Tangent Lines, Normal Lines, and Rectilinear Motion

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Tangent Lines, Normal Lines, and Rectilinear
Motion

• Conner Moon
• Evan Haight
• 2nd period
• Mrs. Autrey

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Tangent Lines

• The definition of a tangent line is a line which
  intersects a curve at a point where the slope of
  both the curve and the line are equal. In other
  words, it is a line that touches a curve at one
  point without crossing over.

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Three Things Needed to Find the equation of a
tangent line

1. Derivative
2. Derivative at the point
3. Y-value at the point

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Derivative

• The first step of finding the equation of a
  tangent line is to take the derivative of the
  original equation.

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Derivative at the point

• The second step is to find the derivative at the
  point. For this example we will give the point
  x2.

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Cont
After finding the derivative at the point, your
answer will be the slope of the tangent line.
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Y-value at the point

• Now go back to the original equation and plug in
  your x-value given earlier, x2, to find the
  y-value.

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Now, Slope Intercept Formula

• After finding the slope of the tangent line, and
  your y-value, we plug in the those points to find
  the equation of the tangent line, shown below.

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Example 1
Here we will find the equation of the tangent
line at the point x-2.
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Walk Through
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Solution
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Try Me!!!!

• Find the equation of the line tangent at the
  point x-1

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How did you do?
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Normal Line

• The definition of as normal line is a line that
  is perpendicular to the tangent line at the point
  of tangency. In other words it is a line that
  intersects the tangent line with a slope that is
  the negative reciprocal of the slope of said
  tangent line.

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Things Needed to Find Equation of Normal Line

• Original Equation of the Tangent line
• Slope of Tangent line
• Negative Reciprocal

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Negative Reciprocal

• Ok this sounds a lot harder than it actually is.
  So if the slope of the tangent line is y4, then
  the slope of the normal line, the negative
  reciprocal of the slope of the tangent line, is
  y-1/4.

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Example 1

• Here we will find the equation of the normal line
  at the point x-2

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Walk Through
This is the part where we take the negative
reciprocal of the tangent line, sooo, the slope
of this normal line is 1/8.
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Solution
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Example 2

• Here we will find the equation of the normal line
  at the point y-1

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Try Me!!!!
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How did you do?
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Rectilinear Motion

• In this case when we talk about rectilinear
  motion lets think of it as the motion of a
  particle, and the motion of this particle is
  illustrated by a given expression. Using this
  expression and derivatives we will be able to
  calculate the position, velocity, and
  acceleration of said particle.

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Stages of Rectilinear Motion

• X(t)position
• V(t) / X(t)velocity
• A(t) / X(t)acceleration

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Explanation of a Sign Line

• Staying with the particle idea, a sign line will
  visually show us the positive or negative value
  of a group of numbers, using this information we
  can determine if the particle is moving left or
  right, has a positive or negative velocity, or if
  the acceleration of the particle is positive or
  negative.

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Example 1

• Find when the particle, expressed by the equation
  , is moving to the
  right.

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Walkthrough
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Solution

• As the sign line shows the particle is moving
  right from .

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Try Me!!!!

• Find when the particle, expressed by the equation
  , is moving to the
  left.

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How did you do??
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Now you know how to do tangent and normal lines
and rectilinear motion.

• The end.