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Power series centered at x = 0 x = 0 | https://openstax.org/books/calculus-volume-2/pages/6-key-equations |
â n = 0 â c n x n = c 0 + c 1 x + c 2 x 2 + ⯠â n = 0 â c n x n = c 0 + c 1 x + c 2 x 2 + ⯠| https://openstax.org/books/calculus-volume-2/pages/6-key-equations |
Power series centered at x = a x = a | https://openstax.org/books/calculus-volume-2/pages/6-key-equations |
â n = 0 â c n ( x â a ) n = c 0 + c 1 ( x â a ) + c 2 ( x â a ) 2 + ⯠â n = 0 â c n ( x â a ) n = c 0 + c 1 ( x â a ) + c 2 ( x â a ) 2 + ⯠| https://openstax.org/books/calculus-volume-2/pages/6-key-equations |
Taylor series for the function f f at the point x = a x = a | https://openstax.org/books/calculus-volume-2/pages/6-key-equations |
â n = 0 â f ( n ) ( a ) n ! ( x â a ) n = f ( a ) + f â² ( a ) ( x â a ) + f â³ ( a ) 2 ! ( x â a ) 2 + ⯠+ f ( n ) ( a ) n ! ( x â a ) n + ⯠â n = 0 â f ( n ) ( a ) n ! ( x â a ) n = f ( a ) + f â² ( a ) ( x â a ) + f â³ ( a ) 2 ! ( x â a ) 2 + ⯠+ f ( n ) ( a ) n ! ( x â a ) n + ⯠| https://openstax.org/books/calculus-volume-2/pages/6-key-equations |
binomial series : the Maclaurin series forf(x)=(1+x)r;f(x)=(1+x)r;it is given by(1+x)r=ân=0â(rn)xn=1+rx+r(râ1)2!x2+â¯+r(râ1)â¯(rân+1)n!xn+â¯(1+x)r=ân=0â(rn)xn=1+rx+r(râ1)2!x2+â¯+r(râ1)â¯(rân+1)n!xn+â¯for|x|<1|x|<1 | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
interval of convergence : the set of real numbersxfor which a power series converges | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
Maclaurin polynomial : a Taylor polynomial centered at 0; thenth Taylor polynomial forffat 0 is thenth Maclaurin polynomial forff | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
Maclaurin series : a Taylor series for a functionffatx=0x=0is known as a Maclaurin series forff | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
nonelementary integral : an integral for which the antiderivative of the integrand cannot be expressed as an elementary function | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
power series : a series of the formân=0âcnxnân=0âcnxnis a power series centered atx=0;x=0;a series of the formân=0âcn(xâa)nân=0âcn(xâa)nis a power series centered atx=ax=a | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
radius of convergence : if there exists a real numberR>0R>0such that a power series centered atx=ax=aconverges for|xâa|<R|xâa|<Rand diverges for|xâa|>R,|xâa|>R,thenRis the radius of convergence; if the power series only converges atx=a,x=a,the radius of convergence isR=0;R=0;if the power series converges for al... | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
Taylor polynomials : thenth Taylor polynomial forffatx=ax=aispn(x)=f(a)+fâ²(a)(xâa)+fâ³(a)2!(xâa)2+â¯+f(n)(a)n!(xâa)npn(x)=f(a)+fâ²(a)(xâa)+fâ³(a)2!(xâa)2+â¯+f(n)(a)n!(xâa)n | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
Taylor series : a power series atathat converges to a functionffon some open interval containinga | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
Taylorâs theorem with remainder : for a functionffand thenth Taylor polynomial forffatx=a,x=a,the remainderRn(x)=f(x)âpn(x)Rn(x)=f(x)âpn(x)satisfiesRn(x)=f(n+1)(c)(n+1)!(xâa)n+1Rn(x)=f(n+1)(c)(n+1)!(xâa)n+1for somecbetweenxanda; if there exists an intervalIcontainingaand a real numberMsuch that|f(n+1)(x)|â¤M... | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
term-by-term differentiation of a power series : a technique for evaluating the derivative of a power seriesân=0âcn(xâa)nân=0âcn(xâa)nby evaluating the derivative of each term separately to create the new power seriesân=1âncn(xâa)nâ1ân=1âncn(xâa)nâ1 | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
term-by-term integration of a power series : a technique for integrating a power seriesân=0âcn(xâa)nân=0âcn(xâa)nby integrating each term separately to create the new power seriesC+ân=0âcn(xâa)n+1n+1C+ân=0âcn(xâa)n+1n+1 | https://openstax.org/books/calculus-volume-2/pages/6-key-terms |
Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
