Suppose f′′(x)=−(sin(x)), f′(0)=−2, and f(0)=5

Question

Find f(π/4):

Answer 1

f''(x) = -sin(x)

Integrating

f'(x) = cos(x) + C₁

Integrating

f(x) = sin(x) + C₁x + C₂

Using initial conditions

f'(0) = -2 = cos(0)+C₁ --> C₁ = -3

f(0) = 5 = sin(0) - 3(0) + C₂ --> C₂ = 5

f(π/4) = sin(π/4/) -3(π/4) + 5

= √2/2 -(3/4)π +5 ≈ 3.351

Answer 2