Antiderivatives and Indefinite Integration

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Antiderivatives and Indefinite Integration

• Lesson 5.1

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Reversing Differentiation

• An antiderivative of function f is
• a function F
• which satisfies F f
• Consider the following
• We note that two antiderivatives of the same
  function differ by a constant

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Reversing Differentiation

• General antiderivativesf(x) 6x2
  F(x) 2x3 C
• because F(x) 6x2
• k(x) sec2(x) K(x) tan(x) C
• because K(x) k(x)

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Differential Equation

• A differential equation in x and y involves
• x, y, and derivatives of y
• Examples
• Solution find a function whose derivative is
  the differential given

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Differential Equation

• When
• Then one such function is
• The general solution is

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Notation for Antiderivatives

• We are starting with
• Change to differential form
• Then the notation for antiderivatives is

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Basic Integration Rules

• Note the inverse nature of integration and
  differentiation
• Note basic rules, pg 286

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Practice

• Try these

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Finding a Particular Solution

• Given
• Find the specific equation from the family of
  antiderivatives, whichcontains the point (3,2)
• Hint find the general antiderivative, use the
  given point to find the value for C

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Assignment A

• Lesson 5.1 A
• Page 291
• Exercises 1 55 odd

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Slope Fields

• Slope of a function f(x)
• at a point a
• given by f (a)
• Suppose we know f (x)
• substitute different values for a
• draw short slope lines for successive values of y
• Example

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Slope Fields

• For a large portion of the graph, when
• We can trace the line for a specific F(x)
• specifically when the C -3

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Finding an Antiderivative Using a Slope Field

• Given
• We can trace the version of the original F(x)
  which goes through the origin.

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Vertical Motion

• Consider the fact that the acceleration due to
  gravity a(t) -32 fps2
• Then v(t) -32t v0
• Also s(t) -16t2 v0t s0
• A balloon, rising vertically with velocity 8
  releases a sandbag at the instant it is 64 feet
  above the ground
• How long until the sandbag hits the ground
• What is its velocity when this happens?

Why?
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Rectilinear Motion

• A particle, initially at rest, moves along the
  x-axis at acceleration
• At time t 0, its position is x 3
• Find the velocity and position functions for the
  particle
• Find all values of t for which the particle is at
  rest

Note Spreadsheet Example
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Assignment B

• Lesson 5.1 B
• Page 292
• Exercises 57 93, EOO