Supposef(x) is a function whose derivative isf′(x) = (x+ 1)ex. Iff(1) =e,find an equation of the tangent line to the curvey=f(x) at the point (1,e). Write youranswer in slope intercept form.

Answer 1

I'm going to assume that you wrote this problem correctly. If the f'(x) = (x+1)ex, then you an distribute the ex term into both terms or f'(x) - ex^2 + ex. Think of e as just any other constant. Now integrate by power rule.

f(x) = (e/3)*x^3 + (e/2)x^2 + c. Now use the point to resolve for the c by plugging it in

e = e/3 + e/2 + c

So c = e/6

Therefore f(x) = (e/3)*x^3 + (e/2)x^2 + e/6