we will start from chapter 2 sections 2.1 to 2.8 math 101 : calculus i – dr. faisal fairag

We will start fromChapter 2

Sections 2.1 to 2.8

MATH 101 : CALCULUS I – Dr. Faisal Fairag

Chapter 2 : Limits and Derivatives

The idea of a limit isillustrated by secant

linesapproaching a

tangent line.

Th

e B

ook S

ays:


Slope and Equation of a line


How to find the slop of a line?

(x1, y1)

(x2, y2)

mxx

yy slope

12

12

How to find the equation of a line?

)( 11 xxmyy


What is the difference??

The word tangent is derived from the Latin word tangens, which means “touching”.

Tangent lineSecant line

Is it possible that a line could be tangent and secant in the same time

Secant line and Tangent line


How to find the slope of these lines?? Tangent line

mxx

yy slope

12

12

Secant line and Tangent line


mxx

yy slope

12

12

2.1 THE TANGENT PROBLEM

Th

e B

ook S

ays:


1

1

1.5

2.25Q

tangent line


Th

e B

ook S

ays:


1

1

tangent line


Th

e B

ook S

ays:


1

1

1.2

1.44Q

tangent line

2.22.0

44.0

12.1

144.1

PQm

1.21.0

21.0

11.1

121.1

PQm

1

1

1

1 22

x

x

x

xmPQ


Th

e B

ook S

ays:


1.5)1.5,2.25(2.5

1.2)1.2,1.44(2.2

1.1)1.2,1.21(2.1

1.01)1.01,1.0201(2.01

PQmx Q


Th

e B

ook S

ays:


1

1

x

X^2



PQmx

2 3

1.5 2.5

1.1 2.1

1.01 2.01

1.001 2.001

x

1.0

PQm

2.0 tangent line

secant line

Slope of the tangent line = 2.0



01

0.51.5

0.91.9

0.991.99

0.9991.999

PQmx x

1.0

PQm

2.0 tangent line

secant line

Equation of the tangent line

Slope = 2.0

point = )1, 1(

)( 11 xxmyy )1(21 xy



01

0.51.5

0.91.9

0.991.99

0.9991.999

PQmx x

1.0

PQm

2.0 tangent line

secant line

2lim PQPQm 2

1

1lim

2

1

x

xx

Q

P


0.50.3333

0.90.2632

0.990.2513

0.9990.2501

PQmx

1.50.2

1.10.2381

1.010.2488

1.0010.2499

PQmx

we will start from chapter 2 sections 2.1 to 2.8 math 101 : calculus i – dr. faisal fairag

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