Differential calculus is often used in optimisation problems. That is, if we have a quantity that is varying according to a function (that can be differentiated) we can optimise that quantity by finding maximum and minimum values. We will look at applications involving area, volume, business, finance and motion.
Caution is required when dealing with optimisation problems and we need to undertake a sanity check on the solutions as they may not make sense for the application given. For example, a function may give negative values but if it was representing a volume or an area relationship those negative values would be nonsensical.
The general strategy for solving optimisation problems is:
- Determine the quantity to be maximised or minimised
- Determine a function for this quantity with respect to one variable
- Differentiate the function to be optimised and find and classify maximum and minimum points accordingly
- Complete a sanity check of your results
Worked example
Example 1
The manager of a computer shop has determined that the costs $C$C for storage and handling of new computers as a function of the number of computers $x$x is given by:
| $C$C | $=$= | $15x+\frac{24000}{x}+6000$15x+24000x+6000 |
Determine how many computers the manager should order to minimise costs.
Think: We are asked to find the number of computers that minimises the storage and handling costs $C$C, so we will need to find a local minimum point for the function given.
Do: To find minimum turning points we need to find the first derivative of the function and determine the location of any stationary points. We can then use the second derivative to classify the points.
Finding the first derivative of the cost function gives:
| $C'$C′ | $=$= | $15-\frac{24000}{x^2}$15−24000x2 |
Solving $C'=0$C′=0 to find the location of any stationary points:
| $15-\frac{24000}{x^2}$15−24000x2 | $=$= | $0$0 |
| $\frac{24000}{x^2}$24000x2 | $=$= | $15$15 |
| $x^2$x2 | $=$= | $\frac{24000}{15}$2400015 |
| $x^2$x2 | $=$= | $1600$1600 |
| $x$x | $=$= | $\pm40$±40 |
At this point we can discount $x=-40$x=−40 as we cannot order a negative number of computers.
Let's now find the second derivative to determine the nature of the stationary point at $x=40$x=40:
| $C''$C′′ | $=$= | $\frac{48000}{x^3}$48000x3 |
At $x=40$x=40, $C''$C′′ will be positive and hence this point is a local minimum as the curve is concave up at this point. Therefore, ordering $40$40 computers will minimise storage and handling costs.
Practice questions
Question 1
A manufacturer of mens shirts determines that her profit $P$P varies with respect to the number of shirts produced according to $P=21x-0.2x^{\frac{3}{2}}-500$P=21x−0.2x32−500.
Find the derivative, $P'$P′.
What is the $x$x-value of the stationary point?
Find the second derivative, $P''$P′′.
Which of the following is correct about the stationary point?
The stationary point is a point of inflection because the second derivative at this point is zero.
AThe stationary point is a minimum because the second derivative at this point is positive.
BThe stationary point is a maximum because the second derivative at this point is negative.
CTherefore what number of shirts corresponds to a maximum profit?
Question 2
A pool is being emptied, and the volume of water $V$V litres left in the pool after $t$t minutes is given by the equation $V=1500\left(11-t\right)^3$V=1500(11−t)3, for $0\le t\le11$0≤t≤11.
Find the rate of change of the volume after $t$t minutes, $V'\left(t\right)$V′(t).
Enter each line of working as an equation.At what rate is the volume of water in the pool changing after $10$10 minutes?
Select the time $t$t at which the pool is emptying at the fastest rate.
$t=11$t=11 mins
A$t=5$t=5 mins
B$t=0$t=0 mins
C$t=10$t=10 mins
D
Question 3
A rectangular cereal box with a square base and open top is to have a volume of $256$256 cm3.
If the side lengths of the base measure $x$x cm, and the height of the box measures $h$h cm, express $h$h in terms of $x$x.
Let the surface area of the open box be represented by $S$S.
Find an equation for $S$S in terms of $x$x only, simplifying where possible.
Now find the possible value(s) of $x$x that will minimise the amount of material required to make the box. Give your answer(s) to the nearest integer if necessary.
Complete the table to prove that when $x=8$x=8 cm, the least packaging material is used.
Give both values to the nearest integer.
$x$x $7$7 $8$8 $9$9 $\frac{dS}{dx}$dSdx $\editable{}$ $0$0 $\editable{}$
Motion and optimisation
We can use the techniques of optimisation in the study of motion of a body to find maximum or minimum values for displacement, velocity, and acceleration of a body.
Velocity is the rate of change of displacement and acceleration is the rate of change of the velocity. So given the displacement as a function of time, we are able to determine functions for velocity and acceleration of the body using the following relationships:
For a displacement function $x(t)$x(t) the instantaneous velocity $v(t)$v(t) is the derivative of the displacement function with respect to time:
$v(t)=\frac{dx}{dt}$v(t)=dxdt
The instantaneous acceleration $a(t)$a(t) is the derivative of the velocity function with respect to time:
$a(t)=\frac{dv}{dt}$a(t)=dvdt
Practice questions
Question 4
A particle $P$P starts from rest at point $O$O, and its velocity $t$t seconds after it starts moving is modeled by the equation $v=t^2\left(9-t\right)$v=t2(9−t). After $9$9 seconds it comes to rest and stops moving.
Determine an equation for the acceleration $a$a of the particle after $t$t seconds.
At what time $0
Write each line of working as an equation.
Calculate the greatest velocity that the particle obtains in the first $9$9 seconds.
Graph the acceleration of the particle over the first $9$9 seconds on the axes below:
Loading Graph...What is the magnitude of the greatest acceleration that the particle achieves in the first $9$9 seconds?
Question 5
The height in metres of a projectile above level, flat ground is given by $h=9+8t-t^2$h=9+8t−t2, where $t$t is given in seconds.
What is the initial height of the projectile?
Find the maximum height reached.
When will the projectile reach the ground?
Draw the graph of the height of the projectile over time.
Loading Graph...How far will the projectile travel in the first $5$5 seconds?