Differentiation

Slope of a Curve

A chord is a straight line which cuts the curve at two points.

A tangent is a straight line which touches the curve at one point only.

The slope or gradient, of a curve at a point P is the slope of the tangent to the curve at that point P.


Slope varies as one moves along a non-linear curve.

At A, slope is negative
B, slope is zero
C, slope is positive


The slope of (the graph) of a function is the derivative of the function. If the function is given as a
rule f(x), the derivative is denoted by f'(x). If we write y = f(x) then f'(x) is also denoted by

The slope of the function at P is got by evaluating the slope of the cords PQ1, PQ2 and letting the point Q approach P. The limiting value gives the slope of the tangent at P

Suppose P has coordinates (a, f(a))
and that Q has coordinates (a + Δx, f(a + Δx))




Slope of chord PQ
Now let Q tend to P. Then Δx -> 0,

Slope of PQ -> slope of tangent to curve at P, and

But, slope of tangent to curve at a is f'(a) so we get,



Example. Let f(x) = 2x2 + 1. Find the slope of the graph of f(x) when x = 2

We need to evaluated f'(2), which is

If f(x) = 2x2 + 1, then
This last expression tends to 8 as Δx -> 0,
so we get

Fortunately, we do not need to go through this type of calculation each time we need a derivative; we use rules.

Rules give the derivative function or the derived function; we then insert the appropriate x value to get the derivative (or slope) at a particular point.

Rules of Differentiation.

  • If
,where c is a constant
  • If
,
  • If
,c a constant
  • If
,


Examples. Differentiate the following.
a)b)
c)d)
e)


Solutions
a)b)
c)d)
e)

Marginal Functions

The derivative of certain economic variables such as total revenue, TR, total cost, TC profit etc is called the marginal function.

Marginal Revenue
The marginal revenue is the rate of change of total revenue per unit increase in Q

Marginal Cost
The marginal cost is the rate of change of total cost per unit increase in Q


Example Consider the demand function
Find the marginal revenue function. If the current demand is 15, find the marginal revenue.

Solution Total revenue, TR is
Read Worked Example 6.7, pages 289-291
(Example 6.6, page 246 of 2nd Ed.)


Example The fixed costs of producing a good are 100 and the variable costs are 2 + Q/10
per unit.
a)Find expression for TC and MC.
b) Evaluate MC at Q=30 and hence estimate the change in total cost, 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 ?


Solution
a)
b)
c)
Read Worked Example 6.9 ; page 292, 293
(Example 6.8; page 248 of 2nd Ed.)

Average Functions.

The average revenue, AR, is defined as average revenue per unit for the first Q successive units sold.
or

The average cost, AC, is the total cost divided by the level of output produced.
or
Read Worked Example 6.11 ; pages 297, 298
(Example 6.10; pages 253, 254 of 2nd Ed.)


Example (#2, p298; page 254 of 2nd Ed.) The demand function for a good is,
a)Find expressions for TR, MR and AR. Is the slope of the MR curve twice the slope of the AR curve?
b)Evaluate TR, MR and AR at Q = 10 and Q = 25.


Solution
a)
b)At Q = 10
TR = 933.77 ; MR = 45.94 ; AR = 93.377

At Q = 25. Do as exercise.



Example (#4, page 299; page 255 of 2nd Ed.) A firm has the following average cost function
a)Show that AC decreases indefinitely as Q increases.
b)Write down the equation for total cost. Hence write down the equation for total variable cost and average variable cost. State the value of fixed cost.
c)Write down the equation for marginal costs.

Solution
a)
b)
Thus 10 is fixed cost and 50Q is total variable cost. 50 is average variable cost
c)

Differentiation and Curve Sketching

Higher Derivatives

Differentiating the first derivative gives the second derivative, differentiating the second gives the third, etc.
Notation. If
denotes the first derivative
denotes the second derivative
denotes the third derivative
gives the rate of change of
, as x increases


The sign of dy/dx indicates whether y is increasing or decreasing as x is increasing since;
when y is increasing, slope is positive, i.e.
when y is decreasing, slope is negative, i.e.
At


The sign of indicates
whether slope is increasing or decreasing since.
when slope is increasing,
when slope is decreasing,


Turning Points

At turning points, tangents are horizontal so slopes are zero i.e
at turning points


Example Find the turning point(s) for
a)
b)
a)
At a turning point, , so
b)


Read Worked Example 6.17 ; pages 310, 311
(Example 6.16; pages 261, 262 of 2nd Ed.)


Maximum and Minimum Turning Points

(local) minimum(local) maximum
Slope goes from
negative, to zero, to
positive so
Slope goes from
positive, to zero, to
negative so

Read Worked Example 6.18 ; pages 314, 317
(Example 6.17; pages 266, 267 of 2nd Ed.)



Example (#6, page 316; page 267 of 2nd Ed.) Find the turning points for the curve
Determine which are local maximum and which are local minimum points.

Solution
Turning points are where
Turning points when x = 3 and x = -1
At x = -1,
Thus x = -1 is a local maximum
At x = 3,
Thus x = 3 is a local minimum
Coordinates of these points are (-1, 5) and (3, -27)

Sketch of graph.

Curve Sketching

Example (#10, page 325; page 276 of 2nd Ed.) Sketch the curve
Solution
Step 1. Find derivatives
Step 2. Find turning points
are turning points








Step 2a. Find y-coordinates at each T.P.
Step 3. Nature of turning points
Step 4. Intercepts with coordinate axes
Step 5. Mark above points and plot.

