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EECM 3714

Lecture 3: Unit 3
Non-linear equations
Renshaw, Ch
...
5, 11 & 12
• Cubic Functions & polynomials
• Other non-linear functions
• Logarithmic and exponential functions
• Examples

CUBIC FUNCTIONS, Ch 5, P 153-60
• The cubic function 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
• The cubic function contains a term involving 𝑥 3 , 𝑥 2 and 𝑥 and usually yields an S-shaped curve
...

See Fig 5
...
See Fig 5
...

• Unlike quadratic equations, there is no rule or formula for solving for the roots of cubic equations

• These roots can be solved only by trial-and-error methods
...


ECONOMIC APPLICATION OF CUBIC FUNCTIONS, P 153-60
• Economic application:

• Total cost function: T𝐶 = 𝑎𝑄3 + 𝑏𝑄2 + 𝑐𝑄 + 𝑑
• Cubic function: has (at most) two turning points and (at most) three 𝑥-intercepts
• A plausible shape for a firm's short run TC curve is given in Fig 5
...

• The technology of production is such that at low levels of output it is relatively difficult to
increase output, so the TC curve slopes upward relatively steeply
...

• In the mid-range, between about 1
...
5 units of output, it is relatively easy to
increase output and the TC curve is relatively flat
...
This
introduces idea of a limit or limiting value
...
Thus, there is a discontinuity in the function: y is undefined at x = 0
...

• Economic application: Demand function - See p
...

• E
...
: Inverse demand function: 𝑝 =

𝑐
𝑞𝑑

−𝑎
...
See Fig 5
...

• An exponential function is a function in which a constant base is raised to a

variable power (exponent), e
...
: 𝑦 = 𝑎 𝑥 is exponential function (where a is a
constant)

• Note: 𝑦 = 𝑥 2 is NOT an exponential function, but 𝑦 = 2𝑥 is
...

• 𝑥 is the logarithm, to base 𝑎, of 𝑦

• Thus, the logarithmic function tells us to which power we have to raise the
constant 𝑎 (the base) to get the answer 𝑦 (while the exponential function tells us
the answer we’ll get if we raise the base to the power 𝑥)
...
2
• Thus, given the exponential function: 𝑦 = 10𝑥 , the inverse function “logarithmic
function” is defined as: 𝑥 = log10 (𝑦)
• log10 means the logarithmic to base 10

RULES FOR MANIPULATING LOGS
1
...
log( AB) = log A + log B - KEY RULE
3
...
log( ) = log A − log B (reverses rule 2)
A
A

5
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log(10) = 1 (because 101 = 10)

USING LOGS TO SOLVE PROBLEMS
Example: 2logy + log 𝑥 − 1 − log 𝑥 + 1 = 0
...


Start by rearranging the terms and move x terms to the right-hand side

• 2logy = log 𝑥 − 1 − log 𝑥 + 1
• Then, use the different rules for manipulating logs
1
⇒ log𝑦 = log
2

𝑥+1
𝑥−1

𝑥+1

⇒ log𝑦 = log[

𝑥−1


...


The log cancel out to get an expression of y as a function x:

⇒ 𝑦=

𝑥+1
𝑥−1

1
2


...
05, y = 200, x to be found
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05)x

 log(2) = log[(1
...
05)x
= x log[1
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3010 = 14
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05] 0
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3 and 11
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71828…

• While log10 can just be written as log, log 𝑒 can be written as In
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12
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6 376 − 7 ; Fig
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7 − 8 (378 − 9)

LOG RULES, P
...
381-6
• The natural exponential function can be used to solve problems involving

variables that are growing continuously over time at a constant rate
...


CONTINUOUS GROWTH, P
...

• Since 𝑦 = 𝑎𝑒 𝑟𝑥 : 𝑒 𝑟𝑥 =
• ln 𝑒 𝑟𝑥 = ln

𝑦
𝑎

𝑦
𝑎

⟹ 𝑟𝑥 = ln

𝑦
𝑎

𝑦

∴𝑟=

ln 𝑎
𝑥

• * See Example 12
...
4 (383, 386)

• Time to double: Suppose that 𝑦 = 𝑎𝑒 𝑟𝑥
...

∴

ln 2
𝑟

= 𝑥
...
693: 𝑥 =

0
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3, p
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02𝑡
...
02𝑡
• Therefore 𝑎 = 70; 𝑟 = 0
...
Also, 𝑦 = 77

• 𝑦 = 𝑎𝑒 𝑟𝑡 ⟹ 77 = 70𝑒 0
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02𝑡 = ln 70 + 0
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ln 77 − ln 70 = 0
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02𝑡 = 0
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∴ 𝑡 = 4
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EXAMPLE 2
• Suppose that Gross Domestic Product (GDP) grows continuously
...

• What is the required GDP growth rate?
• How many years would it take for GDP to double if the growth rate was 10% per year?

• Solution:
1
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0693 = 6
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Time taken to double with 10% growth rate
• 𝑎 = 1; 𝑦 = 2; 𝑟 = 10% = 0
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1𝑡 ⟹ ln 2 = ln 𝑒 0
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1𝑡 ln 𝑒
• ∴ 0
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93 ≈ 7



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