#### 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