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sx^2 + 8x + 5 max at x = 1
sx^2 + 8x + 5 2(sx) + 8
2s(1) + 8 = 0 2s + 8 = 0 2s = -8 s = -4
C(x) = .2x^2 + 5.6x + 5
profit = revenue - cost revenue = demand * x revenue = 14x profit = 14x - (.2x^2 + 5.6x + 5) profit = 8.4x - .2x^2 - 5 profit’ = 8.4 - .4x 0 = -.4x + 8.4 .4x = 8.4 x = 21
profit” = -.4
second derivative negative, concave down, so 21 is a max
8.4(21) - .2(21)^2 - 5 176.4 - 88.2 - 5 83.2
maximum profit 83.2$
x = -8 deltax = -.7
deltay = f(x+deltax) - f(x) deltay = f(-8 -.7) - f(8) deltay = f(-8.7) - f(8) deltay = 85.09 - 72 deltay = 13.09
dy/dx = 2x - 2 dy = (2x-2)dx dy = (2(-8)-2)(-.7) dy = (-16-2)(-.7) dy = -18(-.7 dy = 12.6
13.09 - 12.6 = .49
13.09
12.6
.49
3 8 and 10 show work and label
sqrt(4x+10)
f = x^(1/2) g = 4x+10
f’ = 1/2x(-1/2) g’ = 4
f’(g(x)) * g’(x)
1/2(4x+10)^(-1/2) * 4
2(4x+10)^(-1/2)
f= 2x^(-1/2) g= 4x+10
f’ = x^-3/2 g’ = 4
f’(g(x)) * g’(x)
(4x+10)^(-3/2) * 4 4(4x+10)^-3/2
(4)/(4x+10)^3/2)
4/((4x+10)^(3/2))
4/((4(1)+10)^(3/2)) 4/((4+10)^(3/2)) -4/(14)^(3/2)
4/((4(5)+10)^(3/2)) 4/((20+10)^(3/2)) -4/(30)^(3/2)
4/((4(4)+10)^(3/2)) 4/((16+10)^(3/2)) -4/(26)^(3/2)
question 3
-5x^3 - 120x^2 + 8x - 2 -15x^2 - 240x + 8 -30x - 240
-30x - 240 = 0 -30x = 240 x = -8
-30(-9) - 240 270 - 240 30 concave up from -inf, -8
-30(-7) - 240 210 - 240 -30 concave down from -8, inf
-5(-8)^3-120(-8)^2+8(-8)-2 -5(-512) -120(64) - 64 - 2 2560 - 7680 - 64 - 2 -5186 -8, -5186
5x^3-105x^2-x-1 15x^2 -210x - 1 30x - 210
30x - 210 = 0 30x = 210 x = 7
-inf
0 30(0)-210 0-210 -210
7
10 30(10)-210 300-210 90 inf
-inf, 7 concave down 7, inf concave up
5(7)^3 - 105(7)^2 - 7 - 1 5(343) - 105(49) - 8 1715 - 5145 - 8 -3438 7, -3438
2x^2 -12x + 16 4x - 12 4
(3x^2-3)
f’(g(x)) * g’(x) f = x^2 g = 3x^2-3 f’ = 2x g’ = 6x
2(3x^2-3) * 6x 6x^2-6 * 6x 36x^3 - 36x
36x^3 -36x 108x^2 - 36
36x^3 - 36x = 0 36x^3 = 36x x^2 = 1 x = +- 1
108(1)^2 - 36 108 - 36 +
108(-1)^2 - 36 108 - 36 +
(3*(1)^2-3)^2 = 0
(3*(-1)^2-3)^2 = 0
f(3) = 2 f’(3) = 0 f’(x) < 0 if x < 3 f’(x) > 0 if x > 3 f”(x) > 0 for all x
so, y = 2 at x = 3 at x = 3 it is a max or a min it is decreasing from -inf, 3 it is increasing from 3, inf it is concave up (meaning x = 3 is a min)
-2/3(x)^3 - x^2 + 12x - 11 -2x^2 - 2x + 12 -4x - 2
-2x^2 -2x + 12 = 0 -2(x^2 + x + 12) = 0 x^2 + x - 6 = 0 (x+3)(x-2) = 0 -3 and 2
-10 -2(-10)^2 - 2(-10) + 12 -2(100) + 20 + 12 -200 + 28 -172
-3
0 -2(0)^2 - 2(0) + 12 12
2
10 -2(10)^2 - 2(10) + 12 -2(100) -20 + 12 -200 - 20 + 12 -208
decreasing -inf, -4 increasing -4, 3 decreasing 3, inf
decreasing (-inf, -3), (2, inf) increasing (-3, 2)
question 8
-4x - 2 = 0 -4x = 2 x = 2/-4 x = -1/2
-4(-1) - 2 = 2 -4(1) -2 = -6
concave up -inf, -1/2 concave down -1/2, inf
min at -3 max at 2 inflection point at -1/2
px^2 - 9x - 4 make min at -1
2px - 9
2p(-1) - 9 = 0 -2p = - 9 p = -9/-2 p = 4.5
C(X) = (14 * x/2) + 28*(64/x) C(X) = 14x/2 + 1792/x C(X) = 7x + 1792/x C’(X) = 7 - 1792x^-2 0 = 7 - 1792x^-2 1792x^-2 = 7 1792 = 7x^2 256 = x^2 16 = x
3584x^-3 3584(16)^-3 .875 concave up min, so yeah this minimizes costs.
