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Harold’s AP Calculus BCCheat Sheet29 November 2022RectangularPolarParametricPointfx= yx, ya, b?(r , θ) or r ∠ θPoint (a,b) in Rectangular:xt=ayt=b<a,b>t=3rd variable, usually time,with 1 degree of freedom (df)Polar Rect.x=rcosθy=rsinθtanθ=yxRect. Polarr2= x2+y2 r= ± x2+ y2θ=tan-1yxLineSlope-Intercept Form:y=mx+bPoint-Slope Form:y- y0=m (x-x0)General Form:Ax+By+C=0Calculus Form:fx=f'a x+f(0)29908516192500 <x,y> = <x0,y0>+ t<a,b><x,y> = <x0+at,y0+bt>where<a,b> = <x2-x1, y2-y1>x(t)= x0+ tay(t)= y0+ tbm=?y?x= y2-y1x2-x1=baPlanenxx-x0+nyy-y0+ nzz-z0=0Vector Form:n ? r- r0=0r= r0+ sv+twwhere:v and w are given vectors defining the planer0 is the vector of a fixed point on the planeConicsGeneral Equation for All Conics:Ax2+Bxy+Cy2+Dx+Ey+F=0whereLine: A=B=C=0Circle: A=C and B=0Ellipse: AC>0 or B2-4AC<0Parabola: AC=0 or B2-4AC=0Hyperbola: AC<0 or B2-4AC>0Note: If A+C=0, square hyperbolaRotation:If B ≠ 0, then rotate coordinate system:cot2θ=A-CBx=x'cosθ-y'sinθy=y'cosθ+x'sinθNew = (x’, y’), Old = (x, y)rotates through angle θ from x-axisGeneral Equation for All Conics:r=p1-ecosθ where p= a (1-e2)2d a (e2-1)for 0≤e<1e=1e>1p = semi-latus rectumor the line segment running from the focus to the curve in a direction parallel to the directrixEccentricity:Circle e=0Ellipse 0<e<1Parabola e=1Hyperbola e>1Circle(x-h)2+(y-k)2=r2Center: h,kVertices: NAFocus: h,kCentered at Origin:r = a (constant)θ = θ 0, 2π or [0, 360°]Centered at (r0,?):r2+r02-2rr0cosθ-?=R2Hint: Law of Cosinesorr=r0cosθ-?+a2-r02 sin2(θ-?)x(t)=rcos(t)+hy(t)=rsin(t)+ktmin, tmax=[0, 2π]Center: h,kFocus: h,kEllipse(x-h)2a2+(y-k)2b2=1Center: h,kVertices: h±a, k and h, k±bFoci: h±c, kFocus length, c, from center:c2= a2-b2Ellipse:r=a (1-e2)1+ecosθ for 0<e<1where e= ca= a2- b2arelative to center (h, k)Interesting Note:The sum of the distances from each focus to a point on the curve is constant.d1+d2=kxt=acos(t) +hy(t)=bsin(t)+ ktmin, tmax=[0, 2π]Center: h,kRotated Ellipse:xt=acostcosθ-bsintsinθ+hyt=acostsinθ+bsintcosθ+kθ = the angle between the x-axis and the major axis of the ellipseHyperbolax-h2a2-y-k2b2=1Center: h,kVertices: h±a, kFoci: h±c, kFocus length, c, from center:c2= a2+b2Vertical Axis of Symmetry:r=a (e2- 1)1+ecosθ for e>1Eccentricity: e>1where e= ca= a2+ b2a=secθ>1relative to center (h, k)p = semi-latus rectum or the line segment running from the focus to the curve in the directions θ=± π2Interesting Note:The difference between the distances from each focus to a point on the curve is constant.d1-d2=kLeft-Right Opening Hyperbola:x(t)=asec(t)+hy(t)=btan(t)+ktmin, tmax=[-c, c]Vertex: (h, k)Up-Down Opening Hyperbola:x(t)=atan(t)+hy(t)=bsec(t)+ktmin, tmax=[-c, c]Vertex: (h, k)General Form:x(t)=At2+ Bt+Cy(t)=Dt2+ Et+Fwhere A and D have different signsParabolaVertical Axis of Symmetry:x2=4 pyx-h2=4p(y-k)Vertex: h,kFocus: h,k+pDirectrix: y=k-pHorizontal Axis of Symmetry:y2=4 pxy-k2=4p(x-h)Vertex: h,kFocus: h+p,kDirectrix: x=h-pVertical Axis of Symmetry:r=2d1+ecosθEccentricity: e=1and d=2pTrigonometric Form: y=x2rsinθ=r2 cos2θr=sinθcos2θ=tanθsecθInteresting Note:The distances from a point on the curve to the focus is the same as to the directrix.Vertical Axis of Symmetry:xt= 2pt+ hyt=pt2+k (opens upwards)yt=-pt2-k (opens downwards)tmin, tmax=[-c, c]Vertex: h,kHorizontal Axis of Symmetry:yt= 2pt+ kxt=pt2+h (opens to the right)xt=-pt2-h (opens to the left)tmin, tmax=[-c, c]Vertex: h,kProjectile Motion:xt=x0+vxt+12axt2yt=y0+vy0t-16t2 feetyt=y0+vy0t-4.9t2 