Q. 75 - 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  

$f\text{'}\left(x\right)=1-4x^6; f\left(1\right)=3$

Step-by-Step Solution

Answer

The antiderivative can be given as $f\left(x\right)=x-\frac{4}{7}{x}^7+\frac{18}{7}$

Step 1: Given information

We are given the derivative as $f\text{'}\left(x\right)=1-4x^6; f\left(1\right)=3$

Step 2: Find the antiderivative

We know that differentiating a power function decreases the power by one we can start with the function  $f\left(x\right)=x-x^7+c$

On differentiating the function we get, 

$f\text{'}\left(x\right)=1-7x^6$

which is nearly equal

Now we only adjust the coefficient

$f\left(x\right)=x-\frac{4}{7}{x}^7+c$

We are also given 

$$\begin{gathered} f\left(1\right)=3 \\ Substituting we get, \\ 3=\frac{3}{7}+c \\ c=\frac{18}{7} \end{gathered}$$

$f\left(x\right)=x-\frac{4}{7}{x}^7+\frac{18}{7}$

Step 3: Conclusion

The antiderivative can be given as $f\left(x\right)=x-\frac{4}{7}{x}^7+\frac{18}{7}$