05-03-072 The Fundamental Theorem of Calculus
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Calculus I, Section 5.3, #72The Fundamental Theorem of Calculus
Thesine integral function
Si(x) =?
x0sin(t)
tdtis important in electrical engineering. [The integrandf(t) = (sin(t))/tis not defined whent= 0, but we know
its limit is1whent→0. So we definef(0) = 1and this makesfa continuous function everywhere.]1 (a)Draw the graph ofSi.Using WolframAlpha, (b)At what values ofxdoes this function have local maximum values?The function Si(x) has local maximums at those values ofxfor which Si?(x) changes from positive to negative.We compute the derivative of Si(x)
Si ?(x) =d dx? ?x0sin(t)tdt?
sin(x) x and solve the equation 0 = sin(x) x and ifx?= 0, then0 = sin(x)
So the critical numbers for Si(x) arex=nπ, wherenis an integer.Ifx >0, the denominator of Si?(x) =sin(x)
xis positive, so the derivative changes from positive to negative wherever sin(x) changes from positive to negative. From our knowledge of the unitcircle, this happens atx=π,3π,5π,.... continued...1Stewart,Calculus, Early Transcendentals, p. 401, #72.
Calculus IThe Fundamental Theorem of Calculus
Ifx <0, the denominator of Si?(x) =sin(x)xis negative, so the derivative changes from positive to negative wherever sin(x) changes from negative to positive. From our knowledge of the unitcircle, this happens atx=-2π,-4π,-6π,.... Thus, the local maximums for Si(x) occur atx=...,-6π,-4π,-2π, π,3π,5π,.... (c)Find the coordinates of the first inflection point to the rightof the origin. Si ??(x) =x·cos(x)-sin(x)·1 x2 xcos(x)-sin(x) x2 so to find the inflection points, we need to solve the equation xcos(x)-sin(x) x2= 0, and determine if the sign offuncSi??xchanges.Using WolframAlpha,
So the first inflection point to the right of the origin isx≈4.4934 if the sign of the second derivative
changes. Using WolframAlpha or the TI-84, we compute Si ??(4.48)≈ -0.0131 and Si??(4.50)≈0.0064,so the sign changes and there is an inflection point atx≈4.4934. The coordinates of that point are
(4.4934,Si(4.4934)) or (4.4934,1.6556). (d)Does this function have horizontal asymptotes?We want to compute lim x→∞Si(x) and limx→-∞Si(x)