# Ap Calculus BC 2002 Form B Question 4?

##### 1 Answer

a) We're going to have to recall the FTC here.

#g(6) = 5 + int_6^6 f(t) dt#

We know that

#g'(6) = d/dx(int_6^x f(t) dt) = f(6) = 3#

#g''(6) = d/dx(f(6)) = 0# because the tangent is horizontal

b) Since

This will be

c) Concavity is determined by the second derivative.

#g''(x) = f'(x)#

We are looking for places on the given graph where the tangent line's slopes are negative, or when the graph is decreasing (since we seek the intervals where

This will occur on

d) The trapezoidal approximation is simply done by drawing trapezoids on the graph and adding up their areas.

#A = 3(-1)/2 + 3(1)/2 + (1 + 3)(3)/2 + (1 + 3)(3)/2 + 3(1)/2 + 3(-1)/2#

#A = 6 + 6 = 12#

Thus the approximation for

Hopefully this helps!