apcalcApplications+of+Differentiation

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= Table of Contents = = = = = = = flat

Before we learn the various **applications of differentiation**,

first we must do a brief overview of some essential
= **Differentiation Rules**: =

Also, we need to know the basic derivatives of Trigonometric Functions as well as the basic derivatives of Exponential and Logarithmic Functions:


====To get a full understanding of all the differention rules, you should watch the viedo above and visit: http://middletownhighschool.wikispaces.com/apcalcDifferentiation+Rules====

= History of Differentiation = ====The concept of a derivative in the sense of a [|tangent line] is a very old one, familiar to [|Greek] geometers such as [|Euclid], [|Archimedes] and [|Apollonius of Perga]. The modern development of calculus can be credited to [|Isaac Newton] and [|Gottfried Leibniz] who provided independent and unified approaches to differentiation and derivatives. Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as [|Augustin Louis Cauchy] (1789 – 1857), [|Bernhard Riemann] (1826 – 1866), and [|Karl Weierstrass] (1815 – 1897).====



Now since we know everything we need to know about differentiation, we can apply it to the real world. We will now be able to analyze the behavior of families of functions, how to solve related rates problems (how to calculate rates that we can't measure from those that we can), and how to find the maximum or minimum value of a quantity.

= Related Rates =



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= Maximum and Minimum Values = ====First, let's explain exactly what we mean by maximum and minimum values. A function f has an **absolute maximum** (or global maximum) at c if f(c) is greater than or equal to f(x) for all x in D, where D is the domain of f. The number f(c) is called the **maximum value** of f on D. Similarly, f has an **absolute minimum** at c if f(c) is less than or equal to f(x) for all x in D and the number f(c) is called the **minimum value** of f on D. The maximum and minimum values of f are called the **extreme values** of f.==== ====A function f has a **local maximum** (or relative maximum) at c if f(c) is greater than or equal to f(x) when x is near c. Also, f has a **local minimum** at c if f(c) is less than or equal to f(x) when x is near c.====

After understanding maximum and minimum values, we can now learn how to solve optimization problems.
= Optimization Problems = ====﻿Optimization problems involve problems such as maximizing areas, volumes, and profits and minimizing distances, times, and costs. In solving such practical problems the greatest challenge lies in converting the word problem into a mathematical optimization problem by setting up the function that is to be maximized or minimized. Therefore, we must follow these six simple steps:====

**Example 2:** A rectangular box with a square base and no top is to have a volume of 108 cubic inches. Find the dimensions for the box that require the least amount of material.
====The function that is to be minimized is the surface area ( //S//) while the volume ( //V//) remains fixed at 108 cubic inches.media type="youtube" key="-vzPzBJ81gY" height="349" width="425" align="right"==== .
 * [[image:http://media.wiley.com/Lux/19/39419.nfg002.jpg width="265" height="134" align="center"]] ||
 * ====The open-topped box for example 2.==== ||  ||   ||   ||
 * [[image:http://media.wiley.com/Lux/19/39419.nfg002.jpg width="265" height="134" align="center"]] ||
 * ====The open-topped box for example 2.==== ||  ||   ||   ||
 * ====The open-topped box for example 2.==== ||  ||   ||   ||
 * ====The open-topped box for example 2.==== ||  ||   ||   ||

Letting //x// = length of the square base and //h//= height of the box, you find that

 * [[image:http://media.wiley.com/Lux/68/39368.nce017.gif align="center"]] ||
 * [[image:http://media.wiley.com/Lux/68/39368.nce017.gif align="center"]] ||

with the domain of //f(x)// = (0,+∞) because //x//represents a length.

 * [[image:http://media.wiley.com/Lux/70/39370.nce018.gif align="center"]] ||
 * [[image:http://media.wiley.com/Lux/70/39370.nce018.gif align="center"]] ||

hence, a critical point occurs when //x//= 6. Using the Second Derivative Test:

 * [[image:http://media.wiley.com/Lux/72/39372.nce019.gif align="center"]] ||
 * [[image:http://media.wiley.com/Lux/72/39372.nce019.gif align="center"]] ||

least amount of material are a length and width of 6 inches and a height of 3 inches.
= Applications to Economics = ====With our calculus knowledge of differentiation rules, minimum and maximum values, and optimization problems, we can apply differentiation to economics. Before looking at sample problems, we must understand basic calculus applications to economics:====


 * Total Cost || //C//(//x//) ||
 * Marginal Cost || //C'//(//x//) ||
 * Average Cost || [[image:http://www.blc.edu/fac/rbuelow/calc/eqn4-7_284.gif width="81" height="41" align="absMiddle"]] ||
 * Price Function || //p//(//x//) ||
 * Revenue Function || //R//(//x//) = //x// //p//(//x//) ||
 * Marginal Revenue || //R'//(//x//) ||
 * Profit Function || //P//(//x//) = //R//(//x//) - //C//(//x//) ||
 * Marginal Profit || //P'//(//x//) = //R'//(//x//) - //C'//(//x//) ||

