Daniel B. answered • 03/14/23
A retired computer professional to teach math, physics
The integral is the area under the curve.
It can be divided into the following sections.
From -4 to 0 it is a quarter-circle of radius 4, so its area is
π4²/4 = 4π
From 0 to 2 it is a rectangle with sides length 2 and 4, so its area is
2×4 = 8
From 2 to 4 it is a triangle with base 2 and height 4, so its area is
2×4/2 = 4
From 4 to 6 it is a triangle with base 2 and height -4, so its area is
2×(-4)/2 = -4
From 6 to 8 it is a triangle with base 2 and height -4, so its area is
2×(-4)/2 = -4
a)
The value of each g(x) is the sum of the above regions from -4 to x
b) g(x) is increasing where its derivative -- f -- is positive.
That is between -4 and 4.
g(x) is decreasing where its derivative -- f -- is negative.
That is between 4 and 8.
c) g(x) is concave up where its second derivative -- f' -- is positive.
The answer to the question is the interval (6, 8) because that is where both f(t) < 0 and f'(t) > 0.
d) There are a couple definition of "inflection point".
The point x = 6 satisfies both definition, because to the left the derivative -- f --
is decreasing and to the right it is increasing.
Some definitions also identify all the points between 0 and 2 as inflection points,
because the second derivative -- f' -- is 0.
Other definitions do not call those points inflection, but undulation points.
I do not know what definition you used in class.
e) The equation of the tangent is
y - g(6) = g'(6)(x - 6)
You can get g(6) from section a), and
g'(6) = f(6) = -4
f) You can get the range from the results in a)
