Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Derivative Concepts: Slope of Tangent Lines and Function Behavior, Schemes and Mind Maps of Calculus

The concept of derivatives through the slope of tangent lines and the behavior of functions. It covers the limit definition of derivatives, finding derivatives using the definition, and interpreting derivatives as rates of change. Examples are provided for various functions.

Typology: Schemes and Mind Maps

2021/2022

Uploaded on 09/12/2022

torley
torley ๐Ÿ‡บ๐Ÿ‡ธ

4.6

(41)

258 documents

1 / 7

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
2/3/2011
1
Math 103 โ€“Rimmer
3.1/3.2 The Derivative
(
)
(
)
lim
x a
f x f a
m
x a
โ†’
โˆ’
=โˆ’
The limit of the slopes of the secant li
nes is the slope of the tangent line.
(
)
f a
โ€ฒ
๎˜๎˜‚๎˜ƒ๎˜‚๎˜„
( )
to at .f x x a=
๎˜…๎˜†๎˜‡
secant line slope
Math 103 โ€“Rimmer
3.1/3.2 The Derivative
(
)
(
)
0
lim
h
f a h f a
m
h
โ†’
+ โˆ’
=
Another expression for the slope of the
tangent line.
๎˜๎˜‚๎˜ƒ๎˜‚๎˜„
(
)
f a
โ€ฒ
( )
the slope of the tangent line
to at .f x x a=
๎˜…๎˜†๎˜‡
secant line slope
pf3
pf4
pf5

Partial preview of the text

Download Derivative Concepts: Slope of Tangent Lines and Function Behavior and more Schemes and Mind Maps Calculus in PDF only on Docsity!

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

lim x a

f x f a m โ†’ x a

The limit of the slopes of the secant lines is the slope of the tangent line.

f โ€ฒ ( a )

the slope of the tangent line

to f x at x = a.

secant line slope

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

0

lim h

f a h f a m โ†’ h

Another expression for the slope of the tangent line.

f โ€ฒ ( a )

the slope of the tangent line

to f x at x = a.

secant line slope

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

If you zoom in on the point of tangency, the function is

"locally linear" there.

http://www.stewartcalculus.com/tec/

Module 3.

Math 103 โ€“ Rimmer List the following numbers from smallest to largest. 3.1/3.2 The Derivative

โˆ’

g โ€ฒ (^) ( 0 )< 0 < g โ€ฒ( 4 )< g โ€ฒ( 2 )< g โ€ฒ( โˆ’ (^2) )

most steep

least steep

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

The derivative is the

of with respect to when.

f a

y f x x x a

instantaneous rate of

change

2 1

y^ f^ x^ f^ x

x x x

โˆ† โˆ’

โˆ† โˆ’

[ 1 2 ]

This is the of

with respect to over the interval ,.

y f x

x x x

average rate of change =

This is called a

difference quotient

( ) ( )

2 1

2 1

0 2 1

lim lim x x x

y^ f^ x^ f^ x

โˆ† โ†’ (^) x โ†’ x x

โˆ† โˆ’

โˆ† โˆ’

Math 103 โ€“ Rimmer Interpreting the derivative as a rate of change. 3.1/3.2 The Derivative

The cost of producing x ounces of gold from a new gold mine is C (^) ( x )dollars.

What is the meaning of C โ€ฒ ( x )? What are its units?

change in measures the ratio: change in

C C C x x x

โˆ† โ€ฒ = โˆ†

( ) is the rate of change of production cost with respect to

the number of ounces produced, this is called.

C โ€ฒ x

marginal cost

The units for C โ€ฒ ( x )are dollars per ounce.

What does C โ€ฒ (^) ( 800 ) =17 mean?

C โ€ฒ ( 800 )is a ratio so let's turn 17 into a fraction.

( )

17 800 1

C โ€ฒ =

When you are producing 800 ounces of gold and you

increase production by 1 to 801 ounces, cost will increase by $17.

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

0

lim h

f x h f x f x โ†’ h

Let the number a vary.

f โ€ฒ ( x ) can be thought of as a new function, it is called the derivative of f.

If f โ€ฒ ( a )exists, then f is called differentiable at a.

is called , if it is differentiable

for all numbers in ,.

f a b

a b

differentiable on

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

Find the derivative of the function using the definition of the derivative.

2 f x = 4 x โˆ’ 7 x

2 f x + h = 4 x + h โˆ’ 7 x + h

0

lim h

f x h f x f x โ†’ h

( )

2 2 = 4 x + 4 h โˆ’ 7 x + 2 xh + h

2 2 f x + h = 4 x + 4 h โˆ’ 7 x โˆ’ 14 xh โˆ’ 7 h

2 โˆ’ f x = โˆ’ 4 x + 7 x

f ( x + h ) โˆ’ f ( x )=

2

4 h โˆ’ 14 xh โˆ’ 7 h =^ h^ ( 4 โˆ’^14 x^ โˆ’^7 h )

0

lim h

f x h f x f x โ†’ h

0

lim h

h x h

โ†’ h

0

lim 4 14 7 h

x h โ†’

f โ€ฒ^ ( x (^) ) = 4 โˆ’ 14 x

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative

Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV.

The main connection: function: sign of the slope of the tangent line derivative: + above axis, below axis 0 "touches" axis

x x x

โ‡’ โˆ’ โˆ’ โ‡’ โˆ’ โ‡’ โˆ’

(a): sign of the slope of the tangent line โˆ’ โ†’ 0 โ†’ + โ†’ 0 โ†’ โˆ’

(a) โ‡”II

deriv.: below,then 0, then above, then 0, then below

(b): sign of the slope of the tangent line

  • โ†’ dne โ†’ โˆ’ โ†’ dne โ†’ +

(b) โ‡”IV

deriv.: above, then jump to below, then jump to above

(c): sign of the slope of the tangent line โˆ’ โ†’ 0 โ†’ +

(c) โ‡”I

deriv.: below,then 0, then above

(d): sign of the slope of the tangent line

  • โ†’ 0 โ†’ โˆ’ โ†’ 0 โ†’ + โ†’ 0 โ†’ โˆ’

(d) โ‡”III

deriv.: above, then 0, then below, then 0, then above, then 0, then below

Math 103 โ€“ Rimmer 3.1/3.2 The Derivative Animation of the graph of the derivative function

http://www.stewartcalculus.com/tec/ Module 3.