18.01SC | Fall 2010 | Undergraduate

# Single Variable Calculus

2. Applications of Differentiation

## Part A: Approximation and Curve Sketching

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This section describes how differentiation can be used to simplify complex calculations and graph functions.

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### Overview

In this session you will:

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

### Problem Set

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

Section Topic Exercises
2A Approximation 1, 3, 6, 11, 12a, 12d, 12e
2B Curve sketching 2a, 2e, 2h, 4, 6a, 6b, 7a, 7b

#### Solutions

Solutions to Applications of Differentiation problems (PDF)

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

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### Overview

Linear approximation is a powerful application of a simple idea. Very small sections of a smooth curve are nearly straight; up close, a curve is very similar to its tangent line. We calculate linear approximations (i.e. equations of tangent lines) near x=0 for some popular functions; we can then change variables to get approximations near x=a.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction to Linear Approximation

Clip 2: Linear Approximation to ln x at x=1

Clip 3: Linear Approximation and the Definition of the Derivative

Clip 4: Approximations at 0 for Sine, Cosine and Exponential Functions

### Worked Example

Comparing Linear Approximations and Calculator Computations

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Approximations at 0 for ln(1+x) and (1+x)r

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### Overview

When using linear approximation, we replace the formula describing a curve by the formula of a straight line. This makes calculation and estimation much easier.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Curves are Hard, Lines are Easy

Clip 2: Linear Approximation of a Complicated Exponential

Clip 3: Question: Can We Use the Original Formula?

### Worked Example

Product of Linear Approximations

### Lecture Video and Notes

#### Video Excerpts

Clip 1: GPS Time Dilation Example

Clip 2: Relative Error

### Recitation Video

#### Implicit Differentiation and Linear Approximation

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### Overview

Linear approximation uses the first derivative to find the straight line that most closely resembles a curve at some point. Quadratic approximation uses the first and second derivatives to find the parabola closest to the curve near a point.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: The Formula for Quadratic Approximation

Clip 2: Explaining the Formula by Example

Clip 3: Quadratic Approximation at 0 for Several Examples

### Worked Example

Comparing Quadratic Approximations and Calculator Computations

### Recitation Video

#### Quadratic Approximation of an Exponential Function

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### Overview

In this session we see some examples of quadratic approximation, then finish compiling a “library” of quadratic approximations of key functions.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: List of Approximations

Clip 2: Example: ln((1+1/k)k)

Clip 3: Example: Approximation of a Complicated Function

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Approximating ln(1+x) and (1+x)r

### Worked Example

Finding a Formula for the Best Degree n Approximation

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### Overview

The first derivative of a function tells us whether its graph slopes up or down or is level. The second derivative tells us how that slope is changing. By combining this information with what we know about asymptotes, intercepts and plotting points we can sketch a pretty good graph of the function.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction to Curve Sketching

Clip 2: Example: 3x-x3

Clip 3: Graphing a Rational Function

### Recitation Video

#### Sketching a Curve

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### Overview

This session outlines a step by step method of graphing a function then provides an example of this method. Finally, you are asked to turn things around and find the equation of a function given a description of its graph.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: General Strategy for Graph Sketching

Clip 2: Example: x/ln x

### Worked Example

Graph Features

Use the mathlet below to complete this worked example.

### Mathlet

Concave and Convex Mathlet

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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