#### To determine

**(a)**

**To find:**

The marginal cost function

#### Answer

The marginal cost function = 2-0.04x+0.00021x2

#### Explanation

**1. Concept:**

Use differentiation

**2. Formula:**

i. C'x=ddx(Cx)

ii. Power rule:

ddxxn=nxn-1

iii. Sum rule:

ddxfx+gx=ddxfx+ddx(gx)

iv. Constant multiple rule:

ddxCf(x)=Cddx(fx)

v. Difference rule:

ddxfx-gx=ddxfx-ddx(gx)

vi. Constant function rule:

ddxC=0

**3. Given:**

Cx=920+2x-0.02x2+0.00007x3

**4. Calculation:**

Marginal cost function is the derivative of cost function with respect to the amount of commodity produced

C'x=ddxCx=ddx(920+2x-0.02x2+0.00007x3)

By using sum and difference rule,

C'x=ddx(920)+ddx(2x)-ddx(0.02x2)+ddx(0.00007x3)

By using constant multiple rule and constant function rule,

C'x=0+2ddx(x)-0.02ddx(x2)+0.00007ddx(x3)

By using power rule,

C'x=2-0.022x+0.000073x2

⇒C'x=2-0.04x+0.00021x2

**Conclusion:**

Therefore, the marginal cost function = 2-0.04x+0.00021x2

#### To determine

**(b)**

**To find:**

C'100 and explain its meaning

#### Answer

C'100=0.1 and it means that the change in cost with respect to the amount of commodity produced at x = 100 is 0.1. Also it is the approximate cost of producing the 101^{st} unit.

#### Explanation

**1. Concept:**

Plug x = 100 in the answer from part a and explain

**2. Given:**

C'x=2-0.022x+0.000073x2

**3. Calculation:**

C'x=2-0.022x+0.000073x2

For x = 100

C'100=2-0.022100+0.0000731002=2-4+2.1=0.1

C'100 means that the change in cost with respect to the amount of commodity produced at x = 100 is 0.1. Also it is the approximate cost of producing the 101^{st} unit.

**Conclusion:**

C'100=0.1 and it means that the change in cost with respect to the amount of commodity produced at x = 100 . Also it is the 0.1 and is the approximate cost of producing the 101^{st} unit.

#### To determine

**(c)**

**To compare:**

C'100 with the cost of producing the 101^{st} item

#### Answer

The cost of producing the 101^{st} commodity is slightly larger than C’(100)

#### Explanation

**1. Concept:**

Using part (b) and comparing with C (101)

**2. Given:**

C'100=0.1

Cx=920+2x-0.02x2+0.00007x3

**3. Calculation:**

The cost of producing the 101^{st} item isC(101) – C(100)

C100=920+2*100-0.021002+0.000071003=990

C101=920+2*101x-0.021012+0.000071003=990.10107

C101-C100=990.10107-990=$ 0.101107

We have C'(100)= 0.1

So, the cost of producing the 101^{st} item is slightly larger than C’(100). The difference is negligible.

**Conclusion:**

The cost of producing the 101^{st} commodity is slightly larger than C’(101).