Introduction to Derivatives Tangents

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Introduction to DerivativesTangents

• Section 3.1

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Tangent Definition

• From geometry
• a line in the plane of a circle
• intersects in exactly one point
• We wish to enlarge on the idea to include
  tangency to any function, f(x)

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Slope of Line Tangent to a Curve

• Approximated by secants
• two points of intersection
• Let second point get closerand closer to
  desiredpoint of tangency

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Slope of Line Tangent to a Curve

• Recall the concept of a limit from previous
  chapter
• Use the limit in this context

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The Slope Is a Limit

• Consider f(x) x3 Find the tangent at x0
  2
• Now finish

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Calculator Capabilities

• Able to draw tangent line
• Steps
• Specify function on Y screen
• F5-math, A-tangent
• Specify an x (where to place tangent line)

• Note results

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Difference Function

• Creating a difference function on your calculator
• store the desired function in f(x)x3 -gt f(x)
• Then specify the difference function(f(x dx)
  f(x))/dx -gt difq(x,dx)
• Call the functiondifq(2, .001)

• Use some small value for dx
• Result is close to actual slope

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Definition of Derivative

• The derivative is the formula which gives the
  slope of the tangent line at any point x for f(x)
• Note the limit must exist
• no hole
• no jump
• no pole

A derivative is a limit !
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Finding the Derivative

• We will (for now) manipulate the difference
  quotient algebraicly
• View end result of the limit
• Note possible use of calculatorlimit ((f(x dx)
  f(x)) /dx, dx, 0)

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Related Line the Normal

• The line perpendicular to the function at a point
• called the normal
• Find the slope of the function
• Normal will have slope of negative reciprocal to
  tangent
• Use y m(x h) k

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Using the Derivative

• Consider that you are given the graph of the
  derivative
• What might theslope of the original function
  look like?
• Consider
• what do x-intercepts show?
• what do max and mins show?
• f (x) lt0 or f (x) gt 0 means what?

f (x)
To actually find f(x), we need a specific point
it contains
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Derivative Notation

• For the function y f(x)
• Derivative may be expressed as

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Assignment

• Lesson 3.1
• Page 139
• Exercises 3, 5, 7, 9, 10, 11, 15, 23, 25, 27,
  31, 33, 35, 45, 47