What is Calculus Three Basic Concepts

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What is Calculus?Three Basic Concepts

• Lesson 2.1

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Limit

• A mathematical tool
• It studies the tendency of a function
• as its variable approaches some value

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Derivative

• Defined as a limit
• Computes variety of values
• rates of change
• slopes of tangent lines of curves
• Known as "differential calculus"

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Integral

• Found by taking a special limit
• Often a limit of a sum of terms
• Computes things such as
• area
• volume
• arc length
• Known as "integral calculus"

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Limit

• Consider the sequence
• As n gets very large, what value does the
  fraction approximate?
• We say the limit of the fraction approaches 1 as
  n gets very large

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The Derivative

• Consider a function   f(x) y
• We seek the slope of the tangent line at point
  x a P (a,f(a))
• An approximation of the tangent is the secant PQ
• Slope of the secant is

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The Derivative

• Now allow h to get very small
• The secant becomes a very close approximation of
  the tangent
• Then the slope of the tangent line at x a is

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The Integral

• Consider the function f(x) y
• We seek the area under the curve between points
  x0 and x3
• An approximation is the sum of the areas of the
  three boxes

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The Integral

• We can get a closer approximation by increasing
  the number of partitions at x0, x1, x2, ... xn
  where n is very large
• The limit of the sum as n -gt infinity is the
  actual area under the curve

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Mathematical Modeling

• Steps
• Make assumptions about the real world
• View the real world problem
• variables
• formulas
• relationships
• This abstraction becomes the model
• Simplify the math and derive mathematical facts
  from the model
• Use the resulting facts to make predictions about
  the real world
• compare predictions to real world events
• fine tune the model

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Example of Mathematical Modeling

• Gather data
• Plot on graph

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Example of Mathematical Modeling

• Observe and compare to known functions
• Which is it??
• linear
• quadratic
• exponential
• logarithmic
• trigonometric

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Assignment

• Lesson 2.1
• Page 81
• Exercises 3, 5, 9, 13, 15, 19, 21, 29, 33, 37,
  39