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63 Cards in this Set
- Front
- Back
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(f + g)(x) =
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f(x) + g(x)
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(f - g)(x)=
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f(x) - g(x)
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(fg)(x)=
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f(x)g(x)
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(f/g)(x)=
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f(x)/g(x)
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(f o g)(x)=
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f(g(x))
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d/dx c =
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0
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power rule
f(x)=x^n |
f'(x)=nx^(n-1)
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product rule
F(x)=f(x)g(x) |
F'(x)=f(x)g'(x) + g(x)f'(x)
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quotient rule
F(x)=f(x)/g(x) |
F'(x) = [g(x)f'(x) - f(x)g'(x)]/[g(x)]^2
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f(x) = x^-n
f'(x)= |
-nx^-n-1
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(cf)'=
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cf'
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(f-g)'=
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f'-g'
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(f+g)'=
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f'+g'
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(fg)'=
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fg' + gf'
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d/dx tanx
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sec^2x
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d/dx (cscx)=
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-cscxcotx
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d/dx (sec x)
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secxtanx
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d/dx (cotx)
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-csc^2x
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chain rule
F(x) = f(g(x)) F'(x)= |
f'(g(x))g'(x)
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d/dx e^x
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e^x
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d/dx e^u
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e^u *u'
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inverse functions
if g=f^-1 g'(a)= |
1/f'(g(a))
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put log_a_x = y into exponential form
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a^y = x
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put ln x = y into exponential form
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e^y = x
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ln e =
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1
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d/dx (lnx)
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1/x
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d/dx ln u
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1/u * u'
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d/dx log_a_x
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1/xlna
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d/dx a^x
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a^x ln a
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d/dx(sin^-1 x)=
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1/root(1-x^2)
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d/dx (cos^-1 x)
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-1/root(1-x^2)
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d/dx(tan^-1 x)=
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1/(1+x^2)
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d/dx (csc^-1 x)=
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-1/[x*root(x^2-1)]
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d/dx (sec^-1 x) =
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1/[x*root(x^2 - 1)]
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d/dx (cot^-1 x) =
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-1/(1+x^2)
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mean value theorem
if f is a differentiable function on the interval [a,b], then there exists a number c between a and b such that |
f'(c) = [f(b)-f(a)/(b-a)
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integral of 1/x
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ln|x| + C
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integral of sinx
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-cosx +C
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integral of cosx
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sin x + C
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integral of sec^2x
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tan x +C
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integral of csc^2x
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-cot x +C
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integral of secxtanx
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secx+C
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integral of cscxcotx
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-csc x +C
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integral of [1/(x^2+1)]
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tan^-1x + C
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integral of [1/root(1-x^2)
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sin^-1 x + C
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integral of e^x
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e^x + C
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integral of tanx
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ln |sec x| + C
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mvt for integrals
if f is continuous on [a,b], then there exists a number c in [a,b] such that integral from a to be of f(x) |
=f(c)(b-a)
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integral of u dv =
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uv - integral of vdu
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how to evaluate
integral of sin^m x cos^n x dx if the power of cosine is odd |
save one cosine factor and use cos^2 x = 1 - sin^2 x
to express the remaining factors in terms of sine |
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how to evaluate
integral of sin^m x cos^n x dx if the power of sine is odd |
save one sine factor and use
sin^2 x = 1 - cos^2 x to express the remaining factors in terms of cosine |
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if powers of both sine and cosine are even, use the half-angle identities
sin^2 x = cos^2 x = |
1/2(1-cos2x)
1/2(1+cos2x) |
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sinxcosx =
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1/2 sin2x
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how to evaluate integral of tan^m x sec^n x dx
if the power of secant is even |
save a factor of sec^2 x and use sec^2 x = 1 + tan^2 x to express the remaining factors in terms of tan x
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how to evaluate integral of tan^m x sec^n x dx
if the power of tangent is odd |
save a factor of secxtanx and use tan^2 x = sec^2 x - 1 to express the remaining factors in terms of sec x
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integral of sec x
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ln |secx + tanx| + C
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sinAcosB=
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1/2[sin(A-B) + sin(A+B)]
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sinAsinB=
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1/2[cos(A-B) - cos(A + B)]
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cosAcosB=
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1/2[cos(A-B) + cos(A+B)]
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what do you substitute for x in
root(a^2 - x^2) |
x=asintheta
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what do you substitute for x in
root(a2 +x2) |
x=atantheta
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what do you substitute for x in
root(x2-a2) |
x=asectheta
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integral of
dx/(x2 + a2) = |
(1/a)tan^-1(x/a) + C
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