# Calculus .pdf

31 May 2023                  1 sur 18

### Calculus .pdf

• 1. and it's applications! and it's applications! C A L C U L U S By: Aadya Gupta Iram Zaheer
• 2. It all began when.... Sir Isaac Newtom made his breakthrough in the field of physics which focused on gravity and laws of motion. Newton started by trying to describe the speed of a falling object. When he did this, he found that the speed of a falling object increases every second, but that there was no existing mathematical explanation for this. Newton saw a void that needed to be filled "Calculus is the most powerful weapon of thought yet devised by the wit of man. -Sir Isaac Newton
• 3. The issue of movement and the rate of change had not yet been explored to any significant degree in the field of mathematics. He began work on this right away, incorporating planetary ellipses into his theory in an attempt to explain the orbit of the planets. He found that by using calculus, he could explain how planets moved and why the orbits of planets are in an ellipse. This is one of Newton’s great epiphanies: that the gravitational force that holds us to the ground is the same force that causes the planets to orbit the Sun and the Moon to orbit Earth
• 4. What is Calculus? The word Calculus comes from Latin meaning "small stone" Because it is like understanding something by looking at small pieces. Derived from the Latin "ca/x' (counter) - ancient Babylonians would use pebbles to represent units, tens, hundreds, etc, on a primitive abacus. Later, defined as as the instantiates rate of change of one variable in respect to another variable is called derivation of function. Geometrically the derivative is the slop of curve at the point on the curve. The process of finding a derivative is called differentiation. The réverse process is called anti-differentiation. The fundamental theorem of calculus states that anti-differentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
• 5. Types of calculus: Differential Calculus- a branch of mathematics concerned mainly with the study of the rate of change of functions with respect to their variables especially through the use of derivatives and differentials. 1. >First Branch of mathematical analysis, devised by Isaac Newton and G.W. Leibniz, are concerned with the problem of finding the rate of change of a function with respect to the variable on which it depends. Thus it involves calculating derivatives and using them to solve problems involving non- constant rates of change
• 6. Types of calculus: 2. Intergral Calculus- a branch of mathematics concerned with the theory and applications (as in the determination of lengths, areas, and volumes and in the solution of differential equations) of integrals and integration was the first know use of intergral calculus circa 1741 >Integral calculus is concerned with the theory and applications of Integrals while differential calculus focuses on rates of change, such as slopes of tangent lines and velocities, integral calculus deals with total size or value, such as lengths, areas, and volumes.
• 8. Calculus in the field of engineering: Calculus initially developed for better navigation system. Engineers use calculus for building skyscrapers, bridges. In robotics calculus is used how robotic parts will work on given command. Electrical and Computer engineers use calculus for Calculus is used to improve safety of vehicles. system design.
• 9. Calculus in Science Isaac Newton developed the use of calculus in his laws of motion and gravitation. Astronomical science deeply depend on calculus as they use it to determine the paths of satellites, rockets, and space vehicles, as well as to convey signals in the global positioning system when compressed data is sent. Astronauts use calculus to determine how the spaceship itself moves. For example, calculus is used to determine the rate of speed required for the spaceship to successfully reach space from the Earth.
• 10. Calculus in Biology Calculus is used to study the interaction of different species within an ecosystem calculus used The concentration of drugs in a living organism can be answered with calculus. Physical attributes, such as backbone length and skull length measured by calculus. Calculus is used for measuring growth rate of bacteria and certain species.
• 11. Derivatives in Physics In physics, the derivatives of the displacement of a moving body with reference to time is the velocity of the body, and the derivative of velocity with reference to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body better known as F = ma
• 12. Derivatives in Mathematics: The most common use of the derivatives in Mathematics is to study functions such as: I. Extreme values of functions II.The mean value theorem IlI.Concavity & curve sketching
• 13. A tech company which wants to produce a cooling system for their processors that can cool down the processors from a temperature of 50°C to 27°C in just half an hour when the temperature outside is 20°C but they don't know what kind of materials they should use or what the surface area and the geometry of the shape are. So what should they do? They simply have to use the general formula of Newton's law of cooling Computer Processor Manufacture
• 14. And by substituting the numbers they get 27 = 20 + (50 - 20) e^-0.5k Solving for k we get k=2.9 so they need a material with k=2.9 (k is a constant that is related to the heat capacity, thermodynamics of the material and also the shape and the geometry of the material) Computer Processor Manufacture
• 15. In an automobile there is always an odometer and a speedometer. These allow the driver to determine his speed and his distance that he has traveled. Electronic versions of these gauges simply use derivatives to transform the data sent to the electronic motherboard from the tires to miles per hour(MPH) and (KM). Automobiles
• 16. Economic research uses calculus functional relationships. Relation of income, market prediction etc. Differentiation is used to find optimum solutions of economics. Calculus also used for maximising the efficiency or products or to minimize their costs. Credit card companies use calculus for monthly payments. Calculus in Economics
• 17. Business and politicians often conduct surveys with the help of calculus. Investment plans do not pass unless approved by mathematicians Doctors often use calculus in the estimation of the progression of the illness. Global mapping is done with the help of calculus. Calculus also used to solve paradoxes. Other fun facts: