Lecture 04 differentiation w4

Definition of a Derivative: The Slope of a Function -Non-linear functions: >» Slope of a non-linear function is the slope of the line or tangent that touches the function at that

Lecture 3: Differentiation | (Vi phan)

Definition of a Derivative:

The Slope of a Function

¢ The gradient of a line is defined as the change in y as a result of a change in x

¢ Linear Function: y = a + bx

° ais the intercept

¢ bis the slope (impact of a unit change in x on the level of y):

Ay _ y27=

Where (x,, y;) and (x,, y,) are two points connecting the line

9/27/2021 INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

Lecture 3: Differentiation | (Vi phan)

Lecture 3: Differentiation | (Vi phan)

Lecture 3: Differentiation | (Vi phan)

Lecture 3: Differentiation | (Vi phân) — DO

(c) E (-1,3) and H (49,3)

Definition of a Derivative:

The Slope of a Function

-Non-linear functions:

>» Slope of a non-linear function is

the slope of the line (or tangent)

that touches the function at that

point

Definition of a Derivative:

The Slope of a Function

¢ Illustration of a tangent (dud’ng tiếp tuyến)

Definition of a Derivative:

The Slope of a Function

¢ Non-linear functions:

VY Slope of a function is the slope of the

line (or tangent) that touches that curve

at that point

Y Example: Consider the total cost curve y

= x2

Definition of a Derivative:

Example

eConsider the total cost curve y = x?

° Derive the slope by considering a move from the point (x, y) to (x + Ax, y + Ay)

° Sub the point (x + Ax, y + Ay) into the function to find the slope i.e., Ay/ Ax:

any given point are

functions of x and Ax

The Derivative of a Function

¢ The derivative of the function f(x) is written as f'"(x)

eThe process of differentiation involves letting Ax

become arbitrarily small i.e letting A—>0 thus allowing us pinpoint the actual tangent

WY Lox 4Ax if Ax>0

°Example: Ax

Ay _ 2x

Notation:

If the function is expressed as f(x) derived function is f’(x)

If the function is expressed in terms of y derived function is dy/dx

9/27/2021 INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

Rules for differentiation

¢ The Constant Rule

elf y=c where cis a constant

Rules for differentiation

Practice Problem

1 Differentiate

(a) y=4x° (b) y = 2/x

Rules for differentiation

(a) 4(3x°) = 12x°

(b) —2/x* because 1/x = x', which differentiates to —x~

Rules for differentiation

° The Power Function Rule

elf y= ax" where a and n are constants

Rules for differentiation — DO IT !

Practice Problem

4 Differentiate

Rules for differentiation — DO IT !

Rules for differentiation

¢ The Sum/Difference Rule

Rules for differentiation — DO IT !

Rules for differentiation

¢ The Product Rule

If y= u.v where u and v are function of x (u=f(x) and v=g(x) ) then:

dx dx dx

Rules for differentiation — DO IT !

Rules for differentiation

¢ The Quotient Rule

lf y = u/v where u and v are function of x

(u=f(x) and v=g(x)) then:

Rules for differentiation — DO IT !

Rules for differentiation

¢ The Chain Rule

The Chain Rule

If y is a function of v and v is a function of x then yisa function of x and can be differentiated:

Rules for differentiation — DO IT !

Practice Problem

1 Differentiate

|

(a) y= (3x-4) (b) y=(x° + 3x45) () y= 5G (d) y= (4x -3)

Application |: Marginal Functions (Revenue, Costs and Profit)

Practice Problem

1 If the demand function is

P=60-@Q

find an expression for TR in terms of O

(1) Differentiate TR with respect to O to find a general expression for MR in terms of

O Hence write down the exact value of MR at Q = 50

(2) Calculate the value of TR when

(a)Q=50 (b)@=5I

tion to the exact value of MR obtained in part (1)

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Application |: Marginal Functions (Revenue, Costs and Profit)

Practice Problem

2 If the total revenue function of a good is given by

10000 — 4Q7

write down an expression for the marginal revenue function If the current demand is

30, find the approximate change in the value of TR due to a

(a) 3 unit increase in O

(b) 2 unit decrease in O

9/27/2021 INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

Application |: Marginal Functions (Revenue, Costs and Profit)

Application |: Marginal Functions (Revenue, Costs and Profit) — DO IT |

1 A firm’s demand function is given by

P =100-4,/0 -30

(a) Write down an expression for total revenue, TR, in terms of O

(b) Find an expression for the marginal revenue, MR, and find the value of MR when

Q=9

(c) Use the result of part (b) to estimate the change in TR when O increases by 0.25

units from its current level of 9 units and compare this with the exact change in TR

9/27/2021 INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

Application |: Marginal Functions (Revenue, Costs and Profit) — DO IT |

3 The fixed costs of producing a good are 100 and the variable costs are 2 + Q/10 per unit

(a) Find expressions for TC and MC

(b) Evaluate MC at Q = 30 and hence estimate the change in TC brought about by a

2 unit increase in output from a current level of 30 units

(c) At what level of output does MC = 22?

Application |: Elasticity (D6 co gian}

¢ Price elasticity of demand

Application |: Elasticity (D6 co gian}

° Points to note:

°Ed is negative for a downward sloping demand curve

eInelastic demand if |Ed| <1

¢Unit elastic demand if |Ed|] =1 eElastic demand if |Ed| >1

Application |: Elasticity (D6 co gian}

Is the demand inelastic, unit elastic or elastic at these prices?

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Application |: Elasticity (D6 co gian)- DO IT

find the price elasticity of demand when Q = 4 Estimate the percentage change in price

needed to increase demand by 10%

9/27/2021 INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

The demand equation for a manufacturer’s product is

—— p= 50(151 _ a)9-92v4+T19 ¬

(a) Find the value of dp/dg when 150 units are demanded

b) Using the result in part (a), determine the point elasticity of |

demand when 150 units are demanded At this level, is demand elastic,

inelastic, or of unit elasticity?

(c) Use the result in part (b) to approximate the price per unit if

emand decreases from 150 to 140 units

(d) If the current demand is 150 units, should the manufacturer

increase or decrease price in order to increase revenue? (Justify your answet }

9/27/2021 INTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM