ECO 3401

1
Lesson 4
2
Limits

• Getting closer and closer to something but yet
  not touching it.
• Ex Find the limit
• Solution Let f(x)x21

x approaches
x approaches
3
Limit

• Graph

2
(1, 2)
4
Informal Definition of limit

• Let f be a function and let a and L be real
  numbers. Assume that f (x) is defined for all
  xa. Suppose that as x takes values very close
  (but not equal) to a (on both sides of a), the
  corresponding values of f (x) are very close
  (and possibly equal) to L and that the values of
  f (x) can be made as close as you want to L for
  all values of x that are close enough to a. Then
  the number L is the limit of the function f (x)
  as x approaches a, which is written

5
Algebraic Properties of Limits

• Let a, k, A, and B be real numbers, and let f
  be a function such that
• and
• Then

6
Algebraic Properties of Limits
(The limit of a constant is the constant itself)
(The limit of a sum or difference is the sum or
difference of the limits)
(The limit of a product is the product of the
limits)
(The limit of a quotient is the quotient of the
limits)
7
Algebraic Properties of Limits
8
Left-hand and Right-hand Limits

• One sided Limits

Limit from the Left
Limit from the Right
9
Continued

• The first of these two limits is read as the
  limit of f(x) as x approaches c from the left is
  L. The second is read as the limit of f(x) as x
  approaches c from the right is L.
• For the limit of a function to exist as x gt c, it
  must be true that both one-sided limits exists
  and are equal.

10
Example

• Find the limit as x gt 0 from the left and the
  limit as x gt 0 from the right for the function
• Limit from the left
• Limit from the right

2
Since the function approaches different limits
from the left and the right, the limit doesnt
exist
-2
11
Direct Substitution

• Evaluating a limit
• Ex
• Substitute 2 into the function (23)5.
• Ex
• You cannot always use direct substitution
  !!!!!!!!

12
Continued

• Ex
• In this case direct substitution fails because
  both the numerator and the denominator are zero
  when x-2
• You must change the original function into an
  equivalent function
• Factor Numerator

13
Continued

• Evaluate the limit
• The limit is 0
• If you get any nonzero over 0 THERE IS NO
  LIMIT!!!!
• Ex

14
Limits and Asymptotes

• Unbounded Behavior A Limit can fail to exist
  when f(x) increases or decreases without bound as
  a x approaches c.
• Example Evaluate the limit (if possible)
• From the left
• From the right
• Because f is unbounded as x approaches 2, the
  limit does not exist.

15
Graph

• Graph

2
16
Modeling Average Cost

• A small business invests 5,000 in a new product.
  In addition to this initial investment, the
  product will cost 0.50 per unit to produce. Find
  the AC per unit if 1,000 units are produced, if
  10,000 units are produced, and if 100,000 units
  are produced. What is the limit of the average
  cost as the number of units produced increases?
• Solution
• From the given information you can model the
  total cost C(in dollars) by
• TC0.5x5000
• Where x is the number of units produced.

17
Continued

• Therefore, the average cost function is
• If 1000 units are produced, then the AC per unit
  is 5.50
• AC.5(5000/1000)
• If 10,000 units are produced, then the AC per
  unit is 1
• AC.5(5000/10000)
• If 100,000 units are produced , then the AC per
  unit is 0.55
• AC.5(5000/100000)

18
Continued

• As x approaches infinity, the limit for AC per
  unit is
• In words, the cost per unit is approaching .50
  as the number of units produced increases without
  bound.
• This example points out one of the major
  problems of small business. That is, it is
  difficult to have competitively low prices when
  the production is low.

19
Calculus

• Derivatives is basically nothing more than a set
  of rules for finding slopes and rates of change.
• In a business context finding the slope of a
  curve at a point can mean among other uses,
• 1.  Finding slope of Total Cost function at
  certain level of output. MC is the slope of TC
  and the derivative of TC
• 2.  Finding slope of Total Revenue function at a
  certain level of out put. MR is the slope of TR
  and the derivative of TR
• Knowing both MC (derivative of TC) and MR
  (derivative of TR) enables the manager to
  determine how much output a firm must produce to
  earn its maximum possible profit