metersvx=vcosθvy=vsinθGeneral Form:x=At2+ Bt+Cy=Dt2+ Et+Fwhere A and D have the same sign1st Derivativef'x=limh→0fx+h-f(x)hf'c=limx→cfx-f(c)x-cf'x=dydx=y'=Dxdydx=dydθdxdθ=drdθsinθ+ rcosθdrdθcosθ-rsinθHint: Use Product Rule fory=rsinθx=r cos θdydx=dydtdxdt , provideddxdt≠02nd Derivativef''x=ddxdydx= d2ydx2=y''=Dxxd2ydx2=ddxdydx=ddθdydxdxdθd2ydx2=ddxdydx=ddtdydxdxdt=ddtdydtdxdtdxdtIntegralFundamental Theorem of Calculus:Fx=abfx dx =Fb-F(a)Riemann Sum:S=i-1nf(yi)(xi-xi-1)Left Sum:S= 1nfa+fa+1n+fa+2n+…+f(b-1n)Middle Sum:S= 1nfa+12n+fa+32n+…+f(b-12n)Right Sum:S= 1nfa+1n+fa+2n+…+f(b)Inverse Functionsff-1 (x)=f-1 f(x)=xInverse Function Theorem:f-1f'a= 1f'aif y=sin θif y=cosθif y=tanθif y= csc θif y=secθif y=cotθthen θ=sin-1ythen θ=cos-1ythen θ=tan-1ythen θ=csc-1ythen θ=sec-1ythen θ=cot-1yor θ=arcsinyor θ=arccos yor θ=arctanyor θ=arccscyor θ=arcsecyor θ=arccotyArc LengthL= ab1+[f'x]2 dxProof:?s= x-x02+ y- y02?s = (?x)2+ (?y)2ds= dx2+dy2ds= dx2+dy2dx2dx2ds= dx2+dydx2dx2ds= dx2 1+dydx2ds= 1+dydx2dxL= dsL= r2+ drdθ2 dθCircle:L=s=rθProof:L=fraction of circumference?π?(diameter)L=θ2ππ (2r)=rθL= αβdxdt2+ dydt2 dtL= αβdxdt2+ dydt2+ dzdt2 dtProof:ds= dx2+dy2ds= dx2dt2dt2+dy2dt2dt2ds= dxdt2+dydt2dt2ds= dxdt2+dydt2 dtL= dsPerimeterSquare: P = 4sRectangle: P = 2l + 2wTriangle: P = a + b + cCircle: C = πd = 2πrEllipse: C≈π(a+b)Ellipse: C≈2πa2+b22C≈π 3a+b-(3a+b)(a+3b)C≈π a+b1+3h10+4-3hEllipse: C=4a0π21-k2sin2θ dθh=a-b2a+b2 & k2=1-b2a2AreaSquare: A = s?Rectangle: A = lwRhombus: A = ? abParallelogram: A = BhTrapezoid: A=B1+ B22 hKite: A = d1 d22Triangle: A = ? BhTriangle: A = ? ab sin(C)Triangle using Heron’s Formula:A= ss-as-bs-cwhere s=a+b+c2Equilateral Triangle: A = ?3s2Frustum: A=13B1+B22hCircle: A = πr?Circular Sector: A = ? r?θEllipse: A = πabA= αβ12[f(θ)]2dθwhere r=f(θ)Proof:Area of a sector:A=s dr= r ?θ dr=12r2 ?θwhere arc length s=r ?θA=αβgt f't dtwhere ft=x and gt=yorx(t) = f(t) and y(t) = g(t)Simplified:A=αβyt dx(t)dt dtProof:abfx dxy = f(x) = g(t)dx =dftdtdt=f’(t) dtLateral Surface AreaCylinder: SA = 2πrhCone: SA = πrlSA= 2πabfx 1+[f'x]2 dxFor rotation about the x-axis:SA= 2πy dsFor rotation about the y-axis:SA= 2πx dsds= r2+drdθ2 dθr=fθ, α≤ θ≤βFor rotation about the x-axis:SA= 2πy dsFor rotation about the y-axis:SA= 2πx dsds= dxdt2+dydt2 dtif x=ft, y=gt, α≤ t≤βTotal Surface AreaCube: SA = 6s?Rectangular Box: SA = 2lw + 2wh + 2hlRegular Tetrahedron: SA = 2bhCylinder: SA = 2πr (r + h)Cone: SA = πr? + πrl = πr (r + l)Sphere: SA = 4πr?Ellipsoid: SA ≈4πapbp+apcp+bpcp31p Where p ≈1.6075, Relative Error≤1.061% (Knud Thomsen’s Formula)Surface of RevolutionFor revolution about the x-axis:A=2π abf(x)1+dydx2 dxFor revolution about the y-axis:A=2π abx1+dxdy2 dyFor revolution about the x-axis:A=2πrαβcosθ r2+ drdθ2 dθFor revolution about the y-axis:A=2πrαβsinθr2+drdθ2dθFor revolution about the x-axis:A=2πabyt dxdt2+ dydt2 dtFor revolution about the y-axis:A=2πabxt dxdt2+ dydt2 dtVolumeCube: V = s?Rectangular Prism: V = lwhCylinder: V = πr?hTriangular Prism: V = BhTetrahedron: V = ? BhPyramid: V = ? Bh = ? lwhCone: V = ? Bh = ? πr?hSphere: V=43πr3Ellipsoid: V = 43 πabcVolume of RevolutionDisk MethodV=ab(area of circle) dthicknessRotation about the x-axis:V= abπ fx2dxRotation about the y-axis:V= cdπx2dyWasher MethodRotation about the x-axis:V= abπ {[f(x)]2-[gx]2} dxV=VOuter Disk-VInner DiskShell MethodV=abcircumference hight dxRotation about the y-axis:V=ab2πx fx dxRotation about the x-axis:V=cd2πy gy dy ................
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