Cambridge Additional Mathematics

150 151

C(150)

chord C(151)

()canswer

tangent

()banswer

1

390 Applications of differential calculus (Chapter 14)

Example 16 Self Tutor

The cost in dollars of producingxitems in a factory each day is given by

C(x)=0|:000 13x^3 {z+0: 002 x^2 }

labour

+5|{z}x

raw materials

+ 2200|{z}

fixed costs

a Find C^0 (x), which is called the marginal cost function.

b Find the marginal cost when 150 items are produced. Interpret this result.

c Find C(151)¡C(150). Compare this with the answer inb.

a The marginal cost function is

C^0 (x)=0:000 39x^2 +0: 004 x+5dollars per item.

b C^0 (150) =$ 14 : 38

This is the rate at which the costs are increasing with

respect to the production levelxwhen 150 items are

made per day.

It gives an estimate of the cost of making the 151 st item

each day.

c C(151)¡C(150)¼$ 3448 : 19 ¡$ 3433 : 75

¼$ 14 : 44

This is the actual cost of making the 151 st item each day, so the answer inbgives a good

estimate.

4 Seablue make denim jeans. The cost model for making

xpairs per day is

C(x)=0: 0003 x^3 +0: 02 x^2 +4x+ 2250 dollars.

a Find the marginal cost function C^0 (x).

b Find C^0 (220). What does it estimate?

c Find C(221)¡C(220). What does this represent?

d Find C^00 (x) and the value ofxwhen C^00 (x)=0.

What is the significance of this point?

5 The total cost of running a train from Paris to Marseille is

given by C(v)=^15 v^2 +

200 000

v

euros wherevis the

average speed of the train in km h¡^1.

a Find the total cost of the journey if the average speed is:

i 50 km h¡^1 ii 100 km h¡^1.

b Find the rate of change in the cost of running the train

at speeds of:

i 30 km h¡^1 ii 90 km h¡^1.

c At what speed will the cost be a minimum?

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\390CamAdd_14.cdr Thursday, 10 April 2014 3:53:42 PM BRIAN