chapter 4: differentiation

Chapter 4: Differentiation Advanced Mathematics Chapter 4: Differentiation Advanced Mathematics Chapter 4: Differentiation Advanced Mathematics Chapter 4: Differentiation Advanced Mathematics Chapter 4: Differentiation Advanced Mathematics

Chapter 4: Differentiation

Nguyen Thi Minh Tam ntmtam.vnua@gmail.com

November 17, 2020

1 4.1 The derivative of a function

2 4.2 Rules of differentiation

3 4.3 Marginal functions

4 4.5 Elasticity

4.1 The derivative of a function

The derivativeof a function f at x = a, denoted by f0(a), is

f0(a) = lim

x →a

f (x ) − f (a)

x − a = lim∆x →0

∆y

∆x provided this limits exist

The derivative of f , denoted by f0 or df

dx, is a function that assign to x the number f0(x )

The second derivativeof f , denoted by f00 or d

2f

dx2, is the derivative of the f0

f00= (f0)0

4.2 Rules of differentiation

The Derivatives of the constant and power functions

(c)0 = 0 if c is a constant, (xn)0= nxn−1

Note

a)  1

x

0

= −1

x2 b) (√x )0 = 1

2√x

Constant Multiple, Sum, Difference, Product and Quotient Rules

(ku)0 = ku0, for any constant k (u + v )0 = u0+ v0

(u − v )0 = u0− v0 (uv )0= u0v + uv0

u v

0

= u

0v − uv0

v2

Example 1 Find dy

dx for each following function:

a) y = 9x5+ 3

x b) y = 3√x − 18

x + 13 c) y = (2x3+ 1)(√x − 1)

4.3 Marginal functions

Marginal revenue

The marginal revenue, MR, is defined by

MR = d (TR)

dQ

If Q changes by a small amount ∆Q, then the corresponding change in TR is

∆TR ≈ MR × ∆Q

Note MR gives the approximate change in TR when Q increases

by 1 unit

Example 2 Given the demand function

P = 60 − Q a) Write down an expression for the marginal revenue function b) If the current demand is 50, estimate the change in the value

of TR due to a 2-unit increase in Q

Marginal cost

The marginal cost, MC, is defined by

MC = d (TC)

dQ

If Q changes by a small amount ∆Q, then the corresponding change in TC is

∆TC ≈ MC × ∆Q

Note MC gives the approximate change in TC when Q increases

by 1 unit

Example 3 Find the marginal cost given the average cost function

AC = 15

Q + 2Q + 9

If the current output is 15, estimate the effect on TC of a 3-unit decrease in Q

Marginal product of labour

When Q is a function of one input L, we define the marginal product of labour, MPL, by

MPL = dQ

dL

MPL gives the approximate change in Q when using 1 more unit of L

Example 4 A Cobb-Douglas production function is given by

Q = 5L1/2K1/2 Assuming that capital, K , is fixed at 100, write down a formula for

Q in terms of L only Calculate the marginal product of labour when

a) L = 1

b) L = 10000

Verify that the law of diminishing marginal productivity holds in this case

The law of diminishing marginal productivity: once the size of the workforce has reached a certain threshold level, the marginal product of labour will get smaller

Consumption and savings

Assume that national income (Y ) is only used up in

consumption (C ) and savings (S ) then

Y = C + S

To analyse the effect on C and S due to variations in Y we use the conceptsmarginal propensity to consume, MPC, and

marginal propensity to save, MPS, which are defined by

MPC = dC

dY, MPS =

dS dY MPC + MPS = 1

Example 5 If the savings function is given by

S = 0.02Y2− Y + 100 calculate the values of MPS and MPC when Y = 40 Give a brief interpretation of these results

4.5 Elasticity

Price elasticity of demand

The price elasticity of demandis a measure of the responsiveness of demand to price change It is usually defined as

E = percentage change in demand percentage change in price Demand is said to be

- inelastic if |E | < 1

- unit elastic if |E | = 1

- elastic if |E | > 1

The elasticity formula:

E = P

Q ×

∆Q

∆P The arc elasticityof demand between points (Q1, P1) and (Q2, P2) is defined by

1

2(P1+ P2) 1

2(Q1+ Q2) ×

∆Q

∆P The price elasticity at a point (point elasticity) is

E = P

Q × dQ dP

Example 6 Given the demand function

P = 100 − 2Q a) Calculate the arc elasticity as P falls from 20 to 10

b) Find the elasticity when the price is 50 Is demand inelastic, unit elastic or elastic at this price?

Note dQ

dP =

1

 dP

dQ



Example 7 Given the demand function

P = −Q2− 10Q + 150 a) Find the price elasticity of demand when Q = 4

b) Estimate the percentage change in price needed to increase demand by 10%

Price elasticity of supply

Theprice elasticity of supplyis define by

E = percentage change in supply percentage change in price

Example 8 If the supply equation is

Q = 150 + 5P + 0.1P2 calculate the price elasticity of supply

a) averaged along an arc between P = 9 and P = 11; b) at the point P = 10

Relationship between elasticity and marginal revenue

MR = P



1 + 1 E



If −1 < E < 0, then MR < 0 ⇒ the revenue function is decreasing in regions where demand is inelastic

If E < −1, then MR > 0 ⇒ the revenue function is increasing

in regions where demand is elastic

Exercise 4.3, 4.3*1, 3-6 (page 301-302), 1, 3, 6, 8 (page 302-303)

Exercise 4.5, 4.5*1, 4, 5, 8 (page 326-327), 1, 6 (page 327, 328)