Math Review and Lessons in Calculus

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Math Review and Lessons in Calculus
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Agenda

• Rules of Exponents
• Functions
• Inverses
• Limits
• Calculus

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Rules of Exponents

• x0 1
• xa xb x(ab)
• x2 x3 x5
• xa / xb x(a-b)
• x5 / x2 x3
• x-a 1 / xa
• x-5 1 / x5

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Rules of Exponents Cont.
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Rules of Exponents Cont.

• xaya (xy)a
• x2y2 (xy)2
• 52 42 (5 4)2

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Exponent Example

• Assume that x1½ x2 ¼ 1.
• Solve x2 as a function of x1.

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Exponent Example Cont.
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Functions

• A function describes how a set of unique
  variables is mapped/transformed into another set
  of unique variables.
• Examples
• Linear y f(x) ax b
• Quadratic y f(x) ax2 bx c
• Cubic y f(x) ax3 bx2 cx d
• Exponential y f(x) aex
• Logarithmic y f(x) aln(x)

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Functions Cont.

• A function is comprised of independent and
  dependent variables.
• The independent variable is usually transformed
  by the function, whereas the dependent variable
  is the output of the function.
• In the examples above y is a variable that is
  usually known as the dependant variable, while x
  is considered the independent variable.

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Functions Cont.

• Functions can map more than one independent
  variable into the dependent variable.
• Example
• y f(x1, x2) ax1 bx2 c
• It is possible to represent a function visually
  by using a graph.

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Some Rules for Functions

• Functions can be summed and then evaluated.
• Functions can be subtracted from each other and
  then evaluated.
• Functions can be multiplied by each other and
  then evaluated.
• Functions can be divided by each other and then
  evaluated assuming the denominator is not equal
  to zero.

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Composition of Functions

• When one function is evaluated inside another
  function, it is said that you are performing a
  composition of functions.
• Mathematically, we represent a composition of
  functions in two ways
• f(g(x))
• (f?g)(x)
• Note this is NOT multiplication of the two
  functions.

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Example of Composition

• Suppose f(x) 2x 1 and g(x) 3x3, then
• f(g(x))2(3x3) 1 6x 6 1 6x7
• g(f(x))3(2x1) 3 6x 3 3 6x6
• Note that f(g(x)) does not necessarily equal
  g(f(x)).

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Inverse Functions

• An inverse function is a function that can map
  the dependent variable into the independent
  variable.
• In essence, it is a function that can reverse the
  independent and dependent variables.
• Example
• y ax b has as its inverse function x y/a
  b/a.

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Example of Finding the Inverse of a Linear
Equation

• y 5x 10
• Subtract 10 from both sides
• y 10 5x
• Divide both sides by 5
• (y/5) 2 x
• x (y/5) 2
• This is the inverse of the first equation above.

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Quadratic Formula

• The quadratic formula is an equation that allows
  you to solve for all the x values that would make
  the following equation true
• ax2 bx c 0

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Using the Quadratic Formula to Find the Inverse
of a Quadratic Equation

• Suppose you had the following equation
• y qx2 rx s
• If we transform the above equation into the
  following, we can use the quadratic equation to
  find the inverse
• qx2 rx s-y 0

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Using the Quadratic Formula to Find the Inverse
of a Quadratic Equation Cont.

• Define a q, b r, and c s-y
• Substituting these relationships into the above
  equation gives the following
• ax2 bx c 0

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Inverse of a Quadratic Equation Example

• Suppose y 2x2 4x 8
• By rearrangement 2x2 4x 8 y 0

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Finding the Intersection of Two Equations

• At the point where two equations meet, they will
  have the same values for the dependent and
  independent variables for each equation.
• Example
• y 5x 7 and y 3x 11 intersect at y 17
  and x 2.

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Finding the Intercepts of a Curve or Line

• The vertical intercept is where the curve crosses
  or touches the y-axis.
• To find the vertical intercept, you set x 0,
  and solve for y.
• The horizontal intercept is where the curve
  crosses or touches the x -axis.
• To find the horizontal intercept, you set y 0,
  and solve for x.

