Introduction to Calculus

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Introduction to Calculus

• Applications of Differentiation

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Overview

• Marginal Analysis in Economics
• The First Derivative
• The Second Derivative
• Curve Sketching

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Marginal Analysis in Economics
Marginal analysis is the study of the rate of
change of economic quantities. Applications of
marginal analysis include

• Cost functions
• Revenue Functions
• Profit functions

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Cost Functions

• Total Cost function, C(x) is the cost of
  producing a quantity of x goods.
• Cost functions are made up of fixed costs and
  variable costs.
• Fixed costs are incurred even if no items are
  produced.
• Variable costs are dependent on the number x of
  items produced.

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In Economics, cubic functions (3rd order
polynomials) are selected so that they have
certain properties.
1. C(x) gt 0 and C(x) gt 0 for x gt 0.
y
C(x)
2. The slope of the cost function is large
initially, then becomes smaller, and then becomes
larger again when too many items are produced.
x
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The marginal cost function is the derivative of
the total cost function, MC(x) C(x), where x
is the quantity produced.
The average cost function is the quotient of the
total cost function and the quantity produced,
AC(x) C(x)/x.
The derivative of the average cost function is
called the marginal average cost function. We can
use the quotient rule of differentiation to find
it.
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Revenue Functions
The demand function shows the relationship
between the price per item and the quantity
demanded by consumers of the item. p D(x) the
price per unit consumers are willing to pay when
x items are available on the market.
The revenue function shows the revenue generated
when a specific number of items are sold.
R(x) xp xD(x)
The marginal revenue function is the derivative
of the revenue function. MR(x) R(x) D(x)
xD(x)
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Profit Function
The profit function shows the profit generated
when a specific number of items are sold.
P(x) R(x) C(x)
The marginal profit function is the derivative of
the profit function.
P(x) R(x) C(x)
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Applications of the First Derivative
The first derivative is useful in determining the
intervals where a function is increasing or
decreasing.

• If f(x) gt 0 for each value in an interval (a,b),
  then f is increasing on (a,b).
• If f(x) lt 0 for each value in an interval (a,b),
  then f is decreasing on (a,b).
• If f(x) 0 for each value in an interval (a,b),
  then f is constant on (a,b).

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Step 1 Find all the values of x for which f(x)
0 or f is discontinuous and identify the open
intervals determined by these numbers.
Step 2 Select a test point c in each interval
found in Step 1 and determine the sign of f(c)
in that interval.

• If f(c) gt 0, f is increasing on that interval

• If f(c) lt 0, f is decreasing on that interval

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Relative Extrema
A function f has a relative maximum at x c if
there exists an open interval (a, b) containing c
such that f(x) lt f(c) for all x in (a, b).
A function f has a relative minimum at x c if
there exists an open interval (a, b) containing c
such that f(x) gt f(c) for all x in (a, b).
A function f has a critical number at x c if
f(c) 0 or f(c) does not exist.
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The First Derivative Test
Step 1 Find all critical numbers of f.
Step 2 Determine the sign of f(x) to the left
and right of each critical number.

• If f(x) changes from positive to negative then
  f(c) is a relative maximum.

• If f(c) changes from negative to positive, then
  f(c) is a relative minimum.

• If f(c) does not change sign, then f(c) is not a
  relative extremum.

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Applications of the Second Derivative

• The first derivative of a function tells us where
  the graph is rising and falling. The second
  derivative tells us in what direction the graph
  of the function curves or bends.

y f(x)
y g(x)
The graph of f is concave up while the graph of g
is concave down.
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Concavity and the Second Derivative

• A curve is concave up if its slope is increasing,
  in which case the second derivative is positive.
• A curve is concave down if its slope is
  decreasing, in which case the second derivative
  is negative.
• A point where the graph of the function changes
  concavity, from concave up to concave down or
  vice versa, is called a point of inflection. At a
  point of inflection the second derivative is
  either zero or undefined.

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Step 1 Find all the values of x for which f(x)
0 or f is undefined and identify the open
intervals determined by these numbers.
Step 2 Select a test point c in each interval
found in Step 1 and determine the sign of f(c)
in that interval.

• If f(c) gt 0 then f is concave upward on that
  interval.

• If f(c) lt 0 then f is concave downward on that
  interval.

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Finding Points of Inflection
Step 1 Compute f(x).
Step 2 Determine the numbers in the domain of f
for which f(x) 0 or f(x) does not exist .
Step 3 Determine the sign of f(x) to the left
and to the right of each number c found in Step
2. if there is a change in the sign of f(x) as
we move across x c, then (c, f(c)) is a point
of inflection.
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The Second Derivative Test
Step 1 Compute f(x) and f(x).
Step 2 Find all the critical numbers of f at
which f(x) 0.
Step 3 Compute f(c) for each critical number c.

• If f(c) lt 0, then f has a relative maximum at c.

• If f(c) gt 0 then f has a relative minimum at c.

• If f(c) 0 the test is inconclusive.

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Asymptotes
The line x a is a vertical asymptote of the
graph of a function f if either
The line y b is a horizontal asymptote of the
graph of a function f if either
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Curve Sketching

• Important Features of a Graph
• The domain of the function
• The x- and y- intercepts
• Relative extrema
• Points of Inflection
• Concavity and changes in concavity
• Horizontal Asymptotes
• Vertical Asymptotes

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Example
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(0.451, 0.631) Relative Maximum
ConcaveUp
ConcaveDown
(2.215, -2.113) Relative Minimum
(1.333, -0.741) Point of Inflection