Sketch-the-curve-y-x-3-a-Find-the-equation-of-the-tangent-to-the-curve-at-A-1-1-b-Find-the-coordinates-of-point-B-where-the-tangent-meets-the-curve-again-c-Calculate-the-area-between-the

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Question Number 208052 by necx122 last updated on 03/Jun/24

$$Sketchthecurvey=x^3.$$$$\left(a\right)Findtheequationofthetangent$$$$tothecurveatA(1,1).$$$$\left(b\right)FindthecoordinatesofpointB,$$$$wherethetangentmeetsthecurveagain.$$$$\left(c\right)Calculatetheareabetweenthe$$$$tangentBandthearcABofthecurve.$$

Commented by Frix last updated on 03/Jun/24

$$\mathrm{Tangent}t:y=ax+b$$$$f:y=x^3\Rightarrow f^{\prime }:y=3x^2$$$$P\in f:P=\left(\begin{array}{c}p\\ p^3\end{array}\right)$$$$P\in t:p^3=ap+b\mathrm{but}a=3p^2\Rightarrow b=-2p^3\Rightarrow$$$$t:y=3p^{2}x-2p^3$$$$$$$$f\left(x\right)=x^3$$$$t\left(x\right)=3p^{2}x-2p^3$$$$d\left(x\right)=f\left(x\right)-t\left(x\right)=x^3-3p^{2}x+2p^3$$$$d\left(x\right)={(x-p)}^2(x+2p)$$$$\mathrm{This}\mathrm{has}\mathrm{a}\mathrm{double}\mathrm{zero}\mathrm{at}x=p\mathrm{and}\mathrm{a}\mathrm{single}$$$$\mathrm{zero}\mathrm{at}x=-2p.\mathrm{Its}\mathrm{extreme}\mathrm{is}\mathrm{at}x=-p$$$$\mathrm{with}d(-p)=4p^3\Rightarrow \mathrm{for}p<0\mathrm{we}\mathrm{have}\mathrm{a}$$$$\mathrm{local}\min ,\mathrm{for}p>0\mathrm{a}\mathrm{local}\max \Rightarrow$$$$4p^3⩽d\left(x\right)⩽0;p<0\wedge p⩽x⩽-2p$$$$\Rightarrow \mathrm{Area}\mathrm{is}-\underset{p}{\stackrel{-2p}{\int }}d\left(x\right)dx=\underset{-2p}{\stackrel{p}{\int }}d\left(x\right)dx$$$$0⩽d\left(x\right)⩽4p^3;p>0\wedge -2p⩽x⩽p$$$$\Rightarrow A\mathrm{rea}\mathrm{is}\underset{-2p}{\stackrel{p}{\int }}d\left(x\right)dx$$$$\Rightarrow \mathrm{Area}\mathrm{for}p\ne 0\mathrm{is}\mathrm{always}\underset{-2p}{\stackrel{p}{\int }}d\left(x\right)dx$$

Commented by Frix last updated on 03/Jun/24

$$y=x^3$$$$y^{\prime }=3x^2$$$$\mathrm{Tangent}\mathrm{at}x=p:y=3p^{2}x-2p^3$$$$\mathrm{Intersection}$$$$x^3-(3p^{2}x-2p^3)=0$$$${(x-p)}^2(x+2p)=0$$$$x_1=p\left(\mathrm{obvious}\right)$$$$x_2=-2p$$$$\mathrm{The}\mathrm{area}\mathrm{is}$$$$\underset{-2p}{\stackrel{p}{\int }}(x^3-(3p^{2}x-2p^3))dx=\frac{27p^4}{4}$$

Commented by necx122 last updated on 03/Jun/24

$$Thankyousirfrix.However,I^{\prime }m$$$$botheredabouthowyoucameabout$$$$they-interceptforthetangentas-2p^3.$$$$Also,insketchingthecurveand$$$$calculatingtheareaboundedbythe$$$$arentwesupposedtocalculateforall$$$$possibleregionsbothaboveandbelow$$$$thex-axiswhichmayindicateadifferent$$$$valuefortheareaboundedbythecurve$$$$andtheline.$$$$Thankyouforyourhelp.Ialways$$$$appreciate.$$$$$$$$$$$$$$$$$$$$$$

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