Welcome To Calculus (Do not be afraid, for I am with you)

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Welcome To Calculus(Do not be afraid, for I am
with you)

• The slope of a tangent line
• By Mr. Kretz

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Background Questions To Ponder

• If you drive 150 miles in 3 hours, whats your
  average speed?
• 50 mph is your AVERAGE RATE OF CHANGE
• What is the generic math term for AVERAGE RATE OF
  CHANGE?
• SLOPE of a line AVERAGE RATE OF CHANGE
• Did you really drive 50 mph constantly on your
  journey?
• NOthat was your AVERAGE RATE OF CHANGE
• Fill in the blank When driving, I looked at my
  speedometer and it read 65 mph. At this instant,
  65 mph was my _______________________ rate of
  change.

INSTANTANEOUS
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• Tell me something about your INSTANTANEOUS RATE
  OF CHANGE 1/2 hour into the trip.
• Pretty Fast (Got pulled over by a cop about 15
  minutes later)
• Tell me something about your INSTANTANEOUS RATE
  OF CHANGE 2 hours into the trip.
• Went backwards to hit that skunk again?

150
150 miles in 3 hours AVE RATE of 50 mph
Distance In Miles
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2
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Time In Hours
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Lets find the slope of the tangent line to y
x2 when x 2 (Instantaneous Rate of Change)
The Slope Of The Red Line
Click here to see a visual animation of zooming
in on a tangent line
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We begin by setting up whats called a secant
line through (2,4) and (2h,(2h)2)
(2h)2 - 4
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As h gets smaller, the secant line approaches the
tangent line, and the average rate of change
becomes the instantaneous rate of change
As h gets closer and closer to zero, we approach
our tangent line
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When h approaches zero, our slope equation
becomes.
The slope of tangent line
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Lets Evaluate The Limit
The Slope of the Tangent Line at (2,4) 4
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(xh,f(xh))
(xh,f(xh))
y f(x)
(xh,f(xh))
(xh,f(xh))

(xh,f(xh))
(x,f(x))
CLICK TO CONTINUE
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Find the slope of the tangent line for any point
(x,f(x)) for f(x)x2
Start with the slope of a secant
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Now lets make it a tangent

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For example
At (3,9) the slope of the tangent is
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At (-4,16) the slope of the tangent is
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Sample Problems
Click Here To View Text Book Exercises
Click here to go to the 1st example
Click here to go to the 2nd example
Click here to go to the limit animation
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SUMMARY
To find the equation of the tangent line, simply
find f (a), that is your slope. Now use your
point of tangency (a,f(a) and your slope, m f
(a) in the point slope form of a linear
equation.
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Whew!!!!
That Was Easy!