STK121 SUT 1.2.1 Rate of Change (2021)

STK 123
SECTION A
Optimization Techniques
Study Unit 2.3

DIFFERENTIATION:
RATES OF CHANGE
QST chapter 2 section 2.4
1
Learning Objectives of SU 2.3 (from Study Guide)
The student must be able to
• calculate the average rate of change of a function over

an interval as the length of the interval decreases
• explain the relationship between the average rate of

change and the slope of the tangent to a function at a
certain point
4
Introduction:
• Calculate changes in business and economics:
– change in turnover of a retailer
– change in gold price
– change in Import and Export figures of SA
– change in salaries of employees in public sector
• Of importance is the rates at which these changes take place

• For a function 𝒚𝒚 = 𝒇𝒇 𝒙𝒙 = 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄
the rate of change is zero
• For linear functions 𝒚𝒚 = 𝒇𝒇(𝒙𝒙) = 𝒂𝒂 + 𝒃𝒃𝒃𝒃
the rate of change is constant (= slope of line)
𝒃𝒃 = 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 = ∆𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 (change in rate)
• For any other non-linear function
the rate of change is not constant (next slide)
5
Example 2.4.1:
How to determine the rate of change of a non-linear function
The relationship between cost of repairing an appliance 𝒚𝒚 (R)
and number of months 𝒙𝒙 is represented by the function
𝒚𝒚 = 𝒇𝒇 𝒙𝒙 = 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝒙𝒙 + 𝟒𝟒𝟒𝟒
fig 2.4.1
• 𝒚𝒚 increases as 𝒙𝒙 increases
• This increase happens at a

faster and faster rate
∴ ∆𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 (rate of change) is not constant

Example 2.4.1 (continued): 𝒚𝒚 = 𝒇𝒇 𝒙𝒙 = 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝒙𝒙 + 𝟒𝟒𝟒𝟒
Consider an increase from 𝒙𝒙 = 𝟐𝟐 𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝟓𝟓:
• 𝒙𝒙 = 𝟐𝟐: 𝒇𝒇 𝟐𝟐 = 𝒚𝒚 = 𝟐𝟐 × 𝟐𝟐𝟐𝟐 −𝟐𝟐 + 𝟒𝟒𝟒𝟒 = 𝟒𝟒𝟒𝟒
• 𝒙𝒙 = 𝟓𝟓: 𝒇𝒇 𝟓𝟓 = 𝒚𝒚 = 𝟐𝟐 × 𝟓𝟓𝟐𝟐 −𝟓𝟓 + 𝟒𝟒𝟒𝟒 = 𝟖𝟖𝟖𝟖
⇒ cost in repairs increases by 𝟖𝟖𝟖𝟖 − 𝟒𝟒𝟒𝟒 = 𝑹𝑹𝑹𝑹𝑹𝑹

when 𝒙𝒙 increases from 2 to 5 months
𝟖𝟖𝟖𝟖−𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑
⇒ average increase ∆𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓: = = 𝑹𝑹𝑹𝑹𝑹𝑹 a month
𝟓𝟓−𝟐𝟐 𝟑𝟑
(which is the slope of the line segment 𝑨𝑨𝑨𝑨 connecting A & B)
This line segment 𝑨𝑨𝑨𝑨 does not really reflect the slope,
direction or steepness of the curve at point A or at point B
Therefore investigate what happens in the vicinity of A
Consider an increase from 𝒙𝒙 = 𝟐𝟐 𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝟑𝟑:
• 𝒙𝒙 = 𝟐𝟐: 𝒇𝒇 𝟐𝟐 = 𝟒𝟒𝟒𝟒 fig 2.4.1
• 𝒙𝒙 = 𝟑𝟑: 𝒇𝒇 𝟑𝟑 = 𝟓𝟓𝟓𝟓
⇒ average increase per month

𝟓𝟓𝟓𝟓−𝟒𝟒𝟒𝟒
∆𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓= = 𝑹𝑹𝟗𝟗
𝟑𝟑−𝟐𝟐
(the slope of line segment 𝑨𝑨𝑨𝑨)
This line segment 𝑨𝑨𝑪𝑪 gives a better indication of what happens

