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

$f'(x)=\dfrac{4x}{1+7x^2}$

Integrating gives

$f(x)=\displaystyle\int\frac{4x}{1+7x^2}\,\mathrm dx$

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

$f(x)=\dfrac27\ln(1+7x^2)+C$

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

$10=\dfrac27\ln1+C\implies C=10$

so that

$\boxed{f(x)=\dfrac27\ln(1+7x^2)+10}$