20
AVERAGE RATE OF CHANGE

• AVERAGE RATE OF CHANGE

f(xh)
f(x)
x
x?x Or xh
Rise/Run
21
Tangent line

• Using a tangent line to measure the SLOPE

f(xh)
f(xh)
f(x)
f(x)
x
xy
xy
x
The Slope of the Tangent line will approximate
the slope of the curve at the point of tangency
f(x),f(xh)

x, xy
22
Derivatives and Slope

• The derivative is the slope of the tangent line.
  The derivative can be written as f (x) or dy/dx
  (dee y over dee x) or y , or ?y/ ?x (delta y
  over delta x)
• Four steps to find derivatives
• 1. Find f(xh)
• 2. Find the difference of f(xh)-f(x)
• 3. Find the ratio f(xh)-f(x)/h
• 4. Take the limit (instantaneous Rate of Change)
• Provided the limits exists

23
Example

• Let f(x)x2
• a. Compute f (x)
• b. Compute f (2) and Interpret your result
• Solution
• a. To find f (x) we use the 4 step process

Expand terms
24
Continued

• Find f (x)
• B. f (2)2(2)4
• This result tells us that the slope of the
  tangent line to the graph of f at point (2,4) is
  4. It also tells us that the function f is
  changing at the rate of 4 units per unit in x at
  x2

Tangent Line
y
f(x)x2
4
2
x
25
Example

• Find a formula for the slope of the graph of f(x)
  x21.
• What is the slope at points (-1,2) and (2,5)
• .

Expand terms
26
Continued

• Using the formula2x, you can find the slope at
  the specified points. At(-1,2), the slope is
  m2(-1)-2 and at
• (2,5), the slope is m2(2)4

f(x)x21
Slope4
Slope-2
2
1
-1
2
27
RULES FOR DIFFERENTIATION
28
1. CONSTANT RULE

• 1. THE CONSTANT RULE
• The derivative of a constant function is zero.
• That is f ( c ) 0, c is a constant.
• Ex f(x)28 then f (x) 0
• if f(x) -2 then f (x) 0

29
2. THE (SIMPLE) POWER RULE

• f (x) xn then f (x) nx(n-1)
• Ex if f(x) x then
• f (x) 1x(1-1) 1
•  
• if f(x) x8 then
• f (x) 8x(8-1) 8x7
•  
• if f(x) x-10, f (x) -10x-11
• if f(x) x5/2 f (x) 5/2x3/2

30
3. THE CONSTANT MULTIPLE RULE

• The derivative of a constant times a
  differentiable function is equal to the constant
  times the derivative of the function.
• if f (x)3x2 f (x) 23x 6x
• if f(x) 5x3 then f (x)15x2
•  if f(x) 3/ x1/2 f (x) 3x-1/2
• f (x) 3(-1/2x-3/2) -3/2(x-3/2).

31
4. THE SUM AND DIFFERENCE RULES

• The derivative of the sum or difference of two
  differentiable functions is the sum or difference
  of their derivatives.
• Ex f(x) g(x) f (x) g (x)
• and
• f(x) - g (x) f (x) - g (x)
•  

32
Continued

•  1.   f(x) 4x5 3x4 - 8x2 x 3
• f (x) 20x4 12x3 - 16x 1 0
•  
• 2.   Find the value of the derivative of the
  function at the indicated point. f (x) 1/x
  (1,1)
•  

33
Continued

• 3.   f(t) 4 - 4/3t (1/2, 4/3)

34
Continued

• 4. Y(2x1)2 (0,1)
• At (0,1) y4

35
Continued

• 5.   Find an equation of the tangent line to the
  graph of the function at the indicated point.
• Y-2x45x2-3 (1,0)
• At (1,) the slope is my-8(13)102
• The equation of the tangent line is
• Y-02(x-1) y2x-2 

36
Main Concepts

• 1.  A derivative is the slope of a tangent line
  to a function.
• 2.  Evaluating a derivative means finding the
  value of the derivative of the function yf(x) at
  a point.
• 3.  The derivative is an instantaneous rate of
  change
• 4.  If a derivative of a function is positive at
  a point the slope will be positive at the same
  point.

