18.01SC | Fall 2010 | Undergraduate

Single Variable Calculus

1. Differentiation

Part A: Definition and Basic Rules

« Previous | Next »

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

» Session 1: Introduction to Derivatives
» Session 2: Examples of Derivatives
» Session 3: Derivative as Rate of Change
» Session 4: Limits and Continuity
» Session 5: Discontinuity
» Session 6: Calculating Derivatives
» Session 7: Derivatives of Sine and Cosine
» Session 8: Limits of Sine and Cosine
» Session 9: Product Rule
» Session 10: Quotient Rule
» Session 11: Chain Rule
» Session 12: Higher Derivatives
» Problem Set 1

« Previous | Next »

« Previous | Next »

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.

« Previous | Next »

« Previous | Next »

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.

Lecture Videos and Notes

Video Excerpts

Worked Example

Secants and Tangents

Please use the mathlet below to complete the problem.

Mathlet

Secant Approximation

Recitation Video

Definition of Derivative

View video page

Graphing a Derivative Function

View video page

« Previous | Next »

« Previous | Next »

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

Recitation Video

Quotient Rule

View video page

Worked Example

Quotient Rule Practice

« Previous | Next »

« Previous | Next »

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

Recitation Video

Chain Rule

View video page

Lecture Video and Notes

Video Excerpts

Clip 1: Example: sin(10t)

Worked Example

Do We Need the Quotient Rule?

« Previous | Next »

« Previous | Next »

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

« Previous | Next »

« Previous | Next »

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

Recitation Video

Smoothing a Piecewise Function

View video page

Worked Example

Continuous but not Smooth

« Previous | Next »

« Previous | Next »

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

« Previous | Next »

« Previous | Next »

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

Constant Multiple Rule

View video page

Tangent to Graph of Polynomial

View video page

« Previous | Next »

« Previous | Next »

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)

Recitation Video

Derivatives of Sine and Cosine

View video page

Worked Example

Geometric Derivatives of Sine and Cosine

  Use the mathlet below to complete the worked example.

Mathlet

Creating the Derivative

« Previous | Next »

« Previous | Next »

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 3: Questions and Answers

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

Worked Example

The Function sinc(x)

« Previous | Next »

« Previous | Next »

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

Recitation Video

Product Rule

View video page

Worked Example

Smoothing a Piecewise Polynomial

« Previous | Next »

Course Info

Instructor
Departments
As Taught In
Fall 2010
Level
Undergraduate
Topics
Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos
Download Course
MIT Open Learning
Over 2,500 courses & materials
Freely sharing knowledge with learners and educators around the world. Learn more
© 2001–2024 Massachusetts Institute of Technology
Accessibility
Creative Commons License
Terms and Conditions
Proud member of:
© 2001–2024 Massachusetts Institute of Technology