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Pe
P is starting r is rate
Ce^kt C is starting k is rate
y = Ce^kt k = .041 y = 7700e^.41t t = 5 y = 7700e^(.415) 9452 14400 = 7700e^(.41x) 1.87012987 = e^(.41*x) ln(1.87012987) = .41x ln(1.87012987)/.41 = x 15 years
t = 5
15 years
k = -.047 half life means time to cut in half
compound yearly at -.047 y = Ce^(kt) 1/2(150) = 150e^(-.047(t)) 75 = 150e^(-.047t) .5 = e^(-.047t) -0.69314718056 = -.047t 14.7478123523 = t 14.7 years
y = Ce^(kt) t = 4 200 = 100e^4k 2 = e^4k ln2 = 4k (ln2)/4 = k 0.17328679514 = k
300 = 100e^x(0.17328679514) 3 = e^x(0.17328679514) ln3 = 0.17328679514x ln3/0.17328679514 = x 6.33985000288 = x 6.34 years
as price increases, sales decrease R(x) = demandx Elasticity = deltasales/deltaprice = (deltasales/sales)/(deltap/p) P/X * deltax/deltap P/x * 1/p p = D(x) E = -1/x * D(x)/D’(x)
if E= 2 this means a 1% increase in price will result in a 2% decrease in sales
revenue is increasing when derivative is positive revenue is demand * x revenue is increasing when R’ is positive R’ = D’(x)x + D(x)
revenue is increasing when E>1
revenue increasing when e>1
e = deltasales / deltaprice
P(50) = D(50) = 300 - 2(50) = 200
-1/x * D(X)/D’(x) -1/x * (300-2x)/-2 -1/x * -(300-2x)/2 -1/x * (-150 + x) 150/x - 1
E = 1
1 = 150/x - 1 2 = 150/x 2x = 150 x = 75 so 75
-1/x * D(x)/D’(x) -1/x * (100e^-.5x)/(100e^(-.5x) * -.5) -1/x * 1/-.5 -1/x * -2 2/x 1 = 2/x 1x = 2 x = 2
2
2
y = Ce^(kt) we’re solving for t we dont know C but we can just say 100
y = 100e^(-.000124t)
and we know its - 28% so 72 = 100e^(-.000124t) 72/100 = e^(-.000124t) ln(72/100) = -.000124t ln(72/100)/-.000124 = t 2649.22634655 = t 2649.2 = t
Ce^-kt 50 = 100e^(-.051t) 1/2 = e^(-.051t) ln(1/2) = -.051t (ln(1/2))/-.051 = t 13.5911211874 = t 13.6 = t
Ce
200 = 100e^8k 2 = e^8k ln(2) = 8k ln(2)/8 = k 0.08664339757 = k
300 = 100e^(0.08664339757t) 3 = e^(0.08664339757t) ln(3) = 0.08664339757t (ln(3))/0.08664339757 = t 12.6797000058 = t 12.68 = t
39 = 100e^(-.000124t) 39/100 = e^(-.000124t) ln(39/100) = -.000124t (ln(39/100))/-.000124 = t 7593.61725692 = t 7593.6 = t
minutes = 41 + ce^(-kx) 52 = 41 + ce^(-k0) 52 = 41 + ce^(0) 52 = 41 + c 11 = c
43 = 41 + 11e^(-k11) 2 = 11e^(-k11) 2/11 = e^(-k11) ln(2/11) = -11k ln(2/11)/-11 = k 0.154977099294 = k
x = 41 + 11e^(-(0.154977099294)17) x = 41.7892158549 x = 41.8
ce^-kt 49000 = 66000e^(-5k) (49000/66000) = e^-5k ln(49000/66000) = -5k (ln(49000/66000))/-5 = k 0.0595668887832 = k
