel: Gregory Higby
Description: 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^'(t)=t^2-2 t+3 ;(1,2)

   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.
$$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Applied Calculus for the Managerial, Life, and Social Sciences

Soo T. Tan 10th Edition

Chapter 6, Problem 60

Instant Answer

Solved by Expert Gregory Higby

05/10/2021

Step 1

To find the original function $f(t)$, we need to find the antiderivative of $f^{\prime}(t)$. Show more…

Show all steps

Thanks for your feedback!

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. $$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Play audio

Feedback

Gregory Higby and 90 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In the context of the given problem, finding the antiderivative involves integrating the given derivative to recover the original function up to an unknown constant.

Constant of Integration

When finding an antiderivative, the integration process introduces an arbitrary constant because the derivative of a constant is zero. This constant is determined using additional information provided in the problem, ensuring that the resulting function fits specific conditions.

Initial Condition

An initial condition is a specific value of the function at a given point, which is used to determine the constant of integration. By substituting this value into the antiderivative, the exact function that satisfies both the differential equation and the initial condition can be found.

Please give Ace some feedback