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submit question 2, 9, and 15
19x^3 + ln(x^5) 19x^3 + 5ln(x) 57x^2 + 5lnx 57x^2 + 5/x
chain rule then product rule f = ln(x) g = (9x^3+14x^2)^5(6x+3) f’ = 1/x g’ = product rule
ok, in the product rule now: f = (9x^3+14x^2)^5 g = 6x+3 f’ = damn it chain rule g’ = 6
ok, in the second chain rule now: f = x^5 g = 9x^3+14x^2 f’ = 5x^4 g’ = 27x^2 + 28x
f’(g(x)) * g’(x) 5(9x^3+14x^2)^4 * (27x^2+28x) ^ ok boom
so now we go back up ok, in the product rule now: f = (9x^3+14x^2)^5 g = 6x+3 f’ = 5(9x^3+14x^2)^4 * (27x^2+28x) g’ = 6
f’g + fg’ (5(9x^3+14x^2)^4 * (27x^2+28x))(6x+3) + 6(9x^3+14x^2)
ok now we go back up again chain rule f = ln(x) g = (9x^3+14x^2)^5(6x+3) f’ = 1/x g’ = (5(9x^3+14x^2)^4 * (27x^2+28x))(6x+3) + 6(9x^3+14x^2)
f’(g(x)) * g’(x)
1/((9x^3+14x^2)^5(6x+3)) * (5(9x^3+14x^2)^4 * (27x^2+28x))(6x+3) + 6(9x^3+14x^2)^5
massively overcomplicated option:
ln((9x^3+14x^2)^5) + ln(6x+3) 5ln(9x^3+14x^2) + ln(6x+3) 5 * (27x^2 + 28x)/(9x^3+14x^2) + 6/6x+3 5 * (x(27x+28)/(x^2(9x+14)) + 2/(2x+1) 5 * (27x+28)/(x(9x+14)) + 2/(2x+1) 5(27x+28)/(x(9x+14)) + 2/(2x+1
question 2
e^(-.5x-3)
e^(-.5x-3) * -.5 -.5e^(-.5x-3)
-.5e^(-.5x-3) = 0 never so is it increasing or decreasing?
decreasing, cause -.5 multiply
21x^2 * 3lnx
63x^2 * lnx
f = 63x^2 g = lnx f’ = 126x g’ = 1/x
f’g + fg’
126x(lnx) + 63x^2(1/x) 126x(lnx) + 63x
-5e^(4x^2-8) -5e^(4x^2-8) * 8x -40xe^(4x^2-8)
solve for 4
-40(4)e^(4(4)^2-8) -160e^(4(16)-8) -160e^(64-8) -160e^(56)
revenue = demandx
revenue = 240e^(-.04x) * x 240x * e^(-.04x)
product rule f’g + fg’
f = 240x g = e^(-.04x) f’ = 240 g’ = e^(-.04x) * -.04 g’ = -.04e^(-.04x)
240(e^(-.04x)) + 240x(-.04e^(-.04x)) 240e^(-.04x) - 9.6xe^(-.04x)
240e^(-.04x) - 9.6xe^(-.04x) = 0 240e^(-.04x) = 9.6xe^(-.04x) 240 = 9.6x 25 = x
so, 25, is that a min or a max?