x = 66000e^(-0.0595668887832(13)) x = 30425.6252089 x = 30425.63
35000 = 77000e^(-3k) 35000/77000 = e^-3k ln(35000/77000) = -3k ln(35000/77000)/-3 = k 0.262819120121 = k
x = 77000e^(-10(0.262819120121)) x = 5560.08958556 x = 5560.09
Ce^kt 17 = 9e^k4 17/9 = e^k4 ln(17/9) = k4 ln(17/9)/4 = k 0.15899719168 = k
x = 9e^23(0.15899719168) x = 348.681879332 x = 348.68
429.8 = 400e^10k 429.8/400 = e^10k ln(429.8/400) = 10k ln(429.8/400)/10 = k 0.00718554371004 = k
x = 400e^21(0.00718554371004) x = 465.150479479 x = 465.15
2500e^(.049t)
-1/x * D(x)/D’(x)
chain ruuule f = sqrt(x) g = 192-2x f’ = (1/2)x^-1/2 g’ = -2
f’(g(x)) * g’(x) (1/2)(192-2x)^(-1/2) * -2
-(192-2x)^(-1/2) -1/(192-2x)^(1/2) -1/sqrt(192-2x)
-1/x * (sqrt(192-2x)/(-1/sqrt(192-2x))
multiplying fractions a/(b/c) = a * c/b
sqrt(192-2x) / -1/sqrt(192-2x) sqrt(192-2x) * sqrt(192-2x)/-1 (192-2x)/-1 -(192-2x) -192+2x
-1/x * (-192+2x) (192-2x) / x
x * sqrt(192-2x) product rule
f’g + fg’ f = x g = sqrt(192-2x) f’ = 1 g’ = -1/sqrt(192-2x)
1(sqrt(192-2x) + x(-1/sqrt(192-2x)) sqrt(192-2x) + -x/sqrt(192-2x) sqrt(192-2x)/1 + -x/sqrt(192-2x) multiply first term by sqrt(192-2x) 192-2x/sqrt(192-2x) - x/sqrt(192x-2x) 192-3x/sqrt(192-2x)
now find 0s 192 -3x = 0 192 = 3x 64 = x
-1/x * D / D’ D = 105-2.1x D’ = -2.1
-1/x * ((105-2.1x) /-2.1) -1/x * (-50+x) 50/x - 1
x(105-2.1x) 105x-2.1x^2 -2.1x^2 + 105x -4.2x + 105 = 0 105 = 4.2x 25 = x
-1/x * D/D’
D = 196 - 3.5x D’ = 3.5
-1/x * (196-3.5x)/(3.5) -1/x * (56-x) -56/x - 1
(196-3.5x)x
f’g + fg’ f = 196-3.5x g = x f’ = -3.5 g’ = 1
((-3.5)(x)) + (196-3.5x)(1) -3.5x + 196 - 3.5x 196 - 7x
196 = 7x 28 = x
-1/x * 136e^(-.125x) / ((136e^(-.125x)) * -.125)
-1/x * -1/.125
-1/x * -8
-1/x * -8 8/x
f = x g = 136e^(-.125x) f’ = 1 g’ = 136e^(-.125x) * -.125
f’g + fg’ 1(136e^(-.125x)) + x(136e^(-.125x) * -.125) 136e^(-.125x) + x(-17e^(-.125x) 136e^(-.125x) - 17xe^(-.125x)
e^(-.125x)(-17x+136) = 0 -17x+136 = 0 136 = 17x 8 = x
84e^(-.01x) / 84e^(-.01x) * -.01
1 / -.01 -1/.01 -100
-1/x * 100 100/x
f = 84e^(-.01x) g = x f’ = 84e^(-.01x) * -.01 g’ = 1
f’g+fg’
(84e^(-.01x) * -.01)x + 84e^(-.01x) * 1 (84e^(-.01x) * -.01)x + 84e^(-.01x) (-.84e^(-.01x))x + 84e^(-.01x) -.84xe^(-.01) + 84e^(-.01x)
e^(-.01)(-.84x + 84) = 0 -.84x + 84 = 0 84 = .84x 100 = x
1/x * (9 - 3x^(1/2)) / (-1.5/x^(1/2)) 1/x * (9-3x^(1/2) * (2sqrt(x)/3)) 1/x * (3)(3-x^(1/2)(2sqrt(x)/3) 1/x * (
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