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Q. 74

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 $f\text{'}\left(x\right)=7{x}^{2}+8{x}^{11}-18;f\left(0\right)=-2$

The antiderivative can be given as $f\left(x\right)=\frac{7}{3}{x}^{3}+\frac{8}{12}{x}^{12}-18x-2$

See the step by step solution

Step 2: Given information

We are given the derivative as $f\text{'}\left(x\right)=7{x}^{2}+8{x}^{11}-18;f\left(0\right)=-2$

Step 2: Find the antiderivative

We know that differentiating a power function decreases the power by one we can start with the function $f\left(x\right)={x}^{3}+{x}^{12}-18x+c$

On differentiating the function we get,

$f\text{'}\left(x\right)=3{x}^{2}+12{x}^{11}-18$

which is nearly equal

Now we only adjust the coefficient

$f\left(x\right)=\frac{7}{3}{x}^{3}+\frac{8}{12}{x}^{12}-18x+c$

Also we are given $f\left(0\right)=-2\phantom{\rule{0ex}{0ex}}Substitutingthisinthefunctionweget,\phantom{\rule{0ex}{0ex}}c=-2$

Hence the antiderivative becomes

$f\left(x\right)=\frac{7}{3}{x}^{3}+\frac{8}{12}{x}^{12}-18x-2$

Step 3: Conclusion

The antiderivative can be given as $f\left(x\right)=\frac{7}{3}{x}^{3}+\frac{8}{12}{x}^{12}-18x-2$