18.01SC | Fall 2010 | Undergraduate

Single Variable Calculus

2. Applications of Differentiation

Part B: Optimization, Related Rates and Newton's Method

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The study of the greatest and least outputs of a function – highest profits, least materials used, closest approach – is called optimization. In addition to optimization, this part of the course also describes how to use derivatives to relate two rates of change and to estimate the values for which a function’s output is zero.

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Overview

In this session you will:

• Do practice problems
• Use the solutions to check your work

Problems Set

Use Applications of Differentiation (PDF) to do the problems below.

Section Topic Exercises
2C Max-min problems 1, 2, 4, 10, 13
2E Related rates 2, 3, 5, 7
2F Locating zeroes; Newton’s method 1

Solutions

Solutions to Applications Differentiation problems (PDF)

This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.

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Overview

Often, our goal in solving a problem is to find extreme values. We might want to launch a probe as high as possible or to minimize the fuel consumption of a jet plane. Sometimes we’ll find our answer on the boundaries of our range of options – we launch the probe straight up. Sometimes we’ll find the best answer by using a derivative to determine when the graph of a function “levels out” at the top of a peak or bottom of a dip.

Lecture Video and Notes

Video Excerpts

Clip 1: Maxima and Minima Using Graphs

Clip 2: Maximum Area of Two Squares

Recitation Video

Minimum Triangle Area

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Overview

The maximum and minimum values of a function may occur at points of discontinuity, at the endpoints of the domain of the function, or at a “critical point” where the derivative of the function is zero. To determine whether a critical point is a global maximum or minimum we compare the value of the function at that point to its value at the other candidates for minima and maxima.

Lecture Video and Notes

Video Excerpts

Clip 1: Minimal Surface Area of a Box: Direct Solution

Clip 2: Minimal Surface Area of a Box: Implicit Differentiation

Worked Example

Can Design

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Overview

The derivative tells us how a change in one variable affects another variable. Related rates problems ask how two different derivatives are related. For example, if we know how fast water is being pumped into a tank we can calculate how fast the water level in the tank is rising. The chain rule is the key to solving such problems.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Related Rates

Clip 2: Related Rates

Clip 3: Rates: A Conical Tank

Worked Example

Solving an Optimization Problem Using Implicit Differentiation

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Overview

When Professor Jerison holds up a piece of string threaded through a ring, the ring slides to the lowest point possible. How would we use calculus to predict the location of that point? This is a minimization problem with an added twist: the position of the ring is constrained by the endpoints and length of the string.

Lecture Video and Notes

Video Excerpts

Clip 1: Ring on a String

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Overview

Newton’s method uses linear approximation to make successively better guesses at the solution to an equation. Starting from a good guess, Newton’s method can be extremely accurate and efficient. Occasionally it doesn’t work at all.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Newton’s Method

Clip 2: Newton’s Method, Continued

Clip 3: Newton’s Method: What Could Go Wrong?

Cube Root of x

Recitation Video

Using Newton’s Method

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Course Info

Fall 2010
Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos