Calculus Homework

Question

a) Find (f^-1)'(a).

f(x)=x^3+3sinx+2cosx, a=2

b)lim x->infinity [ln(1+x^2)-ln(1+x)]

Answer 1

a) Find (f^–1)'(a).

f(x) = x³ + 3sinx + 2cosx, a = 2

Set f(x) = 2 and solve for x

x³ + 3sinx + 2cosx = 2 when x = 0

Calculate f'(0)

f'(x) = 3x² + 3cosx – 2sinx

f'(0) = 3

(f^–1)'(3) = 1/f'(0) = 1/3

b) ln(1+x²) – ln(1+x) = ln[((1 + x²)/(1 + x)] → ln x as x →∞ and ln x→∞ as x →∞

So [ln(1+x²) – ln(1+x)] →∞ as x →∞