BA201 ENGINEERING MATHEMATICS 2012 57 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's day, one of the biggest problems was poor navigation at sea. Before calculus was developed, the stars were vital for navigation. Shipwrecks occured because the ship was not where the captain thought it should be. There was not a good enough understanding of how the Earth, stars and planets moved with respect to each other. Calculus(differentiation and integration) was developed to improve this understanding. Differentiation and integration can help us solve many types of real-world problems. We use the derivativeto determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.). Derivatives are met in many engineering and science problems , especially when modelling the behaviour of moving objects. Our discussion begins with some general applications which we can then apply to specific problems. NOTES:a.There are now many tools for sketching functions (Mathcad, LiveMath, Scientific Notebook, graphics calculators, etc). It is important in t his section to learnthe basic shapes of each curve that you meet. An understanding of the nature of each function is important for your future learning. Most mathematical modelling starts with a sketch. b.You need to be able to sketchthe curve, showing important features. Avoid drawingx-yboxes and just joining the dots. c.We will be using calculusto help find important points on the curve.
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8/12/2019 Chapter 3 Application of Differentiation
3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH
Introduction to Applications of Differentiation
In Isaac Newton's day, one of the biggest problems was poor navigation at sea .
Before calculus was developed, the stars werevital for navigation.Shipwrecks occured because the ship was notwhere the captain thought it should be. Therewas not a good enough understanding of how theEarth, stars and planets moved with respect toeach other.Calculus (differentiation and integration) wasdeveloped to improve this understanding.Differentiation and integration can help us solvemany types of real-world problems .
We use the derivative to determine the maximum and minimum values of particular
functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).Derivatives are met in many engineering and science problems , especially whenmodelling the behaviour of moving objects.Our discussion begins with some general applications which we can then apply tospecific problems.NOTES:
a. There are now many tools for sketching functions (Mathcad, LiveMath, ScientificNotebook, graphics calculators, etc). It is important in this section to learn thebasic shapes of each curve that you meet. An understanding of the nature ofeach function is important for your future learning. Most mathematical
modelling starts with a sketch.b. You need to be able to sketch the curve, showing important features. Avoid
drawing x-y boxes and just joining the dots.c. We will be using calculus to help find important points on the curve.
8/12/2019 Chapter 3 Application of Differentiation
ONE OF THE most important applications of calculus is to motion in a straightline, which is called rectilinear motion.
In this matter, we must assume that the object is moving along acoordinate line. The object that moves along a straight line with position s = f (t ),
has corresponding velocity dsv
dt , and its acceleration
2
2
dv d sa
dt dt .
If t is measured in seconds and s in meters, then the units of velocity aremeters per second , which we abbreviate as m/sec. The units of acceleration arethen meters per second per second , which we abbreviate as m/sec².
s=0 -The particle at the beginning
- The particle returns back to O again
v=0 -the particle is instantaneously at rest
-maximum displacement
a=0 -constant velocity
-the particle is begin
t=0 -initial velocity
-initial acceleration
8/12/2019 Chapter 3 Application of Differentiation
The process of finding maximum or minimum values is called optimisation . We are trying to dothings like maximise the profit in a company, or minimise the costs, or find the least amount ofmaterial to make a particular object.
These are very important in the world of industry.
Example 1:
If the sum of height, h cm and radius, r cm of a cone is 15 cm . What is the maximum volume ofthe cone?
Solution:
( 1)
(2)
Substitute (1) into (2)
( )
8/12/2019 Chapter 3 Application of Differentiation
1. A particle is moving along a straight line where s is the distance travelled by theparticle in t seconds. Find the velocity and acceleration of the particle by usingthe following equations.
a. 2 36 2 s t t
b. 3 28 48 72 s t t t
c. 2 464 16 s t t
2. For the curve of 3 26 2 y x x , find
a. The coordinates of all the turning points.
b. The maximum and minimum points.
c. Sketch the graph for the above curve.
3. A particles moves along a straight line such that its distance, s meter from a fixed
point O is given by 2 39 6 s t t .
a. Find the velocity of the particles after 2 seconds and the acceleration
after 4 seconds.b. Find the acceleration when the velocity is 9 /m s .
8/12/2019 Chapter 3 Application of Differentiation