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42 Cards in this Set
- Front
- Back
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d/dx [cu] = ?
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= cu'
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d/dx [u +/- v] = ?
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= u' +/- v'
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d/dx [uv] = ?
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= uv' + u'v
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d/dx [u/v] = ?
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= (vu' - uv') / v^2
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d/dx [c] = ?
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= 0 (zero)
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d/dx [u^n] = ?
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= (nu^(n-1)) u'
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d/dx [x] = ?
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= 1
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d/dx [ln u] = ?
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= u' / u
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d/dx [e^u] = ?
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= (e^u) u'
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d/dx [sin u] = ?
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= (cos u) u'
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d/dx [cos u] = ?
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= - (sin u) u'
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d/dx [tan u] = ?
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= ((sec^2) u) u'
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d/dx [cot u] = ?
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= - ((csc^2) u) u'
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d/dx [sec u] = ?
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= (sec u tan u) u'
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d/dx [csc u] = ?
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= - (csc u cot u) u'
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d/dx [arcsin u] = ?
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= u' / (√1-u^2)
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d/dx [arctan u] = ?
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= u' / (1 + u^2)
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Domain?
f(x)=2x^3-4x^2+5 |
Straight quadratic = "All real numbers"
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Domain?
f(x)=3x / (x^2-49) |
Avoid denominator of zero, X<> 7, -7
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Domain?
f(x)=(√x-4) |
No "i", make x >=4
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Limit?
lim x^2-5x+6 / x-3 x > 3 |
simple denominator
No "substitute x=3", factor and reduce |
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Limit?
lim (2x^4+5x^3-1) / (4x^2-5x^4+2 x>∞ |
Quadratic denominator
Take largest power coefficients from numerator and denominator = 2/5 |
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Limit?
lim 2 / (x - 3) ^ 8 x > 3 |
Power at end of denominator. Even numbered power = "never negative", odd = "no limit, goes to -∞ and +∞
Look up "zero" answer? |
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Limit
lim f(x) = 4x+3, x < 1 x^2 + 3, x>=1 x>1 |
set left = right, solve w/ #'s given
if left <> right, the No Limit if left = right, that's the limit. Here, 7 <> 4 no limit |
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Limit?
lim sin3x / 7x x>0 |
TRICK! sinx / x = 1, SO sin3x / 7x =
3 / 7 |
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Limit?
lim cosx-1 / 3x x>0 |
Sort of a trick: Since 1-cosx / x = 0,
therefore this also = 0 |
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Limit?
lim lnx / lnx^4 x >1 |
lnx / lnx^4 = lnx / 4lnx (lnx^P = Plnx)
reduce lnx's and left with 1/4 |
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Limit?
lim 2-e^-4x / 5 x>∞ |
= 2/5 - e^-4x / 5
= 2/5 - 1 / 5e^-4x since e is approaching ∞ then 1/5e^-4x = 0, therefore limit = just 2/5 |
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Find continuous value?
f(x) = kx^2, x<=1 3x-1, x>1 |
one equation = other, substitute 1, solve for k
k(1^2) = 3-1 = K=3-1, k=2, done. |
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using f(x+∆x)-f(x) / ∆x find derivative
f(x)=4x^2-3x+1 |
becomes("-" remember to flip all signs in second half) =
4(x+∆x)^2-3(x+∆x)+1-4x^2+3x-1 / ∆x= 4(x^2+2x∆x+∆x^2) +1-4x^2+3x-1 / ∆x (Check with shortcut!!!) |
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using f(x+∆x)-f(x) / ∆x find derivative
f(x)=4x^2-3x+1 part 2 |
4x^2+8x∆x+4∆x2-3x+3∆x+1-4x^2+3x-1/∆x reduces to 8x∆x+4(∆x)^2-3∆x / ∆x
factor out ∆ from denom, = 8x+4∆x-3, 4∆x = zero as x>∞ f'=8x-3 (Check with shortcut!!!) |
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f(x)=2x^6+5x^3-4x+1
f'=? |
gimme putt
12x^5+15x^2-4 |
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f(x)= 2 / x^5 + √x
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remove fractions & radical, change to 2x^ -5 + x^1/2 and take derivative of each
10x^-6 + 1/2x^-1/2 (notice 2nd switch) |
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Find derivative
f(x)=(sin3x)(2x-1) |
sin3x(2) +(2x-1)(3cos(3x))
Don't Simplify! |
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Find Derivative
f(x)=(2x^2+2) / 7x-1 |
(7x-1)(4x) - (2x^2+2)(7) / (7x-1)^2
Don't simplify |
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Find derivative
f(x)=arcsin(5x)+arctan(x^2) |
(5 / √1-25x^2) + (2x / (1+x^4))
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Find derivative
f(x)=(2x^5-3x^2-1)^5 |
Chain rule:
take the 5, bring it down, add f' of inside equation to end 5(2x^5-3x^2-1)^4 (10x^4-6x) |
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Find derivative
f(x)=e^-2x + ln(3x-1) |
f'(e^u)=e^u u' User derivative of e^u + derivative of ln(x)
f'= 2e^-2x + (3 / (3x-1)) |
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Find eq of tangent to:
f(x) = 2xe^2x |
Split 2x & e^2x, 2x=u, e^2x=v
find derivative of both, f'(e)=e^u u' add derivatives for multiplication rule uv' + u'v = 2x(e^2x) + e^2x(2), solve do y-y1 = m(x-x1) |
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Find instantaneous velocity at t=3 for
s(t)=-16t^2+100t+50 |
Take easy derivative = -32t+100, plug in t=3, solve for -4 = dropping 4 ft/sec
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Find marginal cost for x=30 of
C(x)=4x^2-100x |
Take easy derivative = 8x-100 solve for x=30. +240 means cost is rising, should have stopped production earlier.
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Given x^2+xy-y^2=11
Find dy/dx Find Tangent and Normal lines @ (2,-3) |
Implicit derivative so X^2 = 2x, etc
2x+x(dx/dy)+y(1) |
