Riemann Sum Calculus Question

Question

Find f(x) if f'''(x)=20x^3+12x^2+4 with initial conditions f(0)= 7 and f(1)= -11

f(x)=

Answer 1

f '(x) = ∫f ''(x) dx

So f '(x) = ∫20x³ + 12x² + 4 dx = 5x⁴ + 4x³ + 4x + C₁

f(x) = ∫f '(x) dx = ∫5x⁴ + 4x³ + 4x + C₁ dx = x⁵ + x⁴ + 2x² + C₁x + C₂

Since f(0) = 7, we can say: 7 = (0)⁵ + (0)⁴ + 2(0)² + C₁(0) + C₂ or C₂ = 7

Since f(1) = -11, we can say:

-11 = (1)⁵ + (1)⁴ + 2(1)² + C₁(1) + 7

-11 = 1 + 1 + 2 + C₁ + 7

C₁ = -22

So f(x) = x⁵ + x⁴ + 2x² - 22x + 7