Single Variable Calculus Concepts

1
Single Variable CalculusConcepts Contexts

• Section 2.1
• The Tangent and Velocity Problems

2
The Tangent Problem

• The word tangent comes from the Latin word
  meaning touching.
• This implies the tangent should touch not cross a
  curve.
• The tangent should also have the same direction
  as the curve.
• Euclid said that a tangent is a line that
  intersects a circle once and only once.
• This works well for a circle but not other curves.

3
Find the Equation of a Tangent

• Find the equation of the tangent line to the
  parabola
• At the point P(1,1).
• We need to find the slope of the tangent.
• TEC Exploration.

4
Find Tangent Given Data

• Estimate the slope of the tangent line at the
  point where t 0.04

5
The Velocity Problem

• Consider the speedometer of your car. How do we
  define instantaneous velocity?
• Lets consider a falling object.
• Galileo discovered that the distance fallen by
  any free falling body is proportional to the
  square of the time it has been falling.
• This approach neglects wind resistance.

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Find The Velocity

• If the distance fallen after t seconds is
  denoted by s(t) and measured in meters, than
  Galileos law is expressed by the equation
• Find the velocity of the falling object after 5
  seconds.

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Instantaneous Velocity

• The instantaneous velocity when t5 is defined to
  be the limiting value of these average
  velocities.
• If we consider the two points
• Then the slope of the secant line PQ is

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• This is the same as the average velocity over the
  interval a, ah
• Therefore the velocity at time ta (the limit of
  these average velocities as h approaches 0) must
  be equal to the slope of the tangent line at P.

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Find the Slope of the Tangent Line

• Estimate the slope of the tangent line to the
  curve at the given x value.

10
We Need Limits

• To find slopes of tangent lines or instantaneous
  velocities we must be able to take limits .

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Homework

• Problems
• 1, 4, 5, 7, 8, 9