There is always more than one way to parameterize a curve. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The derivative of the parametrically defined curvex=x(t)x=x(t)andy=y(t)y=y(t)can be calculated using the formuladydx=yâ²(t)xâ²(t).dydx=yâ²(t)xâ²(t).Using the derivative, we can find the equation of a tangent line to a parametric curve. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The area between a parametric curve and thex-axis can be determined by using the formulaA=â«t1t2y(t)xâ²(t)dt.A=â«t1t2y(t)xâ²(t)dt. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The arc length of a parametric curve can be calculated by using the formulas=â«t1t2(dxdt)2+(dydt)2dt.s=â«t1t2(dxdt)2+(dydt)2dt. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The surface area of a volume of revolution revolved around thex-axis is given byS=2Ïâ«aby(t)(xâ²(t))2+(yâ²(t))2dt.S=2Ïâ«aby(t)(xâ²(t))2+(yâ²(t))2dt.If the curve is revolved around they-axis, then the formula isS=2Ïâ«abx(t)(xâ²(t))2+(yâ²(t))2dt.S=2Ïâ«abx(t)(xâ²(t))2+(yâ²(t))2dt. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The polar coordinate system provides an alternative way to locate points in the plane. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
Convert points between rectangular and polar coordinates using the formulasx=rcosθandy=rsinθx=rcosθandy=rsinθandr=x2+y2andtanθ=yx.r=x2+y2andtanθ=yx. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
Use the conversion formulas to convert equations between rectangular and polar coordinates. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The area of a region in polar coordinates defined by the equationr=f(θ)r=f(θ)withαâ¤Î¸â¤Î²Î±â¤Î¸â¤Î²is given by the integralA=12â«Î±Î²[f(θ)]2dθ.A=12â«Î±Î²[f(θ)]2dθ. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The arc length of a polar curve defined by the equationr=f(θ)r=f(θ)withαâ¤Î¸â¤Î²Î±â¤Î¸â¤Î²is given by the integralL=â«Î±Î²[f(θ)]2+[fâ²(θ)]2dθ=â«Î±Î²r2+(drdθ)2dθ.L=â«Î±Î²[f(θ)]2+[fâ²(θ)]2dθ=â«Î±Î²r2+(drdθ)2dθ. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The equation of a vertical parabola in standard form with given focus and directrix isy=14p(xâh)2+ky=14p(xâh)2+kwherepis the distance from the vertex to the focus and(h,k)(h,k)are the coordinates of the vertex. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The equation of a horizontal ellipse in standard form is(xâh)2a2+(yâk)2b2=1(xâh)2a2+(yâk)2b2=1where the center has coordinates(h,k),(h,k),the major axis has length 2a,the minor axis has length 2b, and the coordinates of the foci are(h±c,k),(h±c,k),wherec2=a2âb2.c2=a2âb2. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The equation of a horizontal hyperbola in standard form is(xâh)2a2â(yâk)2b2=1(xâh)2a2â(yâk)2b2=1where the center has coordinates(h,k),(h,k),the vertices are located at(h±a,k),(h±a,k),and the coordinates of the foci are(h±c,k),(h±c,k),wherec2=a2+b2.c2=a2+b2. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
The polar equation of a conic section with eccentricityeisr=ep1±ecosθr=ep1±ecosθorr=ep1±esinθ,r=ep1±esinθ,whereprepresents the focal parameter. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
To identify a conic generated by the equationAx2+Bxy+Cy2+Dx+Ey+F=0,Ax2+Bxy+Cy2+Dx+Ey+F=0,first calculate the discriminantD=4ACâB2.D=4ACâB2.IfD>0D>0then the conic is an ellipse, ifD=0D=0then the conic is a parabola, and ifD<0D<0then the conic is a hyperbola. | https://openstax.org/books/calculus-volume-2/pages/7-key-concepts |
d y d x = d y / d t d x / d t = y â² ( t ) x â² ( t ) d y d x = d y / d t d x / d t = y â² ( t ) x â² ( t ) | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
d 2 y d x 2 = d d x ( d y d x ) = ( d / d t ) ( d y / d x ) d x / d t d 2 y d x 2 = d d x ( d y d x ) = ( d / d t ) ( d y / d x ) d x / d t | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
A = â« a b y ( t ) x â² ( t ) d t A = â« a b y ( t ) x â² ( t ) d t | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
s = â« t 1 t 2 ( d x d t ) 2 + ( d y d t ) 2 d t s = â« t 1 t 2 ( d x d t ) 2 + ( d y d t ) 2 d t | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
