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2.1 Notes - Calculus

Notes with practice problems for lecture 2.1: Basic Differentiation Ru...

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Calculus I (MATH M211)

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2 Notes: The Derivative & the Tangent Line Problem

AP CALCULUS AB

IN Standard(s):

C.D: State, understand,

and apply the definition of

derivative.

C.AD: Find the slope of a

curve at a point, including

points at which there are

vertical tangents and no

tangents.

Learning Target(s):

x I can find the slope of the tangent line to a curve at a point.

x I can solve problems involving the slope of a tangent line.

x I know that the derivative of ݂ is the function whose value

at ݔ is

௛՜଴

௙ሺ௫ା௛ሻି௙ሺ௫ሻ

௛

provided this limit exists.

x I can identify that the derivative at a point is the slope of

the line tangent to a graph at that point on the graph.

x I know that the difference quotients

௙ሺ௔ା௛ሻି௙ሺ௔ሻ

௛

and

௙ሺ௫ሻି௙ሺ௔ሻ

௫ି௔

express the average rate of change of a

function over an interval.

Success Criteria:

x I can explain the relationship between

differentiability and continuity.

x I can determine average rates of change of a

function over an interval using difference quotients.

x I can evaluate the instantaneous rate of change of

a function.

x I can use multiple notations for the derivative of a

function.

x I can evaluate derivatives represented graphically,

numerically, analytically, and verbally.

What is a Tangent Line?

Misconceptions about Tangent Lines

Writing a formal definition of what a tangent looks like is often problematic. The following four misconceptions can

be helpful when deciding whether a line is tangent to a curve.

Misconception #1:

A line is tangent to a

curve if the line crosses

the curve at exactly one

point.

Although line ܮ touches curve ܥat one point, line ܮ is

not a tangent line.

Misconception #2:

A tangent line to a curve

must cross the curve only

once.

Line ܮ is tangent to curve ܥ at point ܲ despite the fact

that line ܮ crosses curve ܥ at two other points.

Misconception #3:

A line is tangent to a

curve if the line touches

the curve at one point

but does not cross the

curve.

Again, although line ܮ touches curve ܥ at point ܲ, it is

still not considered a tangent line.

Misconception #4:

A tangent line to a curve

LVDOLQHWKDWMXVW ́JUD]HVμ

the curve at a point but

does not cross the curve.

In the example above, line ܮ is tan gent to curve ܥ at

point ܲ despite the fact that is crosses the curve.

Examples of Tangent Lines drawn to a curve ݂

ሺ ݔ

ሻ at a point ܲ

Note: $OWKRXJKPDQ\WLPHVZHPLJKWVD\WKDWDWDQJHQWOLQHGUDZQWRDFXUYHPD\RQO\ ́WRXFKμWKHFXUYH

one time, that is not entirely true. A secant line, by definition, is a line that can intersect a curve at least twice.

The Tangent Line Problem

Activity: Begin with the graph of ሻݔሺ݂ൌ ݕ

Step 1 /HW·VEHJLQZLWKDQDUELWUDU\SRLQWDQGFDOOLW ൫݂ǡܿ

ሺ ܿ

ሻ ൯. Then, OHW·VFKRRVH

another point on the curve that has a horizontal distance of ݔο away from our

initial ݔ value, ܿ. We can call that new point

൫ ݂ ǡݔο ൅ ܿ

ሺ ݔο ൅ ܿ

ሻ൯ . Place this point

on your curve ² preferably close to the right edge. Then connect those points.

What is the slope of that secant line joining those two points?

Step 2 : 1RZMXVWIRUNLFNVOHW·VOHWRXUVHFRQGെ ݔvalue from above move closer

to our original െ ݔvalue of ܿ. Reposition that point so that it is about half as close

൫ ݂ ǡܿ

ሺ ܿ

ሻ൯ . Then connect the two points.

Do we still see a secant line?

Is there any difference in the way we calculate its slope?

Step 3 : Since we are having such a blast moving our second െ ݔvalue closer to ܿ,

OHW·VPRYHLWone more time so that it almost touches ܿ. Reposition

൫݂ǡݔο ൅ ܿ

ሺ ݔο ൅ ܿ

ሻ ൯ so this it lies right next to ൫݂ǡܿ

ሺ ܿ

ሻ ൯ and then connect the two

points.

