Please help with AP Calc

Question

https://us-static.z-dn.net/files/db8/fc1b84547f2f08b33d5641b90a8639b1.jpg

Answer 1

I'll just help you with FRQ 2 (a):

Given: g(x) = 2x + ∫₀^x f(t) dt.

(a) Compute for g(5). Find g'(x) and evaluate g'(3)

First solve ∫₀⁵ f(t) dt.

∫₀⁵ f(t) dt = ∫₀² f(t) dt + ∫₂⁵ f(t) dt

Using the piecewise function f(x):

∫₀⁵ f(t) dt = ∫₀² 4 dt + ∫₂⁵ (10 - 3t) dt

= [ 4t ]₀² + [ 10t - (3/2)t² ]₂⁵

= 4(2) +10(5) + (3/2)(5)² - 10(2) - (3/2)(2)²

= 8 + 50 + (3/2)(25) - 20 - 6

= 32 + 75/2

= 64/2 + 75/2

= 139 / 2 = 69.5

So to solve g(5):

g(5) = 2(5) + ∫₀⁵ f(t) dt

g(5) = 10 + 69.5

g(5) = 79.5

g'(x) = 2 + (d/dx) ∫₀^x f(t) dt

Using the fundamental theorem of calculus:

g'(x) = 2 + f(x)

g'(-3) = 2 + f(-3) = 2+ (4-(-3+3))^1/2

g'(-3) = 4