Differentiation

In this slide you will get all the information about DEFFERENTIATION

1. SHREE VAISHNAV ACADEMY BAL VIGYAN 2016-17

2. Topic - Differentiation Presented by- Vivek Jain Theme - Design for better tomorrow Sub Theme - Mathematical solution of real life

3. Acknowledgements I would like to express my special thanks of gratitude to my teacher as well as our principal H.C.Tiwari who gave me the golden opportunity to do this wonderful project on the topic DIFFERENTIATION , which also helps me to know about so many things. I am really thankful to them. Secondly I would also like to thank my parents and friends who helped me a lot in finalizing this project within the limited time frame.

4. Content 1.History Differentiation 2. Introduction of Differentiation 3. Reverse of Differentiation 4. Basic Formulas of Differentiation 5. Application of Differentiation 6.Differentiation in Economics 7. Differentiation in physics 8. Radar gun 9. Differentiation in Chemistry 10. Differentiation in Biology 11. Use of Differentiation in odometer & speedometer & Vector Derivative 12. Differentiation in society 13. Results of survey 14. Conclusion

5. Modern Differentiation and derivative usually credited to Sir Issac Newton and Gottfried Leibniz The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287–212 BC) and Apollonius of Perga (c. 262–190 BC).[1] Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents; see Archimedes' use of infinitesimals History of Derivative

6. Introduction of Differentiation - Differentiation is the action of computing a derivative. The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. Slope of graph = Δ y/Δx

7. Reverse of Differentiation (Integration) Integration is a way of adding slices to find the whole.Integration can be used to find areas, volumes, central points and many useful things.  finding an Integral is the reverse of finding a Derivative. Applications of the Indefinite Integral-Displacement from Velocity, and Velocity from AccelerationA very useful application of calculus is displacement, velocity and acceleration.Recall (from Derivative as an Instantaneous Rate of Change) that we can find an expression for velocity by differentiating the expression for displacement: v=​ds/dt

8. Similarly, we can find the expression for the acceleration by differentiating the expression for velocity, and this is equivalent to finding the second derivative of the displacemen. a=dv/dt=d^2s/dt^2 It follows (since integration is the opposite process to differentiation) that to obtain the displacement s of an object at time t (given the expression for velocity v) we would use: s=∫vdt Similarly, the velocity of an object at time t with acceleration a, is given by: v=∫a dt Integration -

9. 1.d(constant) /dx=0 2. d(x) /dx=1 3.d(log x) /dx=1/x 4. d(a x ) /dx=a x log a 5. d(e x ) /dx=e x 6. d(sin x)/dx =cos x 7.d(cos x) /dx = - sin x 8.d(tan x) /dx =sec^2 x 9.d(sec x) /dx =sec x. tan x 10.d(cosec x) /dx = - cosec x.cot x 11.d(cot x) /dx= - cosec^2 x 12.d(f).(g)/dx= f dg/dx +g df/dx 13.d[(f) /(g)] /dx =( g df/dx – f dg/dx ) /g^2 Basic Formulas of Differentiation

10. Application of Differentiation - 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 when modeling the behavior of moving objects. Other use of Differentiation are - 1.It is used in history, for predicting the life of a stone. 2.It is mainly used in daily by pilots to measure the pressure n the air. 3.It is also use in many subjects like Physics, Chemistry, Economics, etc. 4.It is used to solve problems including limits. 5.It is used to find local maxima and local minima

11. Most undergrad level core micro and macro involves fairly simple differentiation, you will do a lot of optimisation and use the chain rule and product rules a lot. One thing you will have to get used to in economics is seeing things written as functions and differentiating them. We always use differentiation to find Marginal Cost . Differentiation in Economics -

12. Mathematical use of Differentiation in Economics The concept of ‘marginals’ (marginal revenue, marginal product, marginal cost) etc is about the most important concept in microeconomics, because all decisions are taken ‘at the margin’. Do you increase production by another unit or just produce at the level you are doing? Well if your marginal revenue (the amount of revenue you will earn by producing another unit of output) is higher than your marginal cost (the amount it will cost you to produce another unit) then go for it. If your marginal cost is higher then you don’t. As you produce more your MR will fall and your MC will rise so you will maximise profits by producing where MR = MC. Basic golden rule of microeconomics.

13. Because MR is basically the ‘change in revenue over the change in output’ you find it by differentiating total revenue with respect to output. Total revenue is price x quantity. So we have TR=PQ MR=d(TR) /dQ So MR=d(PQ) /dQ PQ is P times Q, and TR and MR are ‘total revenue’ and ‘marginal revenue’ Use of Differentiation -

14. Differentiation in physics 1.Velocity : It is the derivative of position with respect to time. v=ds/dt 2.Acceleration :  It is the derivative of velocity with respect to time a=dv/dt 3.Momentum and Force: Momentum (usually denoted p) is mass times velocity, p=mv

15. 4.Total Energy: For so-called "conservative" forces, there is a function V(x) such that the force depends only on position and is minus the derivative of V, namely F(x)=−dV(x)/dx.The function V(x) is called the potential energy. For instance, for a mass on a spring the potential energy is 1/2kx^2where k is a constant, and the force is −kxThe kinetic energy is 1/2mv^2. Using the chain rule we find that the total energy d/dt(1/2mv^2+V(x))=mvdv/dt+V′(x)dx/dt Force (F) is mass times acceleration, sothe derivative of momentum is  dp/dt=d(mv)/dt=mdv/dt=ma

16. Radar gun - When a radar gun is pointed and fired at your care on the highway.The gun is able to determine the speed and distance at which the radar was able to hit a certain section of your vehicle.With the use of derivative it is able to calculate the speed at which the car was going and also report that the car was from the radar gun.