S = 2 Ï â« a b y ( t ) ( x â² ( t ) ) 2 + ( y â² ( t ) ) 2 d t S = 2 Ï â« a b y ( t ) ( x â² ( t ) ) 2 + ( y â² ( t ) ) 2 d t | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
A = 1 2 ⫠α β [ f ( θ ) ] 2 d θ = 1 2 ⫠α β r 2 d θ A = 1 2 ⫠α β [ f ( θ ) ] 2 d θ = 1 2 ⫠α β r 2 d θ | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
L = ⫠α β [ f ( θ ) ] 2 + [ f Ⲡ( θ ) ] 2 d θ = ⫠α β r 2 + ( d r d θ ) 2 d θ L = ⫠α β [ f ( θ ) ] 2 + [ f Ⲡ( θ ) ] 2 d θ = ⫠α β r 2 + ( d r d θ ) 2 d θ | https://openstax.org/books/calculus-volume-2/pages/7-key-equations |
angular coordinate : θθthe angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
cardioid : a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid isr=a(1+sinθ)r=a(1+sinθ)orr=a(1+cosθ)r=a(1+cosθ) | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
conic section : a conic section is any curve formed by the intersection of a plane with a cone of two nappes | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
cusp : a pointed end or part where two curves meet | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
cycloid : the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
directrix : a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
discriminant : the value4ACâB2,4ACâB2,which is used to identify a conic when the equation contains a term involvingxy,xy,is called a discriminant | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
eccentricity : the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
focal parameter : the focal parameter is the distance from a focus of a conic section to the nearest directrix | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
focus : a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
general form : an equation of a conic section written as a general second-degree equation | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
limaçon : the graph of the equationr=a+bsinθr=a+bsinθorr=a+bcosθ.r=a+bcosθ.Ifa=ba=bthen the graph is a cardioid | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
major axis : the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
minor axis : the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
nappe : a nappe is one half of a double cone | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
orientation : the direction that a point moves on a graph as the parameter increases | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
parameter : an independent variable that bothxandydepend on in a parametric curve; usually represented by the variablet | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
parameterization of a curve : rewriting the equation of a curve defined by a functiony=f(x)y=f(x)as parametric equations | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
parametric curve : the graph of the parametric equationsx(t)x(t)andy(t)y(t)over an intervalaâ¤tâ¤baâ¤tâ¤bcombined with the equations | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
parametric equations : the equationsx=x(t)x=x(t)andy=y(t)y=y(t)that define a parametric curve | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
polar axis : the horizontal axis in the polar coordinate system corresponding torâ¥0râ¥0 | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
polar coordinate system : a system for locating points in the plane. The coordinates arer,r,the radial coordinate, andθ,θ,the angular coordinate | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
polar equation : an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
pole : the central point of the polar coordinate system, equivalent to the origin of a Cartesian system | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
radial coordinate : rrthe coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
rose : graph of the polar equationr=acos2θr=acos2θorr=asin2θr=asin2θfor a positive constanta | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
space-filling curve : a curve that completely occupies a two-dimensional subset of the real plane | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
standard form : an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
