Everything Maths Grade 12

7.6 CHAPTER 7. DIFFERENTIAL CALCULUS

We have seen that the coordinates of the turningpoint can be calculatedby differentiating the function
and finding the x-coordinate (speed in the case of the example) for which the derivative is 0.

Differentiating (7.19), we get:

f�(v) =

3

40

v− 6

If we set f�(v) = 0 we can calculate the speed that corresponds tothe turning point.

f�(v) =

3

40

v− 6

0 =

3

40

v− 6

v =

6 × 40

3

= 80

This means that the most economical speed is 80 km/h.

See video: VMhgi at http://www.everythingmaths.co.za

Example 13: Optimisation Problems

QUESTION

The sum of two positivenumbers is 10. One of the numbers ismultiplied by the squareof the

other. If each number isgreater than 0 , find the numbers that make this product a maximum.

SOLUTION

Step 1 : Examine the problem and formulate the equations that are required

Let the two numbers be a and b. Then we have:

a +b = 10 (7.20)

We are required to minimise the product of a and b. Call the product P.

Then:

P = a.b (7.21)

We can solve for b from (7.20) to get:

b = 10−a (7.22)

Substitute this into (7.21) to write P in terms of a only.

P = a(10−a) = 10a−a^2 (7.23)

Step 2 : Differentiate

The derivative of (7.23)is:

P�(a) = 10− 2 a

Step 3 : Find the stationary point

Set P�(a) = 0 to find the value of a which makes P a maximum.