in the vicinity of point A, by taking smaller increases in 𝒙𝒙
Consider an increase from 𝒙𝒙 = 𝟐𝟐 𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝟐𝟐. 𝟐𝟐:
• 𝒙𝒙 = 𝟐𝟐: 𝒇𝒇 𝟐𝟐 = 𝟒𝟒𝟒𝟒
• 𝒙𝒙 = 𝟐𝟐. 𝟐𝟐: 𝒇𝒇 𝟐𝟐. 𝟐𝟐 = 𝟒𝟒𝟒𝟒. 𝟒𝟒𝟒𝟒
⇒ average increase per month
fig 2.4.1
𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒−𝟒𝟒𝟒𝟒
∆𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓= = 𝑹𝑹𝟕𝟕. 𝟒𝟒𝟒𝟒
𝟐𝟐.𝟐𝟐−𝟐𝟐
(the slope of line segment 𝑨𝑨𝑨𝑨)
This line segment 𝑨𝑨𝑫𝑫 gives a much better indication of what

happens in the vicinity of point A. The approximated rate can
be improved by taking smaller and smaller increases in 𝒙𝒙
CONCLUSION:
Instantaneous rate of change at point A:
The limit of the function when the increase in 𝒙𝒙 approaches 0

(the slope of the tangent line to the curve at point A)
𝒇𝒇 𝒙𝒙+𝒉𝒉 −𝒇𝒇(𝒙𝒙)
𝑰𝑰𝑰𝑰𝑰𝑰 = 𝒍𝒍𝒍𝒍𝒍𝒍
𝒉𝒉→𝟎𝟎 𝒉𝒉
𝒉𝒉 tends to 0 (increase in 𝒙𝒙 gets smaller and smaller)

• ℎ tends to 0 (it gets smaller and smaller)
Tangent line: line that touches curve 𝒇𝒇(𝒙𝒙) in a SINGLE POINT 𝒙𝒙

See figure 2.4.3 p 56
Exercises Chapter 2 p 62 nr 1:
𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒊𝒊𝒊𝒊 𝒚𝒚 ∆𝒚𝒚
1. Find the average rate of change ARC = =
𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒊𝒊𝒊𝒊 𝒙𝒙 ∆𝒙𝒙
in the function 𝒚𝒚 = 𝟑𝟑𝒙𝒙𝟐𝟐 over each of the intervals:
a) 𝒙𝒙 = 𝟐𝟐 𝐭𝐭𝐭𝐭 𝒙𝒙 = 𝟑𝟑 b) 𝒙𝒙 = 𝟐𝟐 𝐭𝐭𝐭𝐭 𝒙𝒙 = 𝟐𝟐. 𝟏𝟏
ARC f (3) − f (2) ARC f (2.1) − f (2)
m= m=
3−2 2.1 − 2
27 − 12 13.23 − 12
= =
1 0.1
= 15 = 12.3
c) 𝒙𝒙 = 𝟐𝟐 𝐭𝐭𝐭𝐭 𝒙𝒙 = 𝟐𝟐. 𝟎𝟎𝟎𝟎 d) 𝒙𝒙 = 𝟐𝟐 𝐭𝐭𝐭𝐭 𝒙𝒙 = 𝟐𝟐. 𝟎𝟎𝟎𝟎𝟏𝟏

ARC f (2.01) − f (2) f (2.001) − f (2)
m = m=
ARC
2.01 − 2 2.001 − 2
12.1203 − 12 12.012003 − 12
= =
0.01 0.001
= 12.03 = 12.003 11
Exercises Chapter 2 p 62 nr 3:
3. Find the slope of the tangent line to the curve 𝒚𝒚 = 𝟑𝟑𝟑𝟑𝟐𝟐 in
the point 𝒙𝒙 = 𝟐𝟐: called instantaneous rate of change IRC
f (2 + h) − f (2)
m =IRC
lim m pq = lim
z→ p h →0 h
3(2 + h) 2 − 3(2) 2
= lim
h →0 h
3(4 + 4h + h 2 ) − 12
= lim
h →0 h
12 + 12h + 3h 2 − 12
= lim
h →0 h
h(12 + 3h)
= lim
h →0 h
= lim(12 + 3h)
h →0
= 12 + 3(0)
12
= 12
Example 2.4.2 p 54-57:
The relationship between sales (number of items sold) 𝒚𝒚 and
price 𝒙𝒙 is represented by the function
𝒚𝒚 = 𝒇𝒇 𝒙𝒙 = 𝟏𝟏𝟏𝟏𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎 + 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 − 𝟒𝟒𝟒𝟒𝒙𝒙𝟐𝟐
fig 2.4.2
Example 2.4.2 (continued) : 𝒇𝒇 𝒙𝒙 = 𝟏𝟏𝟏𝟏𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎 + 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 − 𝟒𝟒𝟒𝟒𝒙𝒙𝟐𝟐
What is the change in sales if the price increases from R10 to
R40 an item? ARC for an increase from 𝒙𝒙 = 𝟏𝟏𝟏𝟏 𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝟒𝟒𝟒𝟒:
𝒙𝒙 = 𝟏𝟏𝟏𝟏: 𝒇𝒇 𝟏𝟏𝟏𝟏 = 100 000 + 400 10 − 48 102 = 𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐
𝒙𝒙 = 𝟒𝟒𝟎𝟎: 𝒇𝒇 𝟒𝟒𝟎𝟎 = 100 000 + 400 40 − 48 402 = 𝟑𝟑𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐
𝟑𝟑𝟑𝟑 𝟐𝟐𝟐𝟐𝟐𝟐−𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐

⇒ average change in sales 𝑨𝑨𝑨𝑨𝑨𝑨: = −𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
𝟒𝟒𝟒𝟒−𝟏𝟏𝟏𝟏
(which is the slope of the line segment 𝑨𝑨𝑨𝑨)
Interpretation:
A price increase of R1 brings about
an average increase of -2000 items
(decrease of 2000 items sold)
R30 an item? ARC for an increase from 𝒙𝒙 = 𝟏𝟏𝟏𝟏 𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝟑𝟑𝟑𝟑:
𝒙𝒙 = 𝟏𝟏𝟏𝟏: 𝒇𝒇 𝟏𝟏𝟏𝟏 = 100 000 + 400 10 − 48 102 = 𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐
𝒙𝒙 = 𝟑𝟑𝟎𝟎: 𝒇𝒇 𝟑𝟑𝟎𝟎 = 100 000 + 400 30 − 48 302 = 𝟔𝟔𝟔𝟔 𝟖𝟖𝟎𝟎𝟎𝟎
𝟔𝟔𝟔𝟔 𝟖𝟖𝟖𝟖𝟖𝟖−𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐

⇒ average change in sales 𝑨𝑨𝑨𝑨𝑨𝑨: = −𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
𝟑𝟑𝟑𝟑−𝟏𝟏𝟏𝟏
R11 an item? ARC for an increase from 𝒙𝒙 = 𝟏𝟏𝟏𝟏 𝒕𝒕𝒕𝒕 𝒙𝒙 = 𝟏𝟏𝟏𝟏:
𝒙𝒙 = 𝟏𝟏𝟏𝟏: 𝒇𝒇 𝟏𝟏𝟏𝟏 = 100 000 + 400 10 − 48 102 = 𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐
𝒙𝒙 = 𝟏𝟏𝟏𝟏: 𝒇𝒇 𝟏𝟏𝟏𝟏 = 100 000 + 400 11 − 48 112 = 𝟗𝟗𝟗𝟗 𝟓𝟓𝟓𝟓𝟓𝟓
𝟗𝟗𝟗𝟗 𝟓𝟓𝟓𝟓𝟓𝟓−𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐

⇒ average change in sales 𝑨𝑨𝑨𝑨𝑨𝑨: = −𝟔𝟔𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏−𝟏𝟏𝟏𝟏
This gives a good approximation
of the increase in sales at point A
The exact increase in sales at point A (𝒙𝒙 = 𝟏𝟏𝟏𝟏)

𝒇𝒇 𝟏𝟏𝟏𝟏+𝒉𝒉 −𝒇𝒇(𝟏𝟏𝟏𝟏)
NB formula: p 57 in the middle
Example 2.4.3 p 57 ( IRC of example 2.4.2 at point A 𝒙𝒙 = 𝟏𝟏𝟏𝟏):
𝒇𝒇 𝟏𝟏𝟏𝟏+𝒉𝒉 −𝒇𝒇(𝟏𝟏𝟏𝟏)
𝒇𝒇 𝒙𝒙 = 𝟏𝟏𝟏𝟏𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎 + 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 − 𝟒𝟒𝟒𝟒𝒙𝒙𝟐𝟐
𝒇𝒇 𝟏𝟏𝟏𝟏 = 99200
𝒇𝒇 𝟏𝟏𝟏𝟏 + 𝒉𝒉 = 99200 + 560ℎ − 48ℎ2
𝒇𝒇 𝟏𝟏𝟏𝟏+𝒉𝒉 −𝒇𝒇(𝟏𝟏𝟏𝟏) 99200−560ℎ−48ℎ2 −99200