vertex : a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch | https://openstax.org/books/calculus-volume-2/pages/7-key-terms |
Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
There is always more than one way to parameterize a curve. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The derivative of the parametrically defined curvex=x(t)x=x(t)andy=y(t)y=y(t)can be calculated using the formuladydx=yâ²(t)xâ²(t).dydx=yâ²(t)xâ²(t).Using the derivative, we can find the equation of a tangent line to a parametric curve. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The area between a parametric curve and thex-axis can be determined by using the formulaA=â«t1t2y(t)xâ²(t)dt.A=â«t1t2y(t)xâ²(t)dt. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The arc length of a parametric curve can be calculated by using the formulas=â«t1t2(dxdt)2+(dydt)2dt.s=â«t1t2(dxdt)2+(dydt)2dt. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The surface area of a volume of revolution revolved around thex-axis is given byS=2Ïâ«aby(t)(xâ²(t))2+(yâ²(t))2dt.S=2Ïâ«aby(t)(xâ²(t))2+(yâ²(t))2dt.If the curve is revolved around they-axis, then the formula isS=2Ïâ«abx(t)(xâ²(t))2+(yâ²(t))2dt.S=2Ïâ«abx(t)(xâ²(t))2+(yâ²(t))2dt. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The polar coordinate system provides an alternative way to locate points in the plane. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
Convert points between rectangular and polar coordinates using the formulasx=rcosθandy=rsinθx=rcosθandy=rsinθandr=x2+y2andtanθ=yx.r=x2+y2andtanθ=yx. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
Use the conversion formulas to convert equations between rectangular and polar coordinates. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The area of a region in polar coordinates defined by the equationr=f(θ)r=f(θ)withαâ¤Î¸â¤Î²Î±â¤Î¸â¤Î²is given by the integralA=12â«Î±Î²[f(θ)]2dθ.A=12â«Î±Î²[f(θ)]2dθ. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The arc length of a polar curve defined by the equationr=f(θ)r=f(θ)withαâ¤Î¸â¤Î²Î±â¤Î¸â¤Î²is given by the integralL=â«Î±Î²[f(θ)]2+[fâ²(θ)]2dθ=â«Î±Î²r2+(drdθ)2dθ.L=â«Î±Î²[f(θ)]2+[fâ²(θ)]2dθ=â«Î±Î²r2+(drdθ)2dθ. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The equation of a vertical parabola in standard form with given focus and directrix isy=14p(xâh)2+ky=14p(xâh)2+kwherepis the distance from the vertex to the focus and(h,k)(h,k)are the coordinates of the vertex. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The equation of a horizontal ellipse in standard form is(xâh)2a2+(yâk)2b2=1(xâh)2a2+(yâk)2b2=1where the center has coordinates(h,k),(h,k),the major axis has length 2a,the minor axis has length 2b, and the coordinates of the foci are(h±c,k),(h±c,k),wherec2=a2âb2.c2=a2âb2. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The equation of a horizontal hyperbola in standard form is(xâh)2a2â(yâk)2b2=1(xâh)2a2â(yâk)2b2=1where the center has coordinates(h,k),(h,k),the vertices are located at(h±a,k),(h±a,k),and the coordinates of the foci are(h±c,k),(h±c,k),wherec2=a2+b2.c2=a2+b2. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
The polar equation of a conic section with eccentricityeisr=ep1±ecosθr=ep1±ecosθorr=ep1±esinθ,r=ep1±esinθ,whereprepresents the focal parameter. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
To identify a conic generated by the equationAx2+Bxy+Cy2+Dx+Ey+F=0,Ax2+Bxy+Cy2+Dx+Ey+F=0,first calculate the discriminantD=4ACâB2.D=4ACâB2.IfD>0D>0then the conic is an ellipse, ifD=0D=0then the conic is a parabola, and ifD<0D<0then the conic is a hyperbola. | https://openstax.org/books/calculus-volume-3/pages/1-key-concepts |
d y d x = d y / d t d x / d t = y â² ( t ) x â² ( t ) d y d x = d y / d t d x / d t = y â² ( t ) x â² ( t ) | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
d 2 y d x 2 = d d x ( d y d x ) = ( d / d t ) ( d y / d x ) d x / d t d 2 y d x 2 = d d x ( d y d x ) = ( d / d t ) ( d y / d x ) d x / d t | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
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