Some of 2.3, and all of 2.1Rates of Change, Slopes, and

DerivativesA. What algebra skill you’ll need for 8.2B. Where average rate of change (ARC) is on a graphC. How to find it algebraically (without a graph)D. How we get the difference quotients definition of ARCE. Where instantaneous rate of change (IRC) is on a graphF. How we get the difference quotient definition of IRCG. How to find the IRCH. What a derivative isI. What Leibniz’s Notation looks likeJ. Word Problems

A. What algebra skill you’ll need for 8.2

Negative, positive slopeSlope formulaCombining rational expressions

B. Where average rate of change (ARC) is on a graph The average rate

of change(ARC) between x = 2 and x = 5 is the SLOPE of that red dotted line.

[a secant line]

Let’s find it:

C. How to find it algebraically (without a graph)

Well, since the ARC is really a SLOPE, let’s recall the slope formula:

We can use that f(x) notation because we’ll be dealing with functions. Recall that f(x) simply means the y value that corresponds to x.

Let’s see if we can apply this to a problem:

12

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xx

xfxf

xx

yy

run

rise

12

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2

."2 and 1between change of rate

average thefind ,53)( If"

xx

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xx

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average thefind ,1)( If" :You try

xx

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xx

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D. How we get the difference quotients definition of ARC

You might have been introduced to the famous “difference quotient” in algebra. In case you weren’t, here is what it looks like:

I’ll show you where it came from. In part F, you’ll find out what is so great about it.

h

xfhxf

Let’s take this curve, and label a fixed point x. It would have corresponding y-value labeled f(x).

I am going to jog over a certain distance h on the x-axis. What could I call this new fixed point?

x + h would be that newly created point on the x-axis. Its corresponding y-value would be called f(x + h).

Here is the red, dotted secant line. We need the slope of it.

RISE =

RUN =

E. Where instantaneous rate of change (IRC) is on a graph

It’s the slope of the tangent line. I’ll draw it:

Sometimes they ask you whether the slope of the tangent line is positive, negative, or zero.

F. How we get the difference quotient definition of IRC

Remember the secant line that was h wide?

It was h wide. Imagine you could make h shrink (“approach zero”).

That would get closer and closer to the slope of the tangent line! The IRC!

h

xfhxfIRC

h

0lim

This will be useful because we can use this formula algebraically (no graphing necessary to find or guess at the IRC.)

Let’s see if we can work one…

G. How to find the IRC Find the instantaneous rate of change of this

function at x = 1.2)( xxf

STATE FORMULA

PLUG IN X = 1

f(1 + h) would be _________

_______________________

f(1) would be _________

SIMPLIFY

CANCEL THE h’S

SUB IN ZERO FOR h

.2at 1

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rate ousinstantane theFind:You try2 xxxf

H. What a derivative is

We can evaluate the IRC at any point we want algebraically, but wouldn’t it be easier if we could create a function for the IRC, and then we could plug in any value. It would be more efficient.

The DERIVATIVE is just that. Basically, it is a function for the IRC.

When you see “IRC,” think “Derivative.”

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by found becan ,by noted ,derivative The

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lim:You try

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h

I. What Leibniz’s Notation looks like

8.3section in again up come willThis

.dx

dy wroteLeibniz , writingof Instead

.dx

d wroteLeibniz , writingof Instead

y

xfxf

J. Word Problems

First of all, the units of the ARC or IRC is always consistent with it being a rate. [Like, “something” per “something”]

More particularly, the first “something” is the units of the function f(x), and the second “something” is the units of the x.

“something” per “something”

it.interpret and ,6 at time etemperatur

of IRC theFind .Fahrenheit degrees 1507)(

is hours at time oil theof re temperatu theSuppose rates.

different at cooling and heating requires oil crude Refining

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of IRC theFind .Fahrenheit degrees 122)(

is hours at time oil theof re temperatu theSuppose :You try2

x

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x

Use the limit definition to find an EQUATION (y=mx+b) of the tangent line to the graph of f at the given point:

1,2;1

1 1,1;2

x

xfxxf

The word “differentiable” means…_________________________________

If it is differentiable at a point, then it is continuous at that point.

BUTJust because it is continuous at a point

doesn’t necessarily mean it is differentiable there.

Example:

What suggested HW problems to try?

In section 2.3, try numbers 3-11 odd, 13a, 17a. We will do more of section 2.3 in the future.

In section 2.1, OMIT # 11, 23, 33, 43, 47, 49, 61. With #15-23, the y-value is extra information because you only use the x-coordinate.