To determine
a)
To evaluate:
i. g0
ii. g1
iii. g2
iv. g3
v. g6
Answer
i. g0=0
ii. g1=2
iii. g2=5
iv. g3=7
v. g6=3
Explanation
Given:
gx=∫0xf(t)dt
Calculation:
i. To evaluate g0
The integral of a function over a trivial interval is zero so
g0=∫00f(t)dt =0
ii. To evaluate g1
g1=∫01f(t)dt
Geometrically g(1) is the area under the graph from 0 to 1.
The area is two whole squares
g1=2
iii. To evaluate g2
The area is four whole squares and triangle with base 1 and height 2
g2=∫02f(t)dt=∫01f(t)dt+∫12f(t)dt
g2=2+2+12·2·1=5
iv. To evaluate g3
g3=∫03f(t)dt=g2+∫23ftdt
Simplifying and estimating the integral to the right as the area of a triangle with base 1 and height 4 we have
g3=5+12·4
g3=7
v. Similarly we shall evaluate g6.Now since the the area under the curve from 3 to 6 is below x-axis we shall add negative of the area to get net area, which shall be the value of the integral.
g6=∫06ftdt=g3-∫36ftdt
g6=7-(2+12·2.2)
Simplifying
g6=3
Therefore, g0=0,g1=2,g2=5, g3=7,g6=3
To determine
b)
To find:
On what interval is g increasing
Answer
[0, 3]
Explanation
More area is added to the total area as x increases from 0 to 3
Therefore, g is increasing on the interval [0, 3]
Alternate justification is since f is continuous by fundamental theorem of calculus. If gx=∫0xf(t)dt then g'x=f(x).The function g is increasing when g' is positive. But g'x=f(x). So g is increasing when f(x). The function f(x) is positive in [0,3] so g(x) is increasing in [0,3]Therefore, g is increasing on the interval [0, 3]
To determine
c)
To find:
Where does g have maximum value
Answer
g reaches its maximum at t=3 ,g3=7
Explanation
Calculation:
From the estimated values. So g reaches its maximum when t reaches 3
To estimate g3
g3=∫03f(t)dt
This is the total area between the graph of g and x-axis from x=0 to 3
To evaluate g3
g3=∫03f(t)dt=g2+∫23ftdt
Simplify
g3=5+12·4
g3=7
Therefore, g reaches its maximum at t=3 ,g3=7
To determine
d)
To sketch:
A rough graph of g
Answer
Explanation
From part a) we have
g0=0,g1=2,g2=5, g3=7,g6=3
With the help of the above information, graph is
