Everything Maths Grade 12

CHAPTER 7. DIFFERENTIAL CALCULUS 7.6

(3; 1)

�

x

y

From this drawing, the point (3; 1) is a horizontal point ofinflexion, because the sign of the derivative

does not change from positive to negative.

7.6 Using DifferentialCalculus to Solve Problems

EMCBN

We have seen that differential calculus can be used to determine the stationary points of functions, in

order to sketch their graphs. However, determining stationary points also lends itself to the solution of

problems that require some variable to be optimised.

For example, if fuel usedby a car is defined by:

f (v) =

3

80

v^2 − 6 v + 245 (7.19)

where v is the travelling speed, what is the most economical speed (that means the speed that uses the

least fuel)?

If we draw the graph of this function we find thatthe graph has a minimum. The speed at the minimum

would then give the most economical speed.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100 110 120 130 14 0

fuel consumption (l/h)

speed (km/h)