Rates of Change (The 2nd Pillar of Calculus)

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A picture is worth 1000 words

Trigonometry Review

The basic trig functions
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Exponential Functions

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Close is good enough
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Toolbox of graphs

Rates of Change

Tracking change
Exploring average velocity
Instantaneous velocity
Instantaneous rate of change of any function
Finding tangent line equations
Definition of derivative

The Derivative Function

The derivative function
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The Chain Rule

Two forms of the chain rule
Version 1
Version 2
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A hybrid chain rule

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Derivatives of Logs

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Derivatives in Science

In Physics
In Economics
In Biology

Related Rates

Overview
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Example (ladder)
Example (shadow)

Linear Approximation and Differentials

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An example with negative

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Differentiation Review

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Absolute and Local Extrema

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$c$

$f$ vs. $f'$

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Indeterminate Forms and L'Hospital's Rule

$\frac{0}{0}$ equal?
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Optimization

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Another Example

Newton's Method

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Anti-derivatives

Anti-derivatives and Physics
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Anti-derivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sums Notation
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Definite Integrals

Definition
Properties
What is integration good for?
More Examples

The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

Instantaneous velocity

To get the instantaneous veloctity at a particular time $t=a$ , we compute the average velocity between time $a$ and $a + \Delta t$ , and then take a limit as $\Delta t \to 0$ .

In the animations below, $a=3$ , the $x$ coordinate of point $P$ ; and $a+\Delta t$ is the $x$ -coordinate of point $Q$ .

(Animations from Barry McQuarrie)

The following video shows how this is done, and relates it to the slope of the line tangent to $y=s(t)$ at time $t=a$ . In both cases we get a limit:

$\lim_{\Delta t \to 0} \frac{s(a+\Delta t)-s(a)}{\Delta t}.$ This can also be written as:

$\lim_{x \to a} \frac{s(x)-s(a)}{x-a},$ where

$x=a+\Delta t$ . This quantity (if the limit exists) is called the derivative of

$s(t)$ at time

$t=a$ .