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The concept of the slope of a curve, specifically the slope of the line tangent to the curve and the instantaneous rate of change. It provides examples of how to find the slope of the line tangent to a function using the limit definition and shows how the slope of the curve is equal to the instantaneous rate of change at a given point.
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If we are looking at the graph of y = f (x), then slope of secant line from x = a to x = b = riserun = โ โyx
= f^ (b b) โโ^ fa^ (a)
If we are looking at an object whose position is given by f (t), then average r.o.c. from t = a to t = b = change in positiontime = f^ (t tfinal)^ โ^ f^ (tinitial) Final โ^ tinitial = f^ (b b)^ โโ^ fa^ (a).
I (^) Slope of a Curve (i.e, slope of the line tangent to the curve)
I (^) Instantaneous Rate of Change
mtan = (^) hlimโ 0 f^ (0 +^ h h)^ โ^ f^ (0)
= (^) hlimโ 0
[ (h)^2 โ 3(h)
] โ
[ 02 โ 3(0)
]
h = (^) hlimโ 0 h
(^2) โ 3 h h = (^) hlimโ 0 h โ 3 = โ 3
mtan = (^) hlimโ 0 f^ (x^ +^ h h)^ โ^ f^ (x)
= (^) hlimโ 0
[ (x + h)^2 โ 3(x + h)
] โ
[ x^2 โ 3(x)
]
h = (^) hlimโ 0
(x (^2) + 2xh + h (^2) โ 3 x โ 3 h) (^) โ (x (^2) โ 3 x) h = (^) hlimโ 02 xh^ +^ h
(^2) โ 3 h h = (^) hlimโ 0 2 x + h โ 3 = 2 x โ 3