Application of Derivative Class 12th Best Project by Shubham prasad
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Application of Derivative Class 12th Best Project by Shubham prasad, Student of Nalanda English Medium School Kurud Bhilai Durg Chhattisgarh. Art Integrated Learning on Mathematics branch Application of Derivatives Class 12th Ncert
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This document provides an overview of differentiation and its applications. It defines differentiation as finding the slope of the tangent line to a function's graph at a given point, which provides the instantaneous rate of change. The document then lists the group members working on the topic, outlines the contents to be covered, and gives a brief history of differentiation. It provides definitions and graphical understandings of derivatives, discusses some basic differentiation formulas and their applications in mathematics, sciences, business, physics, chemistry and more. It concludes that derivatives are constantly used to measure rates of change in various everyday and professional contexts.
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MTH 2001: Project 2
Instructions
• Each group must choose one problem to do, using material from chapter 14 in the textbook.
• Write up a solution including explanations in complete sentences of each step and drawings or computer
graphics if helpful. Cite any sources you use and mention how you made any diagrams.
• Write at a level that will be comprehensible to someone who is mathematically competent, but may
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Problems
1. You probably studied Newton’s method for approximating the roots of a function (i.e. approximating
values of x such that f(x) = 0) in Calculus 1:
(1) Guess the solution, xj
(2) Find the tangent line of f at xj,
y = f′(xj)(x−xj) + f(xj) (1)
(3) Find the tangent line’s x-intercept, call it xj+1,
0 = f′(xj)(xj+1 −xj) + f(xj)
xj+1f
′(xj) = xjf
′(xj) −f(xj)
xj+1 = xj −
f(xj)
f′(xj)
(2)
(4) If f(xj+1) is sufficiently close to 0, stop, xj+1 is an approximate solution. Otherwise, return
to step (2) with xj+1 as the guess.
See this animation for a geometric view of the process. It simply follows the tangent line to the curve
at a starting point to its x-intercept, and repeats with this new x value until we (hopefully) find a
good approximation of the solution.
Newton’s method can be generalized to two dimensions to approximate the points (x,y) where the
surfaces z = f(x,y) and z = g(x,y) simultaneously touch the xy-plane. (In other words, it can
approximate solutions to the system of equations f(x,y) = 0 and g(x,y) = 0.) Here, the method is
http://www.fit.edu/mac/group_projects_presentations.php
http://upload.wikimedia.org/wikipedia/commons/e/e0/NewtonIteration_Ani.gif
(1) Guess the solution (xj,yj)
(2) Find the tangent planes to each f and g at this point.
z = f(x,y) =
z = g(x,y) =
(3) Find the line of intersection of the planes.
(4) Find the line’s xy-intercept, call this point (xj+1,yj+1),
xj+1 =
yj+1 =
(5) If f(xj+1,yj+1) < ε and g(xj+1,yj+1) < ε for some small number ε (error tolerance), stop,
(xj+1,yj+1) is an approximate solution. Otherwise, return to step (2) with (xj+1,yj+1) as
the guess.
(a) Find equations of the tangent planes for step (2), an equation for their line of intersection for step
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(b) What assumptions must we make about f and g in order for the method to work? How might
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In mathematics, derivative is defined as the method that shows the simultaneous rate of change. That means it is used to represent the amount by which the given function is changing at a certain point.
In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation.[1] The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
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- 1. NALANDA ENGLISH MEDIUM HIGHER SEC. SCHOOL KURUD BHILAI (C.G.) Session : 2020-2021 Art Integrated Learning On Mathematics Topic : Applications of Derivative Guided By : Mrs. Saida Mam Submitted By : Shubham , Prem, Rajeev, Rohit, Rounak Class – 12th 'A’
- 2. With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature -Alfred North Whitehead Introduction to Derivative 1. The Derivative is the exact rate at which one quantity changes to another. 2. Geomatrically, The derivative is the slope of the curve at the point on curve. 3. The Derivative is often called the 'Instantaneous’ rate of change. 4. The Derivative of function represents an infinitely small change the function with respective one of its variable
- 3. Real Life Applications of Derivative • AutoMobiles - In an automobile there is always an odometer and a speedometer. These two gauges work in tandem and allow the driver to determine his speed and his distance that he has traveled. • Radar Guns - The gun is able to determine the time and distance at which the radar was able to hit a certain section of your vehicle. • Business - You can estimate the profit and loss point for certain ventures. • Graphs - The most common application of derivative is to analyze graphs of data that can be calculated from many different fields. Using derivative one is able to calculate the gradient at any point of a graph.
- 4. Increasing and Decreasing Function Increasing and Decreasing Function :- Let f(x) be a function defined on the interval a<x<b, and let x1 and x2 be two numbers in the interval, Then f(x) is increasing on the interval if f(x2)>f(x1) whenever x2>x1 f(x) is decreasing on the interval if f(x2)<f(x1) whenever x2 >x1
- 5. Increasing and Decreasing Function By Tangent and Normal Tangent line with Positive slope F(x) will be increasing Tangent line with Negative slope F(x) will be decreasing Positive Slopes Negative Slopes f’(x)>0 on a<x<b So, f(x) is increasing f’(x)<0 on a<x<b So, f(x) is increasing
- 6. Local Maxima and Local Minima Relative Local Extrema : A function f(x) has a relative local maximum value at an interior point c of its domain if f(x) ≤ f(c) for all x in some open interval containing c. A function f(x) has a relative local minimum value at an interior point c of its domain if f(x) ≥ f(c) for all x in some open interval containing c.
- 7. ThankYou Submitted By :- Shubham Prasad , Prem raj Singh, Rohit Gupta, Rajeev Sahu, Rounak Kumar Class 12th A