Chapter 2: Q. 77 (page 198)

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 it may be helpful to do some preliminary algebra
$f' (x) = (x^{4} - 8) (1 - 3 x^{5}); f (0) = 2$

Short Answer

Expert verified

The antiderivative can be given as $f (x) = \frac{x^{5}}{5} - \frac{3}{10} x^{10} - 8 x + 4 x^{6} + 2$

Step by step solution

Given information

We are given the derivative as $f' (x) = (x^{4} - 8) (1 - 3 x^{5}); f (0) = 2$

Find the antiderivative

We first simplify the given expression

$f' (x) = x^{4} - 3 x^{9} - 8 + 24 x^{5}$

We know that differentiating a power function decreases the power by one we can start with the function $f (x) = x^{5} - x^{10} - 8 x + x^{6}$

Whose derivative is

$f' (x) = 5 x^{4} - 10 x^{9} - 8 + 6 x^{5}$

Which is nearly equal now we adjust the coefficients

$f (x) = \frac{x^{5}}{5} - \frac{3}{10} x^{10} - 8 x + 4 x^{6} + c$

We are also given that

$f (0) = 2 S u b s t i t u t i n g w e g e t, 2 = c$

Hence the antiderivative becomes

$f (x) = \frac{x^{5}}{5} - \frac{3}{10} x^{10} - 8 x + 4 x^{6} + 2$

Conclusion

The antiderivative can be given as $f (x) = \frac{x^{5}}{5} - \frac{3}{10} x^{10} - 8 x + 4 x^{6} + 2$

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.

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 it may be helpful to do some preliminary algebra
$f' (x) = (x^{4} - 8) (1 - 3 x^{5}); f (0) = 2$

Short Answer

Step by step solution

Given information

Find the antiderivative

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

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 it may be helpful to do some preliminary algebra f'(x)=(x4-8)(1-3x5);f(0)=2

Short Answer

Step by step solution

Given information

Find the antiderivative

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

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 it may be helpful to do some preliminary algebra
$f' (x) = (x^{4} - 8) (1 - 3 x^{5}); f (0) = 2$