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AP Calculus Mr. Manker Difference quotient definition. Finds derivative at a point. −() lim − → ℎ ℎ Write the equation of the tangent line to f(x) at x = 2 if = 3 2 + 7 Write the equation of the tangent line to f(x) at x = 2 if = 3 2 + 7 Use slope-intercept form: y – y1 = m(x – x1) f(2) = 19, so (2, 19) is a point on graph Use derivative to find slope of tan. at x = 2. f’(x) = 6x 6(2) = 12 y – 19 = 12(x – 2) Is = 7 − 4 + 7 increasing or decreasing at x = 1? 2 Is = 7 3 − 4 + 7 increasing or decreasing at x = 1? Rate of change, so find derivative at x = 1: 2 f‘(x) = 21 – 4 f’(1) = 17 The derivative is positive, so the graph is increasing at x = 1. Find f’(2) if f(x) = ln x. We don’t know how to find the derivative of this!! nDeriv(ln x, x, 2) ≈ .500 To graph equation of derivative, replace x value of 2 with variable: nDeriv(ln x, x, x) (Good way to check your derivatives!) = 3 2 − 4 + 7 +ℎ −() lim ℎ ℎ→0 “forward difference quotient” + ℎ − () lim ℎ→0 ℎ 3( + ℎ)2 −4 + ℎ + 7 − (3 2 − 4 + 7) = lim ℎ→0 ℎ 2 2 3 +6ℎ+3ℎ −4−4ℎ+7−3 2 +4−7 = lim ℎ→0 ℎ = lim 6 + 3ℎ − 4 = 6 − 4 ℎ→0 = 5 3 − 7 + 1 Derivative of displacement is velocity: = 15 2 − 7 Derivative of velocity is acceleration: a(t) = 30t See other powerpoint, quiz, and example videos