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variable in exponent
so e^x != 0. it is never 0.
2^(3)(-1) = 2^-3 = 1/8
2^(3)(0) = 2^0 = 1
2^(3)(1) = 2^3 = 8
1000(1+.12/4)^(4*1) 1000(
1000(1+.12/4)^(4*3) 1000(1+.03)^(12) 1000(1.03)^12 1000(1.42576089) 1425.76089 1425.76
5000e^(.08)(5) 5000e^(.4) 7459.12348821 7459.12
3x + e^x
product rule ! f’g + fg’ f = x^2 g = e^x f’ = 2x g’ = e^x
x^2(e^x)
2x(e^x) + x^2(e^x) xe^x(2+x)
e^(x^2+3x) * (2x + 3) (2x+3)e^(x^2+3x)
(7x^6-8)e^(x^7-8x)
e^(3x-x^2) * (3-2x)
y = mx + b
e^(3(3)-3^2) * (3-2(3)) -3
y = -3x + 10
xe
product rule f’g + fg’ f = x g = e^-x f’ = 1 g’ = e^-x * -1 e^-x + x(-e^-x) e^-x - xe^-x e^-x(1-x)
1
e^(-.8x^2-2)
e^(-.8x^2-2) * -1.6x
at 0 this should be 0 right ? cause its -1.6x? ok yeah zero
so then
f’(-1) and f’(1) to get inc dec f’(-1) is positive, so increasing f’(1) is negative, so decreasing
increasing -inf, 0 decreasing 0, inf
quotient rule (fg’ - f’g) / g
f = -8e^x g = -5e^x - 2 f’ = -8e^x g’ = -5e
((-5e^x - 2)(-8e^x) - (-8e^x)(-5e^x)) / (-5e^x-2)^2 (40e^2x + 16e^x - 40e^2x) / (-5e^x-2)^2 (16e^x) / (-5e^x-2)
slope is
(16e^x)/(-5e^(2)-2)^2
4^3x 4^3(0) 4^0 1
78.645
1000(1+.12/4)^(4*1) 1000(
1000(1+.12/4)^(4*3) 1000(1+.03)^(12) 1000(1.03)^12 1000(1.42576089) 1425.76089 1425.76
x(1+.09/12)^(12*17) = 115000 x(1.0075)^(204) = 115000 x4.59188689 = 115000 x = 25044.1708942 25044.17
C(1-r)
1625(1-.3)^5 1625*(.7)^5 273.11375 273.11
13500(1+.08/4)^(422) 13500(1.02)^(88) 77116.77932057 77116.78
Pe^rt 9750e^(.06)(13) 9750e^.78 21269.3545886 21269.35
200e^(.19)(5) 200e^.95 517.14 517
-6x^5 + 3e^x -30x^4 + 3e^x
-6x^5 + 3e^x -30x^4 + 3e^x -120x^3 + 3e^x
e^(-.2x^2+5)
e^(-.2x^2+5) * -.4x
e^(-.8x^2-2)
e^(-.8x^2-2) * -1.6x
at 0 this should be 0 right ? cause its -1.6x? ok yeah zero
so then
f’(-1) and f’(1) to get inc dec f’(-1) is positive, so increasing f’(1) is negative, so decreasing
increasing -inf, 0 decreasing 0, inf
-.4xe^(-.2x^2+5) = 0
at 0 this is 0 cause we multiply by x so theres that
exponential stuff should always be positive -.4 multiplier means we are flipping that
sooooo we are saying increasing from -inf 0 and decreasing from 0, inf
4x^4e^x grrr product rule
f’g+fg’
f = 4x^4 g = e^x f’ = 16x^3 g’ = e
16x^3(e^x) + 4x^4(e^x) e^x(16x^3+4x^4)
double grr double product rule wtf not cool guys f = 16x^3 + 4x^4 g = e^x f’ = 48x^2 + 16x^3 g’ = e
f’g + fg’ (48x^2+16x^3)(e^x) + (16x^3+4x^4)(e^x) (e^x)(48x^2+16x^3+16x^3+4x^4) (e^x)(48x^2+32x^3+4x^4)
-7e^(2x^3-3)
-7e^(2x^3-3) * 6x
-7(6x^2)e^(2x^3-3)
-7(6(2)^2)e^(2(2)^3-3) -7(6(4))e^(2(8)-3) -7(24)e^(16-3) -168e^13
quotient rule grrrr
(f’g - fg’) / g^2 f = -3e^x g = 7e^x - 1 f’ = -3e^x g’ = 7e
-3e^x(7e^x-1) - -3e^x(7e^x) -3e^x(7e^x-1) + 3e^x(7e^x)
3e^x numerator
3e^x / (7e^x-1)
3e^1 / (7e^1 - 1)^2 3e / (7e-1)^2
chain rule
f’(g(x)) * g’
f = 7x^2 g = 7e^3x + 5 f’ = 14x g’ = 7e^3x * 3
14(7e^3x+5) * 7e^3x * 3 14(7e^3x+5) * 21e^3x 294(7e^3x+5)(e^3x)
294(7e^3x+5)(e^3x) = 0 exponentials are always positive so ignore them multiply by 294 is positive always i think this is constant increasing
4900 / (1+99e^-.5x)
quotient rule
f = 4900 g = 1 + 99e^-.5x f’ = 0 g’ = 99e^-.5x * -.5
(f’g - fg’) / g^2 (0(1+99e^-.5x) - 4900(99e^-.5x * -.5))/ (1+99e^-.5x)^2 (-4900(-49.5e^-.5x))/(1+99e^-.5x)^2 -242550e^-.5x/(1+99e^-.5x)
solve for 4
-242550e^-.5(4)/(1+99e^-.5(4))^2 -242550e^-2/(1+99e^-2)^2 -158
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