18.01SC | Fall 2010 | Undergraduate

# Single Variable Calculus

1. Differentiation

## Part A: Definition and Basic Rules

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This section explains what differentiation is and gives rules for differentiating familiar functions.

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### Overview

In this session you will:

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

### Problem Set

Use Differentiation (PDF) to do the problems below.

Section Topic Exercises
1A Graphing 1b, 2b, 3a, 3b, 3e, 6b, 7b
1B Velocity and rates of change 1a, 1b, 1c
1C Slope and derivative 1a, 3a, 3b, 3e, 4a, 4b, 5, 6, 2
1D Limits and continuity 1a, 1c, 1d, 1f, 1g, 3a, 3c, 3d, 3e, 6a, 8a
1E Differentiation formulas, polynomials, products, quotients 1a, 1c, 2b, 3, 4b, 5a, 5c
1J Trigonometric functions 1e, 2
1F Chain rule, implicit differentiation 1a, 1b, 2, 6, 7b, 7c
1J Trigonometric functions, continued 1a, 1k, 1m
1G Higher derivatives 1b, 1c, 5a

#### Solutions

Solutions to Differentiation problems (PDF)

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

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### Overview

This session provides a brief overview of Unit 1 and describes the derivative as the slope of a tangent line. It concludes by stating the main formula defining the derivative.

### Worked Example

Secants and Tangents

Please use the mathlet below to complete the problem.

### Mathlet

Secant Approximation

### Recitation Video

#### Graphing a Derivative Function

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### Overview

This session develops a formula for the derivative of a quotient. We use this to find the derivative of the multiplicative inverse of a function and so of x^{-n}.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Quotient Rule

Clip 2: Example: Reciprocals

### Worked Example

Quotient Rule Practice

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### Overview

We know how to take derivatives of sums, products and quotients of functions. In this session we discover the derivative of a composition of functions.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Chain Rule

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Example: sin(10t)

### Worked Example

Do We Need the Quotient Rule?

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### Overview

The derivative of a function is another function. We can, and sometimes do, take the derivative of the derivative. Once we’ve done that, we can take the derivative again!

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Higher Derivatives

Clip 2: Example: Dnxn

### Worked Example

Repeated Differentiation of Sine and Cosine

Use the mathlet below to complete the worked example.

### Mathlet

Creating the Derivative

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### Overview

In this session we apply the main formula for the derivative to the functions 1/x and x^n. We’ll also solve a problem using a derivative and give some alternate notations for writing derivatives.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Example 1: y=1/x

Clip 2: Harder Problem: Triangles Under the Graph of y=1/x

Clip 3: Notation for Derivatives

Clip 4: Example 2: y=xn

### Worked Example

The Derivative of |x|

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### Overview

We understand slope as the change in y coordinate divided by the change in x coordinate. Considering change in position over time or change in temperature over distance, we see that the derivative can also be interpreted as a rate of change. If we think of an inaccurate measurement as “changed” from the true value we can apply derivatives to determine the impact of errors on our calculations.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction to Rates of Change

Clip 2: Rates of Change

Clip 3: Physical Interpretation of Derivatives

Clip 4: Physical Interpretation of Derivatives, Continued

### Worked Example

Checking Account Balance

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### Overview

The main formula for the derivative involves a limit. This session discusses limits in more detail and introduces the related concept of continuity.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Limits

Clip 2: Continuity

### Worked Example

Continuous but not Smooth

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### Overview

Derivatives are defined in terms of limits of functions. Where those functions are not continuous, limits (and derivatives) may be undefined. This session discusses the different ways functions may be discontinuous and how differentiability is related to continuity.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Jump Discontinuities

Clip 2: Removable Discontinuities

Clip 3: Infinite Discontinuities

Clip 4: Other (Ugly) Discontinuities

Clip 5: Differentiable Implies Continuous

### Worked Example

Limits and Discontinuity

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### Overview

Finding derivatives of varied functions requires a variety of tools. This session uses the main formula to find the derivative of the sum of two functions.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction to Differentiation

Clip 2: Derivative of the Sum of Two Functions

### Recitation Video

#### Tangent to Graph of Polynomial

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### Overview

The sine and cosine functions are used to describe periodic phenomena such as sound, temperature and tides. Trying to differentiate these functions leaves us with two limits to investigate further.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Derivative of sin(x), Algebraic Proof

Clip 2: Derivative of cos(x)

### Worked Example

Geometric Derivatives of Sine and Cosine

Use the mathlet below to complete the worked example.

### Mathlet

Creating the Derivative

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### Overview

When we tried to differentiate the sine and cosine functions we were left with two limits to calculate. In this session Professor Jerison calculates these limits, taking a close look at the unit circle and applying some fundamental ideas from linear approximation.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Limit of sin(x)/x

Clip 2: Limit of (1-cos(x))/x

Clip 4: Derivative of sin(x), Geometric Proof

### Worked Example

The Function sinc(x)

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### Overview

In this session we apply the main formula to a product of two functions. The result is a rule for writing the derivative of a product in terms of the factors and their derivatives.

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Introduction of Product and Quotient Rules

Clip 2: Introduction to General Rules for Differentiation

Clip 3: Product Rule

### Worked Example

Smoothing a Piecewise Polynomial

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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