Do we still see a secant line?

How could we perceive the slope now using a very important concept from Unit 1?

Definition of a Tangent Line with Slope m

If ݂ is defined on an open interval containing ܿ and the if the limit

ο௫՜଴

ο௬

ο௫

ൌ

ο௫՜଴

௙

ሺ ௖ାο௫

ሻ ି௙

ሺ ௖

ሻ

ο௫

݉ ൌ

exists, then the line passing through the point

൫ ݂ǡܿ

ሺ ܿ

ሻ൯ with slope ݉ is the tangent line to the graph of ݂

at the point ൫݂ǡܿሺܿሻ൯.

It is important to note that

ο௫՜଴

௙ሺ௫ାο௫ሻି௙ሺ௫ሻ

ο௫

is the limit definition of a derivative. 7KH ́GHULYDWLYHμRID

function means to find the slope of the tangent line at a pointܿ ൌ ݔ**.** The process of finding a derivative

of a function is called differentiation. A function is ́ differentiableμ at a point ܿ ൌ ݔ if its derivative exists at

ܿ ൌ ݔ. A function is differentiable on an interval

ሺ ܾǡܽ

ሻ if it is differentiable for every െ ݔvalue in

ሺ ܾǡܽ

ሻ Ǥ

Note on notation: The derivative, ݂

ᇱ

ሺ ݔ

ሻ ZKLFKLVUHDG ݂́ prime of ݔ , μ is the most common notation used

to denote the derivative of ݂ ൌ ݕ

ሺ ݔ

ሻ . Others notations used are: ݂

ᇱ

ሺ ݔ

ሻ ݕ ǡ

ᇱ

ௗ௬

ௗ௫

ௗ

ௗ௫

ሺ ݔ

ሻ .

When we read the second notation DERYHZHVD\ ́WKHGHULYDWLYHRIݕ with respect to ݔ_._ μ

BETA

ftp

CtSx

fCd

Ye s

Kpfk'D

Asx

flasx

Example 2 : Given the function ݂ሺݔሻൌ ʹݔ

ଶ

െ ͳǡ find each of the following.

(a) Find the derivative of ݂

ሺ ݔ

ሻ . (b) Find the slope of ݂

ሺ ݔ

ሻ at ݔൌ ʹ

(denoted ݂

ᇱ

ሺ ʹ

ሻ )

ሺ ݔ

ሻ at ݔൌ ʹǤ

(denoted ݂ሺʹሻ)

(d) Find the equation of the tangent line of ሻݔሺ݂ at

ݔൌ ʹǤ

(e) Find each of the following.

ᇱ

ሺ ݔ

ሻ ൌ_______ ݂

ᇱ

ሺ ʹ

ሻ ൌ_______ ݂

ᇱ

ሺ ͳ

ሻ ൌ_______

ᇱ

ሺ െͳ

ሻ ൌ_______ ݂

ᇱ

ሺ െʹ

ሻ ൌ_______ ݂

ᇱ

ሺ Ͳ

ሻ ൌ_______

(f) Sketch ݂

ሺ ݔ

ሻ and the tangent line at ݔൌ ʹǤ (g) Answer each of the following.

On what intervals is ݂ increasing? _____________

On what intervals is ݂ decreasing? _____________

In general, what do you think about the relationship

between a curve and its derivative? (just dating? Engaged? -)

What would the converse of this be?

slim

fyygligf8E

KIfYxying'IIM

Ipyy

yy1m

ftp

mgf

8fYxIi

f

fifty

fly

214

2,

usepoint

slopeform

HEMA

0

8

j

i

increasing

F'A

must

positive

FCA Sdecreasing

Almustbeneg

if f'A

flx S

increasing

ifflded

decreasing

fly

Example 3 : Given the function ݂

ሺ ݔ

ሻ ൌ

ଶ

௫

, find each of the following.

(a) Find the derivative of ݂ሺݔሻ. (b) Find ݂

ᇱ

ሺʹሻ

(d) Find the equation of the tangent line of ሻݔሺ݂ at

ݔൌ ʹǤ

(e) Find ݂

ᇱ

ሺͲሻ (f) Why does our answer for (e) make sense??