240e^(-.04(20)) - 9.6(20)e^(-.04(20)) 240e^-.8 - 192e^-.8 48e^-.8 positive below 25
240e^(-.04(30)) - 9.6(30)e^(-.04(30)) 240e^(-1.2) - 288e^(-1.2) -48e^(-1.2) negative above 25
pos → neg its a max at 25
240e^(-.04x) 240e^(-.04(25)) 240e^-1 88.2910658811 88.29
x^2 * e
f = x^2 g = e^x f’ = 2x g’ = e
f’g + fg’ 2x(e^x) + x^2(e^x)
e^x(2x+x^2)
x^2 * e
f = x^2 g = e^x f’ = 2x g’ = e
f’g + fg’ 2x(e^x) + x^2(e^x)
e^x(2x+x^2)
product rule AGAIN
f = e^x g = 2x+x^2 f’ = e^x g’ = 2 + 2x
f’g + fg’ e^x(2x+x^2) + e^x(2+2x) e^x(2x+x^2+2+2x) e^x(x^2+4x+2)
4.1 percent per year 7700 in 1994
7700e^(1.041t)
4.1 percent per year 7700 in 1994
7700e^(.041t) 7700e^(.041(5)) 7700e^(.205) 9451.94300022 9452
7700e^(1.041t) = 14400 e^(1.041t) = 1.87012987 .041t = ln(1.87012987) t = ln(1.87012987)/1.041 t = 15.2684848208 t = 15
1994 + 15 2009
question 9
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
Ce^(kt)
.5 = e^(-.057t) ln(.5) = -.057t ln(.5)/-.057 = t t = 12.2
1-.69 = Ce^kt .31 = e^(-.000124t) ln(.31) = .000124t ln(.31)/-.000124 = t 9445.02404438 = t 9445 = t
Elasticity function: -1/x * D(x)/D’(x) 1/x * (105.8-2.3x)/(2.3) 1/x * (46-x) (46-x)/x
x(105.8-2.3x) 105.8x - 2.3x
derivative
105.8 - 4.6x = 0 105.8 = 4.6x 23 = x but is it a max
105.8 - 4.6(20) 105.8 - 92 = pos
105.8 - 4.6(30) 105.8 - 138 = neg
pos → neg max at 23
144e^(-.1x)
-1/x * (144e^(-.1x)/(144e^(-.1x) * -.1) -1/x * (144e^(-.1x)/(-14.4e^(-.1x)) -1/x * -10 10/x
144xe^(-.1x)
f = 144x g = e^(-.1x) f’ = 144 g’ = e^(-.1x) *-.1 g’ = -.1e^(-.1x)
f’g + fg’ 144(e^(-.1x)) + 144x(-.1e^(-.1x)) 144e^-.1x + 144x*(-.1e^(-.1x) 144e^-.1x - 14.4xe^(-.1x)
144e^-.1x - 14.4xe^(-.1x) = 0
144e^-.1x = 14.4xe^(-.1x) 10e^-.1x = xe^(-.1x) 10 = x
10
5x^4 - 2x^-1 + 3x^-5 5x^5/5 - 2ln|x| + 3x^-4/4 x^5 - 2ln|x| + 3/x^4 * 1/4 x^5 - 2ln|x| + 3/4x^4 + c
x^3(x-4) x^4 - 4x^3 integrate
x^5/5 - 4x^4/4 x^5/5 - x^4 + C
-7x^2 + 4e
-7x^3/3 + 4e^x + C
-7/3 x^3 + 4e^x + C
now we need to make F(0) = 6 -7/3 * 0^3 + 4e^0 + C
4e^0 + C = 6 4(1) + C = 6 C = 2
-7/3 x^3 + 4e^x + 2
question 15
inside exponent denominator exponent
7t
u = 7t du = 7
we need to get -3 to be 7
-3/7 * integrate: 7e^7t cool so we have our lil ratio
-3/7 * e^7t + c
inside exponent denominator
u = 7x^7 + 6x + 3 du = 49x^6 + 6
great matchinggg!!
u^4 u^5/5 (7x^7+6x+3)^5 / 5 + C
inside exponent denominator denominator
u = 2x^4 + 6x + 5 du = 8x^3 + 6
2 * integral (8x^3+6)/(2x^4+6x+5)
2 / u 2ln|u|
2ln|(2x^4+6x+5)| + c
7x
7ln|x|
f(15) - f(11)
7ln|15| - 7ln|11| = 2.17108449813
u = 4x + 8 du = 4
1/4 * integral 4/4x+8
1/u ln|u| 1/4ln|u|
1/4ln|4x+8|
f(9) - f(5)
1/4ln|4(9)+8| - 1/4ln|4(5)+8| 1/4ln|44| - 1/4ln|28| 0.112996280936 .11
8x^3 - 8
8x^4/4 - 8x 2x^4 -8x
f(5) - f(2) 1194
1/(b-a)
4x+1 integrate
4x^2/2 + 1x
2x^2 + x
1/(b-a) * (2x^2+x) 1/(4-2) * integral(2,4): (2x^2 +x) 1/2 * ((2(4)^2+4)-(2(2)^2+2) 1/2 * ((2(16)+4) - (2(4)+2)) 1/2 * (32+4 - (8 + 2)) 1/2 * (36 - 10) 1/2 * 26 13
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