Q. 77 - Calculus

Question

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  

$f\text{'}\left(x\right)=(x^4-8)(1-3x^5); f\left(0\right)=2$

Step-by-Step Solution

Answer

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

Step 1: Given information

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

Step 2: Find the antiderivative

We first simplify the given expression

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

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

Whose derivative is

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

Which is nearly equal now we adjust the coefficients

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

We are also given that 

$$\begin{gathered} f\left(0\right)=2 \\ Substituting we get, \\ 2=c \end{gathered}$$

Hence the antiderivative becomes

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

Step 3: Conclusion

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