37
5. THE PRODUCT RULE

• The derivative of the product of two
  differentiable functions is equal to the first
  function times the derivatives of the second plus
  the second function times the derivative of the
  first.
•  
• d/dx f(x) g(x) f(x) g (x) g(x) f
  (x)
•  
• Ex f(x) (2x2 - 1)(x33)
•  
• f (x) (2x2 - 1)(3x2) (x3 3) (4x)
• 6x4 -3x2 4x4 12x
• 10x4 - 3x2 12x

38
Product Rule Continued

• Ex f(x) x3 (SQRTx 1)

39
6. THE QUOTIENT RULE
The derivative of the quotient of two
differentiable functions is equal to the
denominator times the derivative of the numerator
minus the numerator times the derivative of the
denominator, all divided by the square of the
denominator   d/dx f(x)/g(x) g(x) f (x) -
f(x) g(x)/ g (x)2  
40
THE QUOTIENT RULE

• Ex

41
THE QUOTIENT RULE

• Ex

42
THE QUOTIENT RULE

• Ex

43
Marginal Analysis

• Marginal Cost function MC(x)C(x)
• Marginal Revenue function MR(x)R(x)
• Marginal Profit function MP(x)P(x)
• Average Cost function AC(x) C(x)/x
• Average Profit function AP(x)P(x)/x

44
Applications

• An important use of rates of change is in the
  field of economics. Economists refer to marginal
  profit, marginal revenue and marginal cost as the
  rates of change of the profit, revenue, and cost
  with respect to the number q of units produced or
  sold. An equation that relates these three
  quantities is PR-C
• where P, R and C represent the following
  quantities
• P total profit, RTotal revenue and Ctotal
  cost
• The derivatives of these quantities are called
  the marginal profit, marginal revenue, and
  marginal cost.

45
Continued

• Ex Find the marginal profit for a production
  level of 50 units
• P 0.0002q3 10q
• MP 0.0006q2 10
• 0.0006(502)10
• 11.50 per unit

46
Applications

• Ex Find the marginal revenue
• A fast-food restaurant has determined that the
  monthly demand for their hamburgers is given by
• p (60,000 -q)/20,000
• Find the MR when q20,000
• TR pq
• (60,000q-q2)/20,000
• MR 60,000-2(20,000)/20,000 1 per unit.
•  
•  

47
Applications

• Ex find the marginal cost for a production level
  10 units
• IF TC0.1q2 3
• MC .2(10) 2
• MC is the approximate cost of one additional unit
  of output.

48
Finding Average Cost

• A company estimates that the cost (in dollars) of
  producing x units of a product can be modeled by
• C800 0.04x 0.0002x2
• Find the the average cost function
• Solution
• C represent the total cost, x represents the
  number of units produced, and AC represents the
  average cost per unit.
• AC C/x
• Substituting the given equation for C produces
• AC (8000.04x 0.0002x2)/(x)
• 800/x 0.04 0.0002x

49
7. CHAIN RULE (14.1)

• If yf(u) is a differentiable function of u, and
  g(x) is a differentiable function of x, then
  yf(g(x)) is a differentiable function of x, and
• d/dx f(g(x)) f (g(x))g(x).
• When applying the Chain rule it helps to think of
  the composite function yf(g(x) or yf(u) as
  having two parts an inside and an outside
• yf(g(x))f(u)

Inside
Outside
50
7. CHAIN RULE

• The Chain Rule tells you that the derivative of
  yf(u) is te derivative of the outer function
  times the derivative of the inner function. That
  is yf (u)u

51
General Power Rule

• d/dx un nu(n-1) u

52
Continued

• Find first derivative

53
Derivatives of Logarithmic Functions

• Derivative of the Natural Logarithmic Function
• Let u be a differentiable function of x

54
Example

• Find the derivative of

Let u2x and u2
55
Example

• Find the derivative of

Let u2x24 and u4x
56
Example

• Find the derivative of

Use the Product rule
57
Derivatives of Exponential Functions

• Let u be a differentiable function of x

58
Example

• Find the derivative of

Let u2x and u2
59
Example

• Find the derivative of

Let ux3 and u3x2
60
Example

• Find the derivative of

Product Rule
61
Example

• Find the derivative of

Product and Difference Rule
Simplify
62
Implicit Differentiation

• Implicit differentiation is used when we have a
  relationship between two variables x and y and we
  are unable or we choose not to solve for y in
  terms of x explicitly.
• Explicitly
• Find the slope of the tangent line to
• (1,3)
• Lets solve for y

63
Continued

• Lets find the first derivative

64
Continued

• The slope of the tangent line when x1 and y3

65
Implicit Differentiation

• Differentiate both sides of the equation with
  respect to x treating y as a differentiable
  function of x.
• Collect the terms with dy/dx on one side of the
  equation
• Solve for dy/dx

66
Continued

• Lets use implicit differentiation for the same
  function,
• Lets solve for dy/dx

67
Continued

• Find the slope of the tangent line to
  (1,3)

68
Continued

• Find dy/dx for

69
Continued

• Find dy/dx for