Teacher didn't explain this well in class, please help.

Question

f(x)={0, if x< -3

{3, if -3 ≤ x < 0

{-5, if 0 ≤ x < 5 

{0, if x ≥ 5

x

g(x) = ∫ f(t)dt

-3

Determine the value of each of the following:

A.) g(−6)=

B.) g(−2)=

C.) g(1)=

D.) g(6)=

E.) The absolute maximum of g(x) occurs when x= which makes g(x)=

Answer 1

g(x) = ∫_{(-3 to x)} f(x)dx

g(-6) = ∫_{(-3 to -6)}f(x)dx = -∫_{(-6 to -3)} 0dx = 0

g(-2) = ∫_{(-3 to -2)} f(x)dx = ∫_{(-3 to -2)} 3dx = 3x_{(-3 to -2)} = -6 - (-9) = 3

g(1) = ∫_{(-3 to 1)} f(x)dx = ∫_{(-3 to 0)} 3dx + ∫_{(0 to 1)}(-5)dx = 9 - 5 = 4

g(6) = ∫_{(-3 to 6)} f(x)dx = ∫_{(-3 to 0)} 3dx + ∫_{(0 to 5)} (-5)dx + ∫_{(5 to 6)}0dx = 9 - 25 + 0 = -16

Maximum value of g is g(0) = 9.

Answer 2

Since is f(x)=0 for x<-3 and x>5, only the intervals -3 to zero and 0 to 5 matter.

The discontinuity at x=0 does not affect the value of the integral at all.

When f(x) is a positive constant the integral of f is a straight line with a positive slope and

if f(x) is a negative constant, the integral will be a straight line with a negative slope.

Therefore, g(x) looks like a triangle pointing downward (i.e. in the negative y direction).

With this information you should be able to draw a graph of g and answer the questions.