#### To determine

**(a)**

**To estimate:**

The average velocity of a car during the first 12 seconds using the midpoint rule.

#### Answer

4523 km/h

#### Explanation

**1) Concept:**

Use the mean value theorem for integrals and the midpoint rule to estimate the average velocity of the car.

**2) Mean Value Theorem for IntegralsandMidpoint rule:**

The Mean Value Theorem: If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-a∫abfx dx

that is,

∫abfx dx=fcb-a

The midpoint rule:

∫abfxdx≈∑i=1nfxi- ∆x=∆x fx1-+…+fxn-

where ∆x=b-an and xi-=12xi-1+xi midpoint of xi-1, xi.

**3) Calculation:**

By using the mean value theorem for integrals:

Here,a=0 and b=12

The average velocity of the car during the first 12 seconds means the average value of v on 0,12.

vave=112-0∫012vx dx

vave=112∫012vx dx

Since we can change the variable in the integration,

vave=112∫012vtdt

Use themidpoint rule.

With n=3, a=0 and b=12 interval width is

∆t=b-an

Substitute values.

∆t=12-03

∆t=4

The three subintervals are 0,4, 4,8, 8,12.

And the midpoints are

t1-=12t0+t4=120+4=124=2

t2-=12t4+t8=124+8=1212=6

t3-=12t8+t12=128+12=1220=10

So, the value of the integral is

∫012vtdt≈M3=∑i=13vti- ∆t

=∆tvt1-+vt2-+vt3-

=4v2+v6+v10

From the given graph,

v2=21

v6=50

v10=66

=421+50+66=548

Therefore,

∫012vtdt~548

Thus,

vave=112∫012vtdt~112548=4523kmh=45.66kmh

**Conclusion:**

The average velocity of the car during the first 12 seconds using the midpoint rule is 4523kmh.

#### To determine

**(b)**

**To find:**

The time where the instantaneous velocity equal to the average velocity

#### Answer

t≈5.2 s

#### Explanation

**1) Concept:**

Use graph to find the time where the instantaneous velocity equal to the average velocity

**2) Calculation:**

As instantaneous velocity is equal to average velocity,

So from part (a), average velocity = 4523 km/h

Therefore,

instantaneous velocity = 4523 km/h

From the graph, at t≈5.2 s,

instantaneous velcotiy= 45.66=4523 km/h

**Conclusion:**

Therefore, the time where the instantaneous velocity equal to the average velocity is 5.2 s