Test4_HW (group 4 and beyond) - Math 1225 Test 4 Packet...

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Calculus
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Chapter 5 / Exercise 5
Calculus
Stewart
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Math 1225Test 4 PacketGroup 1: AntiderivativesAntiderivativesA functionFis called anantiderviativeoffon an intervalIif, for allxI,F0(x) =f(x)Most General AntiderivativeIfFis an antiderivative offon an intervalI, then the most general antiderivative offonIisF(x) +CwhereCis an arbitrary constant.Some Antidifferentiation FormulasFunctionParticular AntiderivativeFunctionParticular Antiderivativecf(x)cF(x)f(x) +g(x)F(x) +G(x)sec2(x)tan(x)sec(x) tan(x)sec(x)xn(n6=-1)xn+1n+ 111-x2arcsin(x)1xln(|x|)11 +x2arctan(x)exexcos(x)sin(x)sin(x)-cos(x)Differential EquationAn equation involving the derivatives (possibly higher order) of a function is known as adifferential equation. In general, the solution to a differential equation will contain arbitraryconstants, but you may be given information to uniquely determine a solution.Drayton Munster. Compiled: November 18, 2015Page 1 of 18
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Calculus
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Chapter 5 / Exercise 5
Calculus
Stewart
Expert Verified
Math 1225Test 4 Packet1. Find the most general antiderivative of the following:
2. Given a plot of the velocity below, sketch a plot of the position function given that theparticle starts at the origin.
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Math 1225Test 4 Packet3. Find the functionfthat satisfies the following conditions (if more than one such functionexists, give the most general form).(a)f00(x) = 0, f(0) = 1;f(1) =-1(b)f0(x) = 2-2xand the liney=xis tangent to the graph off.(c)f0(x) = sin(x) +e2, f(0) = 0.(d)f000(x) = 60x2, f0(0) = 0.Drayton Munster. Compiled: November 18, 2015Page 3 of 18
Math 1225Test 4 PacketGroup 2: Areas and DistancesArea under a CurveTheareaAof the regionSthat lies under the graph of a continuous functionfis the limit ofthe sum of the areas of the approximating rectangles:A= limn→∞Rn=f(x1x+f(x2x+. . .+f(xnxor, in sigma notation,A= limn→∞Rn= limn→∞nXi=1f(xix.

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