18.01SC | Fall 2010 | Undergraduate

Single Variable Calculus

2. Applications of Differentiation

Part C: Mean Value Theorem, Antiderivatives and Differential Equations

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The Mean Value Theorem is the key to proving that our abstract definition of a “derivative” faithfully describes our informal notion of a “rate of change.” The second half of this part of the course introduces notation for and discusses the possibility of reversing the process of differentiation.

» Session 34: Introduction to the Mean Value Theorem
» Session 35: Using the Mean Value Theorem
» Session 36: Differentials > » Session 37: Antiderivatives
» Session 38: Integration by Substitution > » Session 39: Introduction to Differential Equations > » Session 40: Separation of Variables > » Problem Set 5

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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
2G Mean Value Theorem 1b, 2b, 5, 6

Use Integration (PDF) to do the problems below.

Section Topic Exercises
3A Differentials, indefinite integration 1d, 1e, 2a, 2c, 2e, 2g, 2i, 2k, 3a, 3c, 3e, 3g
3F Differential equations: Separation of variables 1c, 1d, 2a, 2e, 4b, 4c, 4d, 8b

Solutions

Solutions to Applications of Differentiation problems (PDF)

Solutions to Integration problems (PDF)

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

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Overview

The mean value theorem tells us (roughly) that if we know the slope of the secant line of a function whose derivative is continuous, then there must be a tangent line nearby with that same slope. This lets us draw conclusions about the behavior of a function based on knowledge of its derivative.

Lecture Video and Notes

Video Excerpts

Clip 1: Description of the Mean Value Theorem

Clip 2: Consequences of the Mean Value Theorem

Recitation Video

Increasing or Decreasing

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Worked Example

Generalizing the Mean Value Theorem

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Overview

We can use the mean value theorem to prove that linear approximations do, in fact, provide good approximations of a function on a small interval. The information the theorem gives us about the derivative of a function can also be used to find lower or upper bounds on the values of that function.

Lecture Video and Notes

Video Excerpts

Clip 1: The Mean Value Theorem and Linear Approximation

Clip 2: The Mean Value Theorem and Inequalities

Worked Example

The Mean Value Theorem and the Sine Function

Recitation Video

Comparing Functions Using the MVT

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Derivative With a Discontinuity

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Overview

The differential dy represents an infinitely small change in the value of y. The derivative dy/dx can be thought of as a ratio of differentials. Using differentials will make our calculations simpler; here we see how they can be used to compute linear approximations.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Differentials

Recitation Video

Computing Differentials

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Lecture Video and Notes

Video Excerpts

Clip 1: Differentials and Linear Approximation

Recitation Video

Linear Approximation with Differentials

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Worked Example

Population Dynamics

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Overview

In this session we ask the question “for what function is this the derivative?” We continue to ask this question throughout this unit and the next.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Antiderivatives

Clip 2: Antiderivative of xa

Recitation Video

Computing Antiderivatives

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Lecture Video and Notes

Video Excerpts

Clip 1: More Antiderivatives

Worked Example

Exploiting Derivative Rules

Lecture Video and Notes

Video Excerpts

Clip 1: Unique Up to a Constant

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Overview

When asked to find the antiderivative of an expression involving familiar functions, we often have an idea of what the answer might be. We can then check and correct our guess by taking the derivative. This “advanced guessing” is related to the technique called “substitution”, in which we attempt to simplify the integrand as a step toward finding the derivative.

Lecture Video and Notes

Video Excerpts

Clip 1: Substitution

Clip 2: Integration by “Advanced Guessing”

Clip 3: More Examples of Integration

Recitation Video

Antidifferentiation by Substitution

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Worked Example

Antiderivative of tan x sec2x

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Overview

In this session we solve an assortment of simple differential equations.

Lecture Video and Notes

Video Excerpts

Clip 1: dy/dx=f(x)

Worked Example

Exponential Growth and Inhibited Growth

Recitation Video

xy’=(x2+x)(y2+1)

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Lecture Video and Notes

Video Excerpts

Clip 1: Differential Equations and Slope, Part 1

Clip 2: Differential Equations and Slope, Part 2

Recitation Video

y"=6x

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Course Info

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Fall 2010
Level
Undergraduate
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Exams with Solutions
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Problem Sets with Solutions
Simulations
Recitation Videos
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