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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, what’s 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? –NO…that 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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150 Distance In Miles Time In Hours 3 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? 12 150 miles in 3 hours = AVE RATE of 50 mph

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Let’s find the slope of the tangent line to y = x 2 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 what’s called a secant line through (2,4) and (2+h,(2+h) 2 ) (2+h,(2+h) 2 ) (2,4) h (2+h) 2 - 4 The slope of that line =

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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 h h 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…. (2+h,(2+h) 2 ) (2,4) h (2+h) 2 - 4 The slope of tangent line =

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Lets Evaluate The Limit == = == = 4 The Slope of the Tangent Line at (2,4) = 4

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(x+h,f(x+h)) (x,f(x)) (x+h,f(x+h)) · y = 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)=x 2 · · Start with the slope of a secant

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Now lets make it a tangent ·

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The formula that will calculate the slope of a tangent line for at any point (x,y) is 2x For example: At (3,9) the slope of the tangent is … 6 At (-4,16) the slope of the tangent is … -8 We say that the derivative of is 2x

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Sample Problems Click Here To View Text Book Exercises Click here to go to the 2nd example Click here to go to the limit animation Click here to go to the 1st example

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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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