Example 4 : Find ݂

ᇱ

ሺͶሻ given ݂ሺݔሻǤ ݂ሺݔሻൌ ξ

FGHIFII

fYA

F'HIME

LEAN

XANAX

IE

112,

FHIISTEIN

F'Atlin

Efilx

fYd E

undefined

MIL

FFEÉFE

EEE

iE

flattest

Example 7 : Sketch the graph of

fx x (). Then find the derivative of ሻݔሺ݂when ݔൌ ͲǤ

Example 8 : Determine if the following statement is True or False. If false, explain or draw a counter-example.

The converse to the theorem above is also true.

Example 9 : Piece-Wise Functions

Determine if the function,

fx , is (a) continuous or dis continuous and (b) differentiable or non-

differentiable. Show all necessary work to justify your conclusions.

2, 1

1, 1

xxx

In general ± when will a function NOT be differentiable?

THEOREM: Differentiability Implies Continuity

If݂ is differentiable at ܿ ൌ ݔ, then ݂ is continuous at ܿ ൌ ݔ_._

-3 -2 -1 1 2 3 4

converse

if f

continuous

then

fis

differentiable

asymptote

gap

hole

sharp

points

Kuspslvertical

tangent

####### W

endpoints

Was this document helpful?

2.1 Notes - Calculus

Course: Calculus I (MATH M211)

19 Documents

Students shared 19 documents in this course

University: Indiana University

Was this document helpful?

2.1 Notes: The Derivative & the Tangent Line Problem

AP CALCULUS AB

IN Standard(s):

C.D.2: State, understand,

and apply the definition of

derivative.

C.AD.1: Find the slope of a

curve at a point, including

points at which there are

vertical tangents and no

tangents.

Learning Target(s):

x I can find the slope of the tangent line to a curve at a point.

x I can solve problems involving the slope of a tangent line.

x I know that the derivative of ݂ is the function whose value

at ݔ is 

௛՜଴ ௙ሺ௫ା௛ሻି௙ሺ௫ሻ

௛ provided this limit exists.

x I can identify that the derivative at a point is the slope of

the line tangent to a graph at that point on the graph.

x I know that the difference quotients ௙ሺ௔ା௛ሻି௙ሺ௔ሻ

௛

and ௙ሺ௫ሻି௙ሺ௔ሻ

௫ି௔ express the average rate of change of a

function over an interval.

Success Criteria:

x I can explain the relationship between

differentiability and continuity.

x I can determine average rates of change of a

function over an interval using difference quotients.

x I can evaluate the instantaneous rate of change of

a function.

x I can use multiple notations for the derivative of a

function.

x I can evaluate derivatives represented graphically,

numerically, analytically, and verbally.

What is a Tangent Line?

Misconceptions about Tangent Lines

Writing a formal definition of what a tangent looks like is often problematic. The following four misconceptions can

be helpful when deciding whether a line is tangent to a curve.

Misconception #1:

A line is tangent to a

curve if the line crosses

the curve at exactly one

point.

Although line ܮ touches curve ܥat one point, line ܮ is

not a tangent line.

Misconception #2:

A tangent line to a curve

must cross the curve only

once.

Line ܮ is tangent to curve ܥ at point ܲ despite the fact

that line ܮ crosses curve ܥ at two other points.

Misconception #3:

A line is tangent to a

curve if the line touches

the curve at one point

but does not cross the

curve.

Again, although line ܮ touches curve ܥ at point ܲ, it is

still not considered a tangent line.

Misconception #4:

A tangent line to a curve

LVDOLQHWKDWMXVW´JUD]HVµ

the curve at a point but

does not cross the curve.

In the example above, line ܮ is tangent to curve ܥ at

point ܲ despite the fact that is crosses the curve.

Examples of Tangent Lines drawn to a curve ݂ሺݔሻ at a point ܲ

Note: $OWKRXJKPDQ\WLPHVZHPLJKWVD\WKDWDWDQJHQWOLQHGUDZQWRDFXUYHPD\RQO\´WRXFKµWKHFXUYH

one time, that is not entirely true. A secant line, by definition, is a line that can intersect a curve at least twice.

am to

2.1 Notes - Calculus

Calculus I (MATH M211)

Indiana University

Comments

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Preview text

Ipyy

ftp

8fYxIi

f

fifty

2,

0

8

j

i

IE

MIL

iE

flattest

2.1 Notes - Calculus

Course: Calculus I (MATH M211)

University: Indiana University

Students also viewed

Related documents