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Q. 73

Expert-verified

Calculus

Found in: Page 198

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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
$f' (x) = 3 x^{5} - 2 x^{2} + 4; f (0) = 1$

The antiderivative can be given as $f (x) = \frac{x^{6}}{2} - \frac{2 x^{3}}{3} + 4 x + 1$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information

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

Step 2: Find the antiderivative

We know that differentiating a power function decreases the power by one we can start with the function $f (x) = x^{6} - x^{3} + 4 x + c$

On differentiating the above function we get,

$f' (x) = 6 x^{5} - 3 x^{2} + 4$ which is nearly equal

Now we only adjust the coefficient

We get, $f (x) = \frac{x^{6}}{2} - \frac{2 x^{3}}{3} + 4 x + c$

We are also given that $f (0) = 1 S u b s t i t t u i n g t h i s i n t h e f u n c t i o n w e g e t, f (0) = c c = 1$

Hence the antiderivative becomes

$f (x) = \frac{x^{6}}{2} - \frac{2 x^{3}}{3} + 4 x + 1$

Step 3: Conclusion

The antiderivative can be given as $f (x) = \frac{x^{6}}{2} - \frac{2 x^{3}}{3} + 4 x + 1$

Most popular questions for Math Textbooks

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