Lesson 4-10b

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Lesson 4-10b

• Anti-Differentiation

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Quiz

• Estimate the area under the graph of f(x) x²
  1 from x -1 to x 2 . Improve your estimate
  by using six right endpoint rectangles.

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Objectives

• Understand the concept of an antiderivative
• Understand the geometry of the antiderivative and
  that of slope fields
• Work rectilinear motion problems with
  antiderivatives

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Vocabulary

• Antiderivative the opposite of the derivative,
  if f(x) F(x) then F(x) is the antiderivative
  of f(x)
• Integrand what is being taken the integral of
  F(x)
• Variable of integration what variable we are
  taking the integral with respect to
• Constant of integration a constant (derivative
  of which would be zero) that represents the
  family of functions that could have the same
  derivative

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Two other Anti-derivative Forms

• Form
• Form

Remember derivative of ex is just ex
Remember derivative of ln x is (1/x)
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Practice Problems
2ex x C
-5 lnx C
ex ?x3 - x C
x ln x C
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How to Find C

• In order to find the specific value of the
  constant of integration, we need to have an
  initial condition to evaluate the function at (to
  solve for C)!
• Example find
  such that F(1) 4.

F(x) x² 3x C
F(1) 4 (1)² 3(1) C 4
C 0 C (boring
answer!)
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Acceleration, Velocity, Position

• Remember the following equation from
  Physicss(t) s0 v0t ½ at² where s0 is
  the initial offset distance (when t0)
  v0 is the initial velocity (when t0) and
  a is the acceleration constant (due to gravity)
• We can solve problems given either s(t) or a(t)
  (and some initial conditions) basically solving
  the problem from either direction!

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Motion Problems

• Find the velocity function v(t) and position
  function s(t) corresponding to the acceleration
  function a(t) 4t 4 given v(0) 8 and s(0)
  12.

2t² 4t 8
v(0) 8 2(0)² 4(0) v0 8 v0
s(0) 12 ?(0)³ 2(0)² 8(0) s0
12 s0
s(t) ?t³ 2t² 8t 12
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Motion Problems

• A ball is dropped from a window hits the ground
  in 5 seconds. How high is the window (in feet)?

a(t) - 32 ft/s²
v(0) 0 -32(0) v0
(ball was dropped) 0 v0
s(5) 0 -16(5)² s0 s0 400 feet
window was 400 feet up
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Slope Fields

• A slope field is the slope of the tangent to F(x)
  (f(x) in an anti-differentiation problem) plotted
  at each value of x and y in field.
• Since the constant of integration is unknown, we
  get a family of curves.
• An initial condition allows us to plot the
  function F(x) based on the slope field.

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Slope Field Example
f(0) -2
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Summary Homework

• Summary
• Anti-differentiation is the reverse of the
  derivative
• It introduces the integral
• One of its main applications is area under the
  curve
• Homework
• pg 358-360 Day 1 1-3, 12, 13, 16
  Day 2 25, 26, 53, 61, 74