ENGINEERING MATHEMATICS 1
Application of Differentiation
In Isaac Newton's day, one of the biggest problems was poor navigation at sea. Shipwrecks occurred 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. 1. 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.). 2. Derivatives are met in many engineering and science problems, especially when modelling the behaviour of moving objects. 3. Calculus is also a base of economics.
4. It is used in history, for predicting the life of a stone. 5. It is used in geography, which is used to study the gases present in the atmosphere. 6. It is mainly used in daily by pilots to measure the pressure in the air. 7. Area under a Curve and area in between the two curves are found by Integration. 8. Volume of Solid of Revolution explains how to use integration to find the volume of an object with curved sides, e.g. wine barrels. 9. Electric Charges have a force between them that varies depending on the amount of charge and the distance between the charges. We use integration to calculate the work done when charges are separated. 10. Average Value of a curve can be calculated using integration.
Picture 1 : Before calculus was developed, the stars were vital for navigation.
One topic about differentiation is rate of change. Rate of change is if two variables both vary with respect to time and have a relation between them, we can express the rate of change of one in terms of the other. Where we need to differentiate both sides with respect to time (d/dt). Our discussion begins with some general applications which we can then apply to specific problems.
Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6cm and height 10 cm.
, r radius, h height, V volume of the cylinder.
1. There the largest volume of cylinder inscribed a right circular core.
2. Radius of core is 6cm.
3. Height of cone is 10cm.
4. Formula value for cylinder is
H = height
1. Using differentiation formulae
2. Use rate of change for the volume of cylinder
1. Obtain the derivative of a function from first principle.
2. Think how to solve the question by using differentiation formulae. 1. We do a discussion among our group members.
2. We also surf the internet to find more information about the topic.
3. Other than that, we also go to the library to find reference books.
See the cone and cylinder from the front elevation and make dimension for the length, to make it easy to calculate.
Use the rate of change method to find the h, height. Compare the value or height and radius of cone. When,
(10 – h) cm
10 – h
10 - cm
Using the formula of cylinder, to find the maximum volume of cylinder inscribed the cone in value of r, radius.
10 - )
10 - )
Differentiate the V, volume to find the critical value or tangent of the function. Find the value of r, radius.
10 - )
= 0r - )
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