Example Problem:
|| Because of our principle that: "If the average cost is a minimum, then marginal cost = average cost," we set the average cost formula equal to the marginal cost formula and attempt to solve. || -11.52974818 - 54.70830486 I, -11.52974818 + 54.70830486 I, 135.5594964 || Here we need a computer algebra system to find the solution to this cubic equation. We see there are two imaginary solutions and one real solution. The answer is approximately, This agrees with our visual estimate above. ||
 * Given the cost function: [[image:http://www.blc.edu/fac/rbuelow/calc/eqn4-7_283.gif width="224" height="24" align="absMiddle"]](a) Find the average cost and marginal cost functions.(b) Use graphs of the functions in part (a) to estimate the production level that minimizes the average cost.(c) Use calculus to find the minimum average cost.(d) Find the minimum value of the marginal cost. || Given Problem, #8, Lesson 4.7 ||
 * [[image:http://www.blc.edu/fac/rbuelow/calc/eqn4-7_287.gif width="402" height="44"]] || To find average cost, we know that we need to use the formula: .[[image:http://www.blc.edu/fac/rbuelow/calc/eqn4-7_284.gif width="81" height="41" align="absMiddle"]] ||
 * [[image:http://www.blc.edu/fac/rbuelow/calc/eqn4-7_286.gif width="177" height="24" align="center"]] || To find the marginal cost, we use the formula: //C'//(//x//) ||
 * We now move on to part (b)[[image:http://www.blc.edu/fac/rbuelow/calc/EX4_7A.JPG width="237" height="225"]] || We plot the graph of the average cost (red), and the graph of the marginal cost (green). By observation, it appears that the value of //x// (number of items produced) where the two graphs intersect is about //x// = 140. ||
 * We now work on part (c)
 * > solve(0=.0008*x^3-.09*x^2-339,x);

= Applications to Physics = ====In physics, the derivative of a function representing the position of a particle along a line at time //t// is the instantaneous velocity at that time. The derivative of the velocity, which is the second derivative of the position function, represents the //instantaneous acceleration// of the particle at time //t//. Below is an exaple applying differentiation to physics, as well as a video.====

**Example 1:** The position of a particle on a line is given by //s(t)// = //t//3 − 3 //t//2 − 6 //t// + 5, where //t// is measured in seconds and //s//is measured in feet. Find

 * 1) ====The velocity of the particle at the end of 2 seconds.====
 * 2) ====The acceleration of the particle at the end of 2 seconds.====

Part (a): The velocity of the particle ismedia type="youtube" key="pFeuGMMiZWw" height="349" width="425" align="right"

 * [[image:http://media.wiley.com/Lux/82/39382.nce024.gif align="absMiddle"]] ||
 * [[image:http://media.wiley.com/Lux/82/39382.nce024.gif align="absMiddle"]] ||

Part (b): The acceleration of the particle is

 * [[image:http://media.wiley.com/Lux/84/39384.nce025.gif align="absMiddle"]] ||  ||
 * [[image:http://media.wiley.com/Lux/84/39384.nce025.gif align="absMiddle"]] ||  ||

= Conclusion = == In conclusion, differentiation, or using calculus and differentiation rules to find derivitives, can be applied to almost anything. It can be applied to related rates, minimum and maximum values of functions, optimization, economics, physics, biology, chemistry, etc. Above are many ways showing how differentiation in calculus is applicable to the real world. ==

=﻿ Works Cited = "Acceleration, velocity, and Position." //Applications//. N.p., 2002. Web. 22 May 2011. []. Dawkins, Paul. "Differentiation Formulas." //Paul's Online Math Notes//. N.p., 2011. Web. 22 May 2011. []. "Differential Calculus." //Wikipedia//. Wikipedia.org, 25 05 2011. Web. 30 May 2011. []. Husch, Lawrence. "Applications of Differentiation." //Visual Calculus//. N.p., 2001. Web. 20 May 2011. []. "Related Rates." //SlideShare//. SlideShare Inc., 2011. Web. 26 May 2011. []. Stewart, James. //Calculus Concepts and Contexts//. 2nd ed. Pacific Grove, CA: Brookes/Cole, 2001. Print. Taylor, John. "Applications of Differentiation." //Aid for Calculus//. N.p., 2000. Web. 27 May 2011. []. Vickers, James. "Derivatives." //Introduction to Mathematical Methods//. N.p., 2005. Web. 18 May 2011. [].