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T ANGENT L INES

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P RE -C ALCULUS VS. C ALCULUS Static vs. dynamic Calculus deals with changes in properties For example: pre-calc math might deal with constant velocities whereas calculus can analyze how an objects velocity changes over time (accelerates) Velocity is the slope of dist/time plot. If straight can use pre-calc, but if the plot is a curve the tangent is constantly changing and you must use calculus

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T HE T ANGENT L INE If you have two points on a curve, P and Q, as Q moves closer and closer to P, the slope of the secant line between the two becomes closer and closer to being the slope of the tangent at point P The Moving Secant Line

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E XAMPLE Find the equation of the line tangent to the parabola y=x 2 at point P slope = m (x, x 2 ) (1, 1) P Q

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E XAMPLE ( CONT ) As Q gets closer and closer to P, the slope of secant PQ gets closer and closer to the slope of the tangent at P Put in values for x as it approaches 1

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E XAMPLE ( CONT ) xm 01.151.5.91.9.991.99.9991.999 As x gets closer to 1, the value of the slope approaches 2 Write equation for line at (1, 1)

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E XAMPLE 2 The flash unit of a camera stores charge on a capacitor and releases it suddenly to set off the flash. The data below describes the charge Q on the capacitor at a time, t, after the flash goes off. Estimate the slope of the tangent at t=0.04sec. t00.020.040.060.080.10 Q10081.8767.0354.8844.9336.76

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E XAMPLE 3 A ball is dropped from a tower. Its distance traveled in meters after t seconds is given by the equation y = 4.9t 2. Find its velocity at 5 seconds (velocity is slope of tangent at that time)

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LIMITS 2. LIMITS The idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating.

LIMITS 2. LIMITS The idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating.

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