=
𝒉𝒉 ℎ
= −𝟓𝟓𝟓𝟓𝟓𝟓 − 𝟒𝟒𝟒𝟒𝟒𝟒
𝒇𝒇 𝟏𝟏𝟏𝟏+𝒉𝒉 −𝒇𝒇(𝟏𝟏𝟏𝟏)
𝑰𝑰𝑰𝑰𝑰𝑰 = 𝒍𝒍𝒍𝒍𝒍𝒍 = 𝒍𝒍𝒍𝒍𝒍𝒍 (−𝟓𝟓𝟓𝟓𝟓𝟓 − 𝟒𝟒𝟒𝟒𝟒𝟒)
𝒉𝒉→𝟎𝟎 𝒉𝒉 𝒉𝒉→𝟎𝟎
𝑰𝑰𝑰𝑰𝑰𝑰 = −𝟓𝟓𝟓𝟓𝟓𝟓 (𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒐𝒐𝒐𝒐 𝟓𝟓𝟓𝟓𝟓𝟓 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊)
OWN WORK: Example 2.4.1 (p 52-54):
𝒇𝒇 𝒙𝒙 = 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝒙𝒙 + 𝟒𝟒𝟒𝟒
Show that the IRC when 𝒙𝒙 = 𝟐𝟐 (𝐚𝐚𝐚𝐚 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐀𝐀) is equal to 7
IRC is the slope of the tangent line that touches curve 𝒚𝒚 = 𝒇𝒇 𝒙𝒙
in the SINGLE POINT 𝒙𝒙 = 𝟐𝟐
𝒇𝒇 𝟐𝟐 =
𝒇𝒇 𝟐𝟐 + 𝒉𝒉 =
𝒇𝒇 𝟐𝟐+𝒉𝒉 −𝒇𝒇(𝟐𝟐)
=
𝒉𝒉
𝒇𝒇 𝟐𝟐+𝒉𝒉 −𝒇𝒇(𝟐𝟐)
𝑰𝑰𝑰𝑰𝑰𝑰 = 𝒍𝒍𝒍𝒍𝒍𝒍 = 𝒍𝒍𝒍𝒍𝒍𝒍 (… … … … … )
𝒉𝒉→𝟎𝟎 𝒉𝒉 𝒉𝒉→𝟎𝟎
𝑰𝑰𝑰𝑰𝑰𝑰 = … (𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒐𝒐𝒐𝒐 … … … )

STK121 SUT 1.2.1 Rate of Change (2021)

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STK121 SUT 1.2.1 Rate of Change (2021)

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STK 123

Study Unit 2.3

QST chapter 2 section 2.4

The student must be able to

• calculate the average rate of change of a function over

• explain the relationship between the average rate of

• Of importance is the rates at which these changes take place

• This increase happens at a

∴ ∆𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓 (rate of change) is not constant

⇒ cost in repairs increases by 𝟖𝟖𝟖𝟖 − 𝟒𝟒𝟒𝟒 = 𝑹𝑹𝑹𝑹𝑹𝑹

⇒ average increase per month

This line segment 𝑨𝑨𝑪𝑪 gives a better indication of what happens

This line segment 𝑨𝑨𝑫𝑫 gives a much better indication of what

Instantaneous rate of change at point A:

The limit of the function when the increase in 𝒙𝒙 approaches 0

𝒉𝒉 tends to 0 (increase in 𝒙𝒙 gets smaller and smaller)

Tangent line: line that touches curve 𝒇𝒇(𝒙𝒙) in a SINGLE POINT 𝒙𝒙

c) 𝒙𝒙 = 𝟐𝟐 𝐭𝐭𝐭𝐭 𝒙𝒙 = 𝟐𝟐. 𝟎𝟎𝟎𝟎 d) 𝒙𝒙 = 𝟐𝟐 𝐭𝐭𝐭𝐭 𝒙𝒙 = 𝟐𝟐. 𝟎𝟎𝟎𝟎𝟏𝟏

𝟑𝟑𝟑𝟑 𝟐𝟐𝟐𝟐𝟐𝟐−𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐

𝟔𝟔𝟔𝟔 𝟖𝟖𝟖𝟖𝟖𝟖−𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐

𝟗𝟗𝟗𝟗 𝟓𝟓𝟓𝟓𝟓𝟓−𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐𝟐𝟐

The exact increase in sales at point A (𝒙𝒙 = 𝟏𝟏𝟏𝟏)

𝒇𝒇 𝟏𝟏𝟏𝟏+𝒉𝒉 −𝒇𝒇(𝟏𝟏𝟏𝟏) 99200−560ℎ−48ℎ2 −99200

𝑰𝑰𝑰𝑰𝑰𝑰 = … (𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒐𝒐𝒐𝒐 … … … )

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