Suggested languages for you:

Americas

Europe

Q. 73

Expert-verified
Found in: Page 198

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# 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$f\text{'}\left(x\right)=3{x}^{5}-2{x}^{2}+4;f\left(0\right)=1$

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

See the step by step solution

## Step 1: Given information

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

## Step 2: Find the antiderivative

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

On differentiating the above function we get,

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

Now we only adjust the coefficient

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

We are also given that $f\left(0\right)=1\phantom{\rule{0ex}{0ex}}Substittuingthisinthefunctionweget,\phantom{\rule{0ex}{0ex}}f\left(0\right)=c\phantom{\rule{0ex}{0ex}}c=1$

Hence the antiderivative becomes

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

## Step 3: Conclusion

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