<< Chapter < Page Chapter >> Page >

Using differential calculus to solve problems

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 used by a car is defined by:

f ( v ) = 3 80 v 2 - 6 v + 245

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 that the graph has a minimum. The speed at the minimum would then give the most economical speed.

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

Differentiating [link] , we get: f ' ( v ) = 3 40 v - 6 If we set f ' ( v ) = 0 we can calculate the speed that corresponds to the 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 · hr - 1 .

Video on calculus - 4

The sum of two positive numbers is 10. One of the numbers is multiplied by the square of the other. If each number is greater than 0, find the numbers that make this product a maximum.

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

    a + b = 10

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

    P = a · b

    We can solve for b from [link] to get:

    b = 10 - a

    Substitute this into [link] to write P in terms of a only.

    P = a ( 10 - a ) = 10 a - a 2

  2. The derivative of [link] is: P ' ( a ) = 10 - 2 a

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

    P ' ( a ) = 10 - 2 a 0 = 10 - 2 a 2 a = 10 a = 10 2 a = 5

    Substitute into [link] to solve for the width.

    b = 10 - a = 10 - 5 = 5
  4. The product is maximised if a and b are both equal to 5.

Michael wants to start a vegetable garden, which he decides to fence off in the shape of a rectangle from the rest of the garden. Michael only has 160 m of fencing, so he decides to use a wall as one border of the vegetable garden. Calculate the width and length of the garden that corresponds to largest possible area that Michael can fence off.

  1. The important pieces of information given are related to the area and modified perimeter of the garden. We know that the area of the garden is: A = w · l We are also told that the fence covers only 3 sides and the three sides should add up to 160 m. This can be written as: 160 = w + l + l

    However, we can use [link] to write w in terms of l : w = 160 - 2 l Substitute [link] into [link] to get: A = ( 160 - 2 l ) l = 160 l - 2 l 2

  2. Since we are interested in maximising the area, we differentiate [link] to get: A ' ( l ) = 160 - 4 l

  3. To find the stationary point, we set A ' ( l ) = 0 and solve for the value of l that maximises the area.

    A ' ( l ) = 160 - 4 l 0 = 160 - 4 l 4 l = 160 l = 160 4 l = 40 m

    Substitute into [link] to solve for the width.

    w = 160 - 2 l = 160 - 2 ( 40 ) = 160 - 80 = 80 m
  4. A width of 80 m and a length of 40 m will yield the maximal area fenced off.

Solving optimisation problems using differential calculus

  1. The sum of two positive numbers is 20. One of the numbers is multiplied by the square of the other. Find the numbers that make this product a maximum.
  2. A wooden block is made as shown in the diagram. The ends are right-angled triangles having sides 3 x , 4 x and 5 x . The length of the block is y . The total surface area of the block is 3600 cm 2 .
    1. Show that y = 300 - x 2 x .
    2. Find the value of x for which the block will have a maximum volume. (Volume = area of base × height.)
  3. The diagram shows the plan for a verandah which is to be built on the corner of a cottage. A railing A B C D E is to be constructed around the four edges of the verandah.
    If A B = D E = x and B C = C D = y , and the length of the railing must be 30 metres, find the values of x and y for which the verandah will have a maximum area.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 12 maths' conversation and receive update notifications?