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1/x + 4x^5 - 3x^6
1/x + 4x^5 - 3x
-3x^6 + 4x^5 + 1/x -18x^5 + 20x^4 + 1/x^2
f’g + fg’
f = x^-1 +2 g = -4x^-1 - 10
f’ = -x^-2 g’ = 4x
-x^-2(-4x^-1-10) + (x^-1 + 2)(4x^-2) 4x^-3 + 10x^-2 + 4x^-3 + 8x^-2 8x^-3 + 18x^-2
-8t/5t+8
f’g - fg’ / g
f -8t g 5t+8
f’ = -8 g’ = 5
-8(5t+8) - -8t(5) -40t -64 + 40t -64 -64/(5t+8)^2
f’g + fg’
f = -3 g = 9
f’ = -6 g’ = -1
-6(9) + -3(-1) -54 + 3 -51
f’(g(x)) * g’(x)
f = x^5 g = 3-7x
f’ = 5x^4 g’ = -7
5(3-7x)^4 * -7
-35(3-7x)^4
problem 3 attempt 2
f’(g(x)) * g’(x)
f = x^(1/3) g = 8x^3 + 7
f’ = 1/3x^(-2/3) g’ = 24x
1/3(8x^3+7)^(-2/3) * 24x^2 8x^2(8x^3+7)^(-2/3)
f = x^(5/3) g = 5x^4 - 3x - 1
f’ = (5/3)x^(2/3) g’ = 20x^3 - 3
(5/3)(5x^4-3x-1)^(2/3)(20x^3-3)
0.6x^2 + 8xy + 9.7y^2 = 5131.8
8xy
f’g + fg’
f 8x g y
f’ 8 g’ dy/dx
8y + 8x(dy/dx)
1.2x^2 + 8y + 8x(dy/dx) + 19.4y(dy/dx) = 0 8x(dy/dx) + 19.4y(dy/dx) = -1.2x - 8y (dy/dx)(8x+19.4y) = -1.2x -8y dy/dx = (-1.2x-8y)/(8x+19.4y)
dy/dx = (-1.2x-8y)/(8x+19.4y)
x = 19 y = 16
dy/dx = (-1.2(19)-8(16))/(8(19)+19.4(16)) dy/dx = -.326
6x^2 + 2xy -2y^3 = 3 12x + 2y + 2x(dy/dx) -6y^2(dy/dx) = 0 2x(dy/dx) -6y^2(dy/dx) = -12x-2y (dy/dx)(2x-6y^2) = -12x-2y dy/dx = (-12x-2y)/(2x-6y^2) dy/dx = (-6x-y)/(x-3y^2)
8x^4 + 5y^2 = 9
32x^3(dx/dt) + 10y^2(dy/dt) = 0 10y^2(dy/dt) = -32x^3(dx/dt) dy/dt = -32x^3/10y^2 * (dx/dt) dy/dt = -16x^3/5y^2 * dx/dt
problem 11 attempt 2
l = 75 w = 65
l’ = 5 w’ = -4
area of rectangle = l*w
rate of change of area of rectangle = l’w + lw’
5(65) + 75(-4)
325 - 300 25
5213
2ft/s
?/s
4588
b
a^2 + b^2 = c^2 4588^2 + b^2 = 5213^2 b = 2475
a’ = ? b’ = 2 c’ = 0
db/dt = 2 da/dt = ? when a is 4588 ft
a^2 + b^2 = c^2 2a(da/dt) + 2b(db/dt) = 0 2a(da/dt) + 2b(2) = 0 2a(da/dt) + 4b = 0 2a(da/dt) = -4b da/dt = -4b/2a da/dt = -4(2475)/2(4588) da/dt = -9900/9176 da/dt = -2475/2294
a
c
cube surface area = 6s^2 volumerate = -60/s s = 20 cube volume = s^3 volumeroc = 3s^2(ds/dt) -60 = 3s^2(ds/dt) -60 = (3(20)^2)(ds/dt) -60/3(400) = ds/dt -60/1200 = ds/dt sideroc = -60/1200 cube surface area = 6s^2 saroc = 12s(ds/dt) saroc = 12(20)(-60/1200) saroc = 240(-60/1200) saroc = -14400/1200 saroc = -12
2x^2 + 12x + 14
4x + 12
4x+12 = 0 4x = -12 x = -3
4(-4)+12 -16+12 -4
4(-2)+12 -8+12 4
dec -inf, -3 inc -3, inf
-3x^3 -18x^2 -36x -22 -9x^2 -36x - 36 -9(x^2-4x-4)
-9(x^2+4x+4) = 0 x^2 +4x +4 = 0 (x+2)(x+2)
-2 critical point
-9(-3^2+4(-3)+4) -9(9-12+4) -9(1) -9
-9(-1^2+4(-1)+4) -9(1-4+4) -9(1) -9
4x^3 + 39x^2 -42x + 9 12x^2 + 78x - 42 6(2x^2 + 13x - 7)
6(2x^2