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39 Cards in this Set
- Front
- Back
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Pythagorean Identities
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sin^2+cos^2=1
tan^2+1=sec^2 1+cot^2=csc^2 |
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tan and cot identities
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tan=sin/cos
cot=cos/sin |
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product rule
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f'g+g'f
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derivatives
sin cos tan |
cos
-sin sec^2 |
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derivatives
sec csc cot |
sectan
-csccot -csc^2 |
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integral
1/sqrt(1-x^2) |
sin^-1
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integral
1/1+x^2 |
tan^-1
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integral
sec^2 csc^2 |
tan
-cot |
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a^2-b^2x^2
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a/bsin
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b^2x^2-a^2
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a/bsec
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a^2+b^2x^2
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a/btan
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area between curves
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integral(a to b) [upper function]^2-[lower function]^2
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volume of revolution- rings
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pi integral[(outer radius)^2-(inner radius)^2]
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volume of revolution- cylinders
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2pi integral[radius*height]
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work
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integral[force]
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arc length
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integral[sqrt(1+(dy/dx)^2)] if y=f(x)
integral[sqrt(1+(dx/dy)^2)] if x=f(y) integral[sqrt(r^2+(dr/dθ)^2)] if r=f(θ) |
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Surface area
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integral[2pi*y*ds] if rotated about x axis
integral[2pi*x*ds] if rotated about y axis ds= [sqrt(1+(dy/dx)^2)] if y=f(x) [sqrt(1+(dx/dy)^2)] if x=f(y) [sqrt(r^2+(dr/dθ)^2)] if r=f(θ) |
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change in x
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(b-a)/n
a=x(0) b=x(n) |
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midpoint rule
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(change in x)[f(x1*)+f(x2*)+...+f(xn*)],
xi* is midpoint[x(i-1), xi] |
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trapezoid rule
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(change in x)/2[f(x0)+2f(x1)+2(fx2)+...+f(xn)]
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simpsons rule
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(change in x)/3[f(x0)+4(f(x1)+2f(x2)+...+2f(x(n-2))+4f(x(n-1))+f(xn)]
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geometric series
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Σ(n=0 to infinity) [ar^n]
=a/1-r |
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Ratio Test
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L=lim(n->infinity)|a(n+1)/a(n)|
Then, 1. if L<1 the series is absolutely convergent 2. if L>1 the series is divergent 3. if L=0 the test is inconclusive |
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Alternating series test
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Σa(n)
a(n)=(-1)^n(b(n) where b(n)>=0 for all n then if, 1. lim(n->infinity) b(n)=0 and, 2.{b(n)} is a decreasing sequence the series Σa(n) is convergent. |
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Comparison Test
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Suppose that we have two series Σa(n) and Σb(n) with a(n), b(n) >= 0 for all n and a(n)<=b(n) for all n.
Then, If Σb(n) is convergent then so is Σa(n). If Σa(n) is divergent then so is Σb(n). |
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Limit Comparison Test
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Suppose that we have two series Σa(n) and Σb(n) with a(n)>=0, b(n)>0 for all n.
Define, c=lim(n->infinity)a(n)/b(n) If c is positive (i.e. c>0 ) and is finite (i.e. c<infinity ) then either both series converge or both series diverge. |
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Root Test
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Suppose that we have the series Σa(n) .
Define, L=lim(n->infinify) |a(n)|^(1/n) Then, 1. if L<1 the series is absolutely convergent (and hence convergent). 2. if L>1 the series is divergent. 3. if L=1 the series may be divergent, conditionally convergent, or absolutely convergent. |
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Taylor Series
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f(x)=Σ(n=0->infinity)[(f^(n)(a)/n!)(x-a)^n]
f(a)+f'(a)(x-a)+(f''(a)/2!)(x-a)^2+(f'''(a)/3!)(x-a)^3+... |
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Horizontal Tangent of Parametric Equation
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dy/dt=0
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Vertical Tangent for Parametric Equations
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dx/dt=0
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Area Under Parametric Curve
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integral(a->b)[y*x']
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Arc Length for parametric equations
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integral(a->b)[sqrt(x'^2+y'^2)]
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Cartesian to polar coordinates
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x=rcos(θ)
y=rsin(θ) x^2+y^2=r^2 tan^-1(y/x)=θ |
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Derivative of polar coordinates
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(dr/dθ)sinθ+rcosθ
----------------------- (dr/dθ)cosθ-rsinθ |
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Area with polar coordinates
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A=integral(a->b)[(1/2)r^2]
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Area enclosed by two curves
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integral(a->b)[1/2(r(0)^2-r(i)^2)]
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Arc length polar cooridinates
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integral[sqrt(r^2+r'^2)]
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Surface area polar coordinates
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integral[2pi*y*ds] rotation around x-axis
integral[2pi*x*ds] rotation around y-axis ds=sqrt(r^2+r'^2) |
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Center of mass coordinates
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x=1/A*integral(a->b)[x(f(x)-g(x))]
y=1/A*integral(a->b)[1/2(f(x)^2-g(x)^2)] A=integral(a->b)[f(x)-g(x)] |