Read Worked Example 6.20 ; pages 318-320
(Example 6.19; pages 269-272 of 2nd Ed.)



Economic Applications Of Maximum and Minimum Points

The first derivatives of economic functions are called marginal functions. Therefore, the optimum value of functions, such as revenue, profit, cost etc, will occur when the corresponding marginal function (first derivative) is zero.
Example (#1, page 340; page 290 of 2nd Ed.) A firm's total revenue is given by the equation
a) Find the value of Q for which TR is maximized. Hence calculate the maximum TR
b) Write down the equations of the average revenue and marginal revenue functions. Describe how AR and MR change before and after the maximum TR
c) Sketch TR, MR and AR on the same diagram. From the graph, write down the values of AR and MR when TR is zero and when TR is a maximum.

Solution
a)
If MR = 0, Q = 15 and TR = 675
b)
For Q < 15 ; AR and MR are positive but
decreasing. At Q = 15, MR = 0
For Q > 15 ; MR < 0 while AR is positive until
Q = 30
c) When TR = 0, AR = 90 and MR = 90
When TR is a maximum (i.e Q = 15)



Example (#3, page 341; page 290 of 2nd Ed.) A shop which sells T-shirts has a demand function and a total cost function given by,
a) Write down the equation for TR and profit.
b) Calculate the number of T-shirts which must be sold to maximize i) profit, ii) total revenue.
c) Write down the equations for MR and MC. Show that MR = MC when profit is maximized.
d) Plot the graphs of
i) TR and TC on the same diagram. From the graph estimate the break-even points. Conform your answer algebraically.
ii) MR and MC on the same diagram. What is the significance of the point of intersection of these two graphs?

Solution
a)
b)
c)
d) i)
ii) Exercise

Read Worked Examples 6.22, 6.23, 6.25; pages 325-335
(Examples 6.21, 6.22, 6,24; pages 277-285 of 2nd Ed.)


Further Differentiation

If
, x > 0


Chain Rule


Composite functions.

To differentiate, use the chain rule.
Write first function as
and second function as
In above examples;
The derivative is
So in case of
If


Example Differentiate the following
i)ii)
iii)iv)
Solution
i)
ii)
iii)
iv)


Product Rule

If u is a function of x and
v is a second function of x


Example Differentiate the following.
i) ii)
iii) iv)

Solution
i)
ii)
iii)
iv)


Quotient Rule

If , then



Example Differentiate the following.
i) ii)
iii) iv)

Solution
i)
ii)
iii)
iv)

Summary

Rules Used
  • Power Rule
  • Exponential Rule
  • loge = ln Rule
  • Chain Rule
  • Product Rule
  • Quotient Rule




Example (#4, page 376; page 319 of 2nd Ed.) A firm's average revenue function is
a) Determine the equation for TR and MR
b) find the value of Q for which TR is a maximum. Calculate the price and TR when TR is a maximum.


Solution
a)
b) TR is a maximum when MR = 0, i.e
When TR is a maximum, i.e Q = 20, then
Price is average revenue so at Q = 20


Example (#6, page 376; page 319 of 2nd Ed.) Given the average cost function
a) Find the minimum AC and graph the AC from Q = 0 to Q = 25.
b) Write down the equation of TC


Solution
a)
b)

Point Elasticity of Demand

Price elasticity of demand.
Point elasticity of demand is then given in terms of dQ/dP as,



Example (#3, p386; page 326 of 2nd Ed.) The demand for family membership of a sports club is given by the equation,
where P is the monthly fee.
a) Derive an expression for the price elasticity of demand in terms of Q. Evaluate εd when Q < 100 and when Q > 100.
b) If the club has 150 members calculate the fee per membership (price). If the fee increases by 10%, use the elasticity to calculate the approximate % change in demand.


Solution
a)
If Q < 100, εd < -1. This means that % change in Q is opposite in direction to that of P and is greater in size since.
If Q > 100, % change in Q is again in opposite direction but less in size than change in prize %.
b)
Read Worked Examples 6.40, and 6.43; pages 380, 384-386
(pages 322-324 of 2nd Ed.)
  • Exercises;
    # 4, 9 pages 386, 387
    (pages 326, 327 of 2nd Ed)

Example Given the demand function
find the price elasticity of demand when P = 51. If this price rises by 2% calculate the corresponding % change in demand.

Solution
We need to evaluate εd when P = 51, so first get corresponding value for Q and then substitute into (1) to get εd.

If P, solve for Q as follows,
Ignore Q = -9. So when P = 51, Q = 5 and from (1)


Review Problems

Example Evaluate the derivative of the following function at x = 2.
a)
b)
Solution
a)
b)



Example
a) The fixed cost of producing a product are 100 and the variable costs are 2 + Q/10 per unit.
i) Find an expression for total costs (TC) and marginal costs (MC)
ii) At what level of output does MC = 22?
b) If the supply equation is
Find the price elasticity of supply if the current price is 80. Estimate the percentage change in supply if the price increases by 5%.

Solution
a)i)
ii)
b)



Example If f(x) = x3 - 3x2 + 2x, find the intervals where f(x) is increasing or decreasing and the coordinates of the maximum or minimum point. Sketch the graph of f(x)