Can someone please help me with this AP Calculus problem?? show work.?

Question

Let f be a continuous fuction whose graph is shown above(the graph start at (-3,3), and goes to the right with a slope of zero until (0,3), then the graph decreases with a slope of -1 until the point (6,-3), where it then goes to the right with a slope of zero to the point (9,-3)) and let h(x)=integral from -2 to x of f(t) dt.
1) Find the zeros of h. Justify your answer.
2) On what interval is h increasing? Show the analysis that leads to your conclusion.
3) What is h(0)?
4) For x is greater than or equal to 0 and less than or equal to 6, express h in terms of x.

Answer 1

The integral gives the area under the curve.
So the integral from -2 to x will be the area under the curve between -2 and x. Keep in mind that the integral is positive when f(x) is above the x-axis and negative when f(x) is below the x-axis.

1. h will equal zero when the area above the x-axis equals the area below the x-axes.
2. h is increasing when f(x) is above the x-axis.
3. h(0) is the area of the triangle formed between x=-2 and x=0
4. find the equation of the line for the interval from 0 to 6. Integrate this from 0 to x to get the area between 0 and x. Then add the area that's on the interval from -2 to 0 which is h(0) from question 3.

Answer 2

1) what are zeros
2) as f(t) is the derivative
+ve gradient is from (0,3)-(-3,3)
3)
4) d^2y/dx^2= -1
dy/dx= -x +c
x=0,y'=3 3=0+c
c=3
h(x)=y= -x^/2 +3x+k
for k u need to know h(0)

Answer 3

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