question 10
question 9
64/16 = 4
profit = revenue - cost profit = 23x - (.8x^2+5.4x +5) profit = 23x - .8x^2 - 5.4x - 5 profit = -.8x^2 + 23x - 5.4x - 5 profit = -.8x^2 + 17.6x - 5 profit’ = -1.6x + 17.6 0 = -1.6x + 17.6 1.6x = 17.6 x = 11
-1.6x + 17.6 -1.6
so, concave down everywhere max at 11
-.8*(11)^2 + 17.6*(11) - 5 91.8$ at 11
x = how many increments of 4 we are doing
72 at 80 each price = 80 + 4x quantity sold = 72-(3x)
price*quantity sold = revenue
(80+4x)*(72-3x) 5760 -240x + 288x -12x^2 -12x^2 + 48x + 5760 -24x + 48
-24x + 48 = 0 48 = 24x 2 = x
80 + 2(4) = 88
x
h
v = x^2 * h 2048 = x^2 * h 2048 = x^2 * 2048/x^2 SA = x^2 + 4(2048/x^2)x SA = x^2 + (8192/x^2)x SA = x^2 + 8192/x SA’ = 2x -8192/x^2 0 = 2x - 8192/x^2 8192/x^2 = 2x 8192 = 2x^3 4096 = x^3 16 = x
SA = 16^2 + 8192/16 SA = 768
1650 = xy extprice = 13.20(2x + 2y) intprice = 22x
extprice = 13.20(2x + 2(1650/x)) intprice = 22x
price = 13.20(2x+2(1650/x)) + 22x price = 26.4x + 43560/x +22x price = 48.4x + 43560/x price’ = 48.4 - 43560/x^2 0 = 48.4 - 43560/x^2 43560/x^2 = 48.4 43560 = 48.4x^2 900 = x^2 30 = x 1650/30 = 55 55 = y
10x^3 - 6x^2 - 9x + 8 (30x^2 - 12x - 9)dx
x^2-2x-8 dy = (2x - 2)dx dy = (2(-2)-2)(-.3) dy = (-4-2)(-.3) dy = (-6)(-.3) dy = 1.8
(-2)^2-2(-2)-8 = 4 + 4 - 8 = 0 (-2.3)^2-2(-2.3)-8 = 5.29 + 4.6 - 8 = 1.89
1.89 - 0 = 1.89 1.8 and 1.89 are pclose.
1.89 - 1.8 = .09
sqrt(36) = 6 x^(1/2) dy = (1/2x^(-1/2))(dx) dy = (1/2(36)^(-1/2)(-1.3) dy = (1/12)-1.3 dy = (1/12)(-13/10) dy = (-13/120)
6 - 13/120
rx^2 - 2x - 9 min at 1
2rx - 2 = 0 2r - 2 = 0 2r = 2 r = 1
1 13
12 10
question 8
question 8
question 8
x = increments of 4
price = 80 + 4x racketssold = 72 - 3x revenue = price*racketssold revenue = (80 + 4x)(72-3x) revenue = 5760 - 240x + 288x - 12x^2 revenue = -12x^2 + 48x + 5760 revenue’ = -24x + 48 0 = -24x + 48 24x = 48 x = 2
now make sure its a max
f’ -24x + 48 f” = -24
f”(2) = -24 ( concave down, its a max )
so, 2 increments of 4 on 80, which means 88 is optimal price.
14 * x/2 + 28 * 64/x
x
h
V = x^2*h 2048 = x^2 * h V = x^2 * 2048/x^2 SA = x^2 + 4hx SA = x^2 + 4(2048/x^2)x SA = x^2 + 8192/x SA’ = 2x - 8192/x^2 0 = 2x - 8192/x^2 8192/x^2 = 2x 8192 = 2x^3 4096 = x^3 16 = x
16^2 + 8192/16 768
now make sure its a min
2x - 8192/x^2 2 - 16384/x
2 - 16384/(768)^3 2 - 16384/452984832 2 - 0.0000361689814815 positive, which means concave up, which means it is a min. min SA 768
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