17. Radar speed guns, like other types of radar, consist of a radio transmitter and receiver. They send out a radio signal in a narrow beam, then receive the same signal back after it bounces off the target object. Due to a phenomenon called the Doppler effect , if the object is moving toward or away from the gun, the frequency of the reflected radio waves when they come back is different from the transmitted waves. From that difference, the radar speed gun can calculate the speed of the object from which the waves have been bounced. This speed is given by the following equation: where  c  is the speed of light ,  f  is the emitted frequency of the radio waves and Δ f  is the difference in frequency between the radio waves that are emitted and those received back by the gun. This equation holds precisely only when object speeds are low compared to that of light, but in everyday situations, this is the case and the velocity of an object is directly proportional to this difference in frequency. Working of Radar gun

18. Use of differentiation in chemistry In chemistry derivative are used to calculate instantaneous rate of reaction The instantaneous rate of reaction i.e. rate of reaction at any instant of time is the rate of change of concentration of any one of reactant or product at that particular instant. Instantaneous rate of reaction = dx/dt Here dx is small change in concentration in small interval of time dt For example , for the reaction R P Instantaneous rate of reaction =-d[R]/dt=+d[P]/dt

19. Use of differentiation in biology Growth of Bacteria: Suppose a droplet of bacterial suspension is introduced into a flask containing nutrients for the bacteria. The bacteria undergo cell divisions and the bacterial density is observed at intervals of time during a short period. The data is then fit to a model describing the bacterial density, N(t), observed at time t. Assume for three different types of bacteria, the growth rates are described by the following differential equations, 1. dN 1 /dt = 2N(t)/t, has solution N(t) = t 2. dN 2 /dt = 2N(t), has solution N(t)= e 2 t 3. dN 3 /dt = 0.2[1 + cos(0.5t)]N(t), has solution N(t) = e 0.2t+0.4 sin(0.5t)

20. Drug Sensitivity: It is extremely important for doctors to understand the characteristics of the drugs they prescribe to patients. The strength of the drug is given by R(M) where M measures the dosage, i.e. the amount of medicine absorbed in the blood, and the sensitivity of the patient’s body to the drug is the derivative of R with respect to M. For a certain drug, the drug strength is described by R(M) = 2M√(10 + 0.5M) where M is given in milligrams. Find R0’(50), the sensitivity to a dose of 50mg. Muscle Contraction: In 1938 Hill hypothesized the relationship between the rate at which a muscle contracts, v, under a given load, p. The Hill Equation is given by (p + a)v = b(p − p) where a, b, p are positive constants.

21. Use of differentiation in odometer and speedometer - In an automobile there is always a odometer and speedometer. These two gauges work in tandem and allow to determine his speed and his distance that he has traveled .Electronic version of this gauges simply use derivatives to transform the data sent to the electronic motherboard from the tires to mile per hour (MPH) and distance (KM). Speedometer is used to find instantaneous velocity

22. A vector derivative is a derivative taken with respect to a vector field. Vector derivatives are extremely important in physics, where they arise throughout fluid mechanics, electricity and magnetism, elasticity, and many other areas of theoretical and applied physics. Vector derivatives Types of vector derivative and their symbol

23. Differentiation in society Basically Differentiation is use in society to compare cost or margin of different things , as well as to compare marks for example – Let cost of car 1 is increased by 10000 INR and cost of car 2 is increased by 20000 INR , then Δ =10000 INR and Δ =20000 INR therefore Δ / Δ =1/2 Then change of cost of car 1 w.r.t. car 2 is 1/2

24. Survey 1 st In survey one we meet shopkeeper who sell various devices which work on the concept of differentiation and we get that maximum no. of shopkeeper do not know any thing about concept of it. Devices like odometer , speedometer etc . Survey 2 nd In our second Survey we went to meet different teacher of mathematics and science and we ask them that how differentiation can help to make tomorrow better and we get many different answers. 1.It is use in science therefore it make our tomorrow better. 2.

25. Conclusion With the help of previous slides we can conclude that Differentiation is use in almost everything . With the use of Differentiation, equipment like odometer, speedometer etc can be improved So, Differentiation is use to find change of one thing with respect to other , So Differentiation can help to make tomorrow better.

26. Bibliography 1.wikipedia.com 2.WolFarm.World.com 3.Fundamental Chemistry 4.Fundamental physics 5. R.D. Sharma 6 . www.Meritnation.com