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Slope of a Line

• From algebra, it is known that the slope (m) of a
  line is defined as the rise over the run, i.e.,
  the change in the y value divided by the change
  in the x value.

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Slope of the Line Cont.

• The slope of a line is constant.
• The slope of a line gives you the average rate of
  change between two points.

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Needed Terminology

• A secant line is a line that passes through two
  points on a curve.
• A tangent line is a line that touches a curve at
  just one point.
• In essence, it gives you the slope of the curve
  at the one point.
• A tangent line can tell you the instantaneous
  rate of change at a point.

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Limits

• A number L is said to be the limit of a function
  f(x) at point t, if as you get closer to t, f(x)
  gets closer to L.

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Example of a Limit
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Finding the Tangent Line

• One way to find the tangent line of a curve at a
  given point is to examine secant lines that have
  corresponding points that get closer to each
  other.
• This is known as examining the limits.

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Graphical Representation of the Secant and the
Tangent Lines
Y
Function y f(x) x2
Secant Line
f(x?x)
Tangent Line
f(x)
X
x
x?x
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Defining the Derivative

• According to Varian, the derivative is the limit
  of the rate of change of y with respect to x as
  the change in x goes to zero.
• Suppose y f(x), then the derivative is defined
  as the following

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Equivalency of Derivative Notation

• There are many ways that are used to represent
  the derivative.
• Suppose that y f(x), then the derivative can be
  represented in the following ways

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Example of Using Limits for a linear Equation
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Example of Using Limits for a Quadratic
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Example of Using Limits for a Cubic
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Differentiation Rules

• Constant Rule
• Power Rule
• Constant Times a Function Rule
• Sum and Difference Rule
• Product and Quotient Rule
• The Chain Rule
• Generalized Power Rule
• Exponential Rule

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Constant Rule

• The constant rule states that the derivative of a
  constant function is zero.

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Power Rule

• Suppose yf(x)xn, then the derivative of f(x) is
  the following
• f(x)nxn-1

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Constant Times a function Rule

• Suppose yf(x)ag(x), then the derivative of f(x)
  is the following
• f(x)ag(x)

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Sum and Difference Rule

• Suppose yK(x)f(x) ? g(x), then the derivative
  of K(x) is
• K(x) f(x) ? g(x)

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The Product Rule

• Suppose yK(x)f(x) g(x), then the derivative
  of K(x) is
• K(x) f(x) g(x) f (x) g(x)

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Quotient Rule

• Suppose yK(x)f(x) / g(x), then the derivative
  of K(x) is
• K(x) f(x) g(x) - f (x) g(x)/ (g(x))2

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Chain Rule

• Suppose yK(x)f(g(x)), then the derivative of
  K(x) is
• K(x) f(g(x))g(x)

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Generalized Power Rule

• Suppose yK(x)f(x)n, then the derivative of
  K(x) is
• K(x) nf(x)n-1 f(x)

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Exponential Rule

• Suppose y K(x) ef(x), then the derivative of
  K(x) is
• K(x)f(x)ef(x)

• K(x) ex K(x) ex
• K(x) e2x K(x) 2e2x

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The Second Derivative

• It is useful in calculus to look at the
  derivative of a function that has already been
  differentiating.
• This is known as taking the second derivative.
• The second derivative is usually represented by
  f(x).

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Second Derivative Cont.

• Suppose that y f(x).
• The second derivative can also be represented as
  the following

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Partial Derivative

• Suppose that y f(x1,x2).
• The partial derivative of y is defined as the
  following

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Example of Taking a Partial Derivative Using
Limits

• Suppose that y f(x1,x2) 5x124x1x23x22 .
• The partial derivative of y w.r.t. x1 is defined
  as the following

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Notes on Partial Derivatives

• The partial derivative has the same rules as the
  derivative.
• The key to working with partial derivatives is to
  keep in mind that you are holding all other
  variables constant except the one that you are
  changing.