1. Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e
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2. Differential Calculus
Chapter 11
Differential Calculus
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Section 11.1
Limits
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f(x) = x2 + x + 1, find
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Example:
Find
Solution:
Quotient property
Polynomial limit
Note that So the limit of as
x approaches 5 is the value of the function at 5:
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Exercises
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Review
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18. One-Sided Limits and Limits Involving Infinity
Section 11.2
One-Sided Limits and Limits Involving Infinity
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21. Find each of the given limits.
Example:
Find each of the given limits.
(a)
Solution:
Since is not defined when the right-hand limit (which requires that does not exist. For the left-hand limit, write the square root in exponential form and apply the appropriate limit properties.
Exponential form
Power property
Polynomial limit
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22. Find each of the given limits.
Example:
Find each of the given limits.
(b)
Solution:
Exponential form
Sum property
Power property
Polynomial limits
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Examples
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Continued on next slide
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Continued from previous slide
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Infinite Limits
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Examples
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Review 1) 2) 3) 5) 4)
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Section 11.3
Rates of Change
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Example:
If find the average rate of change of with respect to x as x changes from –2 to 3.
Solution:
This is the situation described by the expression with The average rate of change is
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Example
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Example
Exercise
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The rate of change of the cost function is called the marginal cost. Similarly, the rate of change of the revenue and profit function are called the marginal revenue and marginal profit.
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Exercises 1) 2)
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37. Tangent Lines and Derivatives
Section 11.4
Tangent Lines and Derivatives
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Secant lines
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40. According to the definition, the slope of the tangent line is
Example:
Let a be any real number. Find the equation of the tangent line to the graph of at the point where
Solution:
According to the definition, the slope of the tangent line is
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41. Hence, the equation of the tangent line at the point is
Example:
Let a be any real number. Find the equation of the tangent line to the graph of at the point where
Solution:
Hence, the equation of the tangent line at the point is
Thus, the tangent line is the graph of
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If f’(x) is defined, then the function f is said to be differentiable at x. The process that produces f’ from f is called differentiation.
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Examples 1) 2) 3)
4) Label f and f’ correctly:
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Existence of the Derivative
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Select the graph of f’
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Exercise
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49. Techniques for Finding Derivatives
Section 11.5
Techniques for Finding Derivatives
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56. From the function p, the revenue function is given by
Example:
The demand function for a certain product is given by
Find the marginal revenue when units and p is in dollars.
Solution:
From the function p, the revenue function is given by
The marginal revenue is
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57. When the marginal revenue is
Example:
The demand function for a certain product is given by
Find the marginal revenue when units and p is in dollars.
Solution:
When the marginal revenue is
or $1.20 per unit. Thus, the next unit sold (at sales of 10,000) will produce an additional revenue of about $1.20.
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Exercise
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59. Derivatives of Products and Quotients
Section 11.6
Derivatives of Products and Quotients
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Derivative of product of 3 functions:
[u(x)v(x) w(x)]’ = u’(x)v(x) w(x) + u(x)v’(x) w(x) + u(x)v(x) w’(x)
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Proof
[f(x)g(x)]’
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Example:
Solution:
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Example:
Find the derivative of
Solution:
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Example
Given that: (sin(x))’ = cos(x) and (cos(x))’ = -sin(x)
Find derivative of
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Proof
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Example
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Complete the table without using the Quotient Rule
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Example: The cost in dollars of manufacturing x hundred items is given by: C(x) = 4x2 + 6x + 5
Find the average cost
Find the marginal average cost
Find the marginal cost
Find the level of production at which the marginal average cost is zero.
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Exercises
Use quotient rule to prove the Power Rule for negative integers n.
i.e. (xn)’ = nxn-1 for negative integers n.
Find derivative of g(x).
y = 1
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Identify each graph.
Sketch the graph of f’
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Review
f(x) = x2
Write equation of the tangent line to the graph of function f(x) = x2 at (1, 1)
Write equation of the tangent line to the graph that passes through the point (-1, -1)
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Section 11.7
The Chain Rule
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Composite function
Let f and g be functions. The composite function, or composition, of f and g is the function whose values are given by f[g(x)] for all x in the domain of g such that g(x) is in the domain of f.
Example: find functions f and g so that f[g(x)] are the followings:
f(x) = g(x) =
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Example: find functions f and g so that f[g(x)] are the followings:
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Proof
c is in the domain of h
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Example:
Use the chain rule to find
Solution:
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Example:
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81. Summary of Differentiation Rules
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Exercises
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83. Exercises: find derivative of functions below.
Find the equation of the tangent line to the graph of f at the given point.
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Label the graph as f or f’:
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85. Derivatives of Exponential and Logarithmic Functions
Section 11.8
Derivatives of Exponential and Logarithmic Functions
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Proof:
Example: y = e2x – 1. Find y’.
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Example:
Let Find
Solution:
Use the quotient rule:
Exercise: for a > 0, y = ax. Find y’.
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Proof:
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Examples: find the derivatives
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If y = ln|u|, then y’ =
Example:
y = loga(x)
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93. Continuity and Differentiability
Section 11.9
Continuity and Differentiability
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Example:
Is the function in the following graph continuous on the given x-intervals?
(a)
Solution:
The function g is discontinuous only at Hence, g is continuous at every point of the open interval
which does not include 0 or 2.
(b)
Solution:
The function g is not defined at so it is not continuous from the right there. Therefore, it is not continuous on the closed interval
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Example:
Is the function in the following graph continuous on the given x-intervals?
(c)
Solution:
The interval contains a point of discontinuity at So g is not continuous on the open interval
(d)
Solution:
The function g is continuous on the open interval
continuous the right at and continuous from the left at
Hence, g is continuous on the closed interval
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Proof:
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Match each function with each graph
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