Chapter 6 Introduction to Calculus

1
Chapter 6 Introduction to Calculus
Bhaskara (1114 1185)
Archimedes of Syracuse (c. 287 BC c. 212 BC )
2
6.0 Calculus

• Calculus is the mathematics of change.
• It has two major branches, differential calculus
  and integral calculus, which are related by the
  Fundamental Theorem of Calculus.
• Differential calculus determines varying rates of
  change. It helps solve problems involving
  acceleration of moving objects (from a flywheel
  to the space shuttle), rates of growth and decay,
  optimal values, graphs of curves, etc.
• Integration is the "inverse" (or opposite) of
  differentiation. It measures accumulations over
  periods of change. Integration can find volumes
  and lengths of curves, measure forces and work,
  etc.
• Calculus has widespread applications in science,
  economics, finance and engineering.
• Although for many years there was a big
  controversy, it is now generally accepted that
  Newton and Leibniz discovered it.

3
6.0 Calculus

• This chapter uses differential calculus to study
  what happens to an equilibrium in an economic
  model when something changes. This is called
  comparative statics.
• Example One commodity market

3
4
6.0 Application to Market National Income
Models Matrix Inversion


5
6.1 Comparative Statics

• Comparative statics a study of different
  equilibrium states associated with different sets
  of values of parameters and exogenous variables.
• Begin by assuming an initial equilibrium is
  given, say (P0,Q0).
• Examples
• Isolated market model (P0,Q0) (shock)? (P1,Q1)
  
• National income model (Y0, C0) (shock)? (Y1, C1)

6
6.1 Comparative Statics

• Static equilibrium analysis
• y f(x)
• Comparative static equilibrium analysis
• y1 - y0 f(x1) - f(x0)
• Where the subscripts 0 and 1 indicate initial
  and subsequent points in time respectively
• Issues
• Quantitative qualitative of change or
• Magnitude direction
• The rate of change --i.e., the derivative (?Y/
  ?G)

7
6.1 Comparative Statics Application

• The Macro-economic model
• Given
• Y C I0 G0
• C a b(Y-T)
• TdtY
• Solving for Y gt Y (a-bd I0 G0)/(1-(
  b(1-t)))
• Question What happens to Y when an something
  (exogenous) changes in the model?

8
6.2 Rate of Change and the Derivative

• Difference quotient
• Let y f(x)
• - Evaluate yf(x) at two points x0 and x1
• y0f(x0) and y1f(x1)
• - Define ?x x1- x0 gt x1 x0 ?x
• ?y y1- y0 gt ?y f(x0 ?x) -f(x0)
• - Then, we define the difference quotient as
• Derivative
• Lets take limits (?x?0) in the difference
  quotient we get the derivative

9
6.2 Rate of Change and the Derivative

• Graphical interpretation of the difference
  quotient
• Secant slope rise/run

f(x)
f(x0?x)
DAlembert saw the tangent to a curve as a limit
of secant lines. As the end point of the secant
converges on the point of tangency, it becomes
identical to the tangent in the limit.
10
6.2 Rate of Change and the Derivative

• Graphical interpretation of the derivative
• Slope tangent of the function at xx0

Jean d'Alembert (17171783)

• As the lim of ?x?0, then the f(x) measures the
  tangent (rise over run) of f(x) at the initial
  point A. Again, the secant becomes the tangent.

11
6.2 Rate of Change and the Derivative Linear
function
12
6.2 Rate of Change and the Derivative Power
function
13
6.2 Rate of Change and the Derivative Power
function
14
Example (continuation) y 3x2
4 (red)Evaluate difference quotient at x3,
when ?x 4 ?y/?x 6x 3?x gt ?y/?x 30 y
30x 67, secant through pts (3, f(3), 7,
f(7)) (blue)
6.2 Rate of Change and the Derivative Power
function
15
Now, evaluate difference quotient at x3 y 3x2
4 (red) f(x3)18 y 18x 31,
tangent at point (3, f(3)) (blue)
6.2 Rate of Change and the Derivative Power
function
16
Figure 6.9 Differential Approximation and Actual
Change of a Function
17
Figure 6.10 Actual and Approximate Change in
Imports
18
Figure 6.11 Differential Approximation for Beta
with Different Functions
19
6.2 Rate of Change and the Derivative Application

• Recall the solution to Y (income) in the
  Macroeconomic model
• We have a linear function in I and G. Assume I is
  fixed, then we have yf(G).
• Comparative static Question
• What happens to Y when G (government spending)
  increases?

19
20
6.3 Concept of limits

• The limit (f(x), x?a, direction) function
  attempts to compute the limiting value of f(x) as
  x approaches a from left or right. (eg., N,
  infinity, undefined)
• If q g(v), what value does q approach as v
  approaches N? Answer L
• As v ? N from either side, q ? L. In this case
  both the left-side limit (v less than N) and the
  right side-limit are equal.
• Therefore, lim q L

Infinitesimals are locations which are not zero,
but which have zero distance from zero.
v?N
v?N
v ? N
21
6.3 Evaluation of a limit

• To take a limit, substitute successively smaller
  values that tend to N from both the left and
  right sides since N may not be in the domain of
  the function
• If v is in both the numerator and denominator
  remove it from either depending on the function
• Formal view of the limit concept for a given
  number L, there can always be found a number
  (L-a1) lt L and and other (La2)gtL, where a1 and
  a2 are arbitrary positive numbers. These numbers
  line in the neighborhood of a point on a line.

22
6.3 Concept of limits

• If q g(v), what value does q approach as v
  approaches N?
• Answer 1 lim q M
• Answer 2 lim q M
• In certain cases, only the limit of one side
  needs to be considered. In taking the limit of q
  as v ? ?, for instance, only the left-side
  limit of q is relevant, because v can approach
  ? only from the left.

v ? ?
v ? - ?
v ? -?
v ? ?
23
6.3 Concept of a limit

• As v approaches a number N, the limit of qg(v)
  is the number L, if, for every neighborhood of L
  that can be chosen, however small, there can be
  found a corresponding neighborhood of N
  (excluding vN) in the domain of the function
  such that, for every value of v in that
  N-neighborhood, its image lies in the chosen
  L-neighborhood.

• Given q (2v 5)/(v 1), find the lim q as v ?
  8.
• Dividing the numerator by denominator

24
6.3 Digression on Inequalities and Absolute

• Rule I (addition and subtraction)
• a gt b result in a k gt b k
• Rule II (multiplication and division)
• a gt b results in ka gt kb when kgt0
• a gt b results in ka lt kb when klt0
• Rule III (squaring)
• a gt b (b?0) results in a2 gt b2
• n absolute value (-n lt n lt n)
• Example Solve the inequity 1-x lt 3 for x

25
6.3 The Concept of Limit

• Let q??y/?x and v ??x such that q f(v) and

• What value does variable q approach as variable v
  approaches 0?

26
6.4 Limit Theorems

• If qavb, then aN b
• If q g(v) b, b
• If q v, then N
• If q vk, then Nk

• Example Find lim (1v)/(2 v) as v?0

27
6.4 Limit Jokes

27
28
6.5 Continuity and Differentiability of a Function

• A continuous function
• When a function qg(v) possesses a limit as v
  tends to the point N in the domain and
• When this limit is also equal to g(N), i.e., the
  value of the function at vN, then the function
  is continuous in N

• Requirements for continuity
• N must be in the domain of the function f, qg(v)
• f has a limit as v ? N
• limit equals g(N) in value

29
6.5 Continuity and Differentiability of a Function

• Almost all the basic functions in mathematical
  econ models are assumed to be continuous.
• For example, a production function is continuous
  if a small change in inputs yields a small change
  in output. (A reasonable assumption.)

29
30
Figure 2.5 Functions That Are Not Continuous
31
6.5 Continuity and Differentiability of a Function

• We say f Rn?R is differentiable at x if f(x)
  exists.
• Not every function has a derivative at every
  point. For example f(x)x.
• x is not differentiable at x0 there is no
  unambiguous tangent line defined at x0.
• We need the function f(.) to be smooth i.e., no
  kinks.
• A function f Rn?R is continuously differentiable
  on an open set U of Rn if and only if for each i
  df/dxi exists for all x in U and is continuous in
  x.

31
32
Figure 6.6 Functions Not Everywhere
Differentiable
33
6.5 Continuity and Differentiability of a Function

• This rational function is not defined at v 2,
  even though the limit exists as v ? 2. It is
  discontinuous and thus does not have continuous
  derivatives --i.e., it is not continuous
  differentiable.

• This continuous function is not differentiable at
  x 3 and therefore does not have continuous
  derivatives, i.e., it is not continuously
  differentiable

34
6.5 Continuity and differentiability of a function

• For a function to be continuous differentiable
• All points in in domain of f defined
• When the limit concept is applied to the
  difference quotient at x x0 as ?x ? 0 from both
  directions. The continuity condition is necessary
  but not sufficient.
• The differentiability condition (smoothness) is
  both necessary and sufficient for whether f is
  differentiable, i.e., to move from a difference
  quotient to a derivative

35
6.6 Resolution of a Controversy Butter Biscuits
or Fruit Chewy Cookies?