Question

The slope f′(x) at each point (x,y) on a curve y=f(x) is given, along with a point (a,b) on the curve. Use this information to find f(x). f′(x) = 4x/(1 + 7x^2) (0,10) NOTE: OF absolute value symbols, | | , are needed for the answer, then use abs(expression). For example, ln|x| must be entered as ln(abs(x))

Answer 1

`f'(x)=(4x)/(1+7x^2)`

Integrating gives

`f(x)=\displaystyle\int(4x)/(1+7x^2)\,\mathrm dx`

To compute the integral, substitute
`u=1+7x^2`, so that
`\frac27\,\mathrm du=4x\,\mathrm dx`. Then

`f(x)=\displaystyle\frac27\int\frac{\mathrm du}u=\frac27\ln|u|+C`

Since
`u=1+7x^2>0` for all
`x`, we can drop the absolute value, so we end up with

`f(x)=\frac27\ln(1+7x^2)+C`

Given that
`f(0)=10`, we have

`10=\frac27\ln1+C\implies C=10`

so that

`\boxed{f(x)=\frac27\ln(1+7x^2)+10}`