Chapter 2: Q. 68 (page 211)

In Exercises 63–68, find a function that has the given derivative and value. In each case you can find the answer with an educated guess-and-check process while thinking about the chain rule.
$f' (x) = \frac{x^{2}}{\sqrt{x^{3} + 1}}, f (1) = 2$

Short Answer

Expert verified

The function is $f (x) = \frac{2}{3} \sqrt{x^{3} + 1} + \frac{4}{3}$

Step by step solution

Step 1. Given Information

The given function is $f' (x) = \frac{x^{2}}{\sqrt{x^{3} + 1}}, f (1) = 2$

We need to find antiderivative to get the function.

Step 2. Finding antiderivative

$f (x) = \int \frac{x^{2}}{\sqrt{x^{3} + 1}} d x f (x) = \int \frac{x^{2}}{\sqrt{t}} \frac{d t}{3 x^{2}} = \frac{1}{3} \int t^{- \frac{1}{2}} d t f (x) = \frac{1}{3} \frac{\sqrt{t}}{\frac{1}{2}} = \frac{2}{3} \sqrt{t} + C f (x) = \frac{2}{3} (\sqrt{x^{3} + 1}) + C$

Step 3. Finding the value of C

Given that $f (1) = 2$

$2 = \frac{2}{3} \sqrt{1} + C C = 2 - \frac{2}{3} = \frac{4}{3}$

The function become $f (x) = \frac{2}{3} \sqrt{x^{3} + 1} + \frac{4}{3}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Exercises 63–68, find a function that has the given derivative and value. In each case you can find the answer with an educated guess-and-check process while thinking about the chain rule.
$f' (x) = \frac{x^{2}}{\sqrt{x^{3} + 1}}, f (1) = 2$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding antiderivative

Step 3. Finding the value of C

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Exercises 63–68, find a function that has the given derivative and value. In each case you can find the answer with an educated guess-and-check process while thinking about the chain rule.f'x=x2x3+1,f1=2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding antiderivative

Step 3. Finding the value of C

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Exercises 63–68, find a function that has the given derivative and value. In each case you can find the answer with an educated guess-and-check process while thinking about the chain rule.
$f' (x) = \frac{x^{2}}{\sqrt{x^{3} + 1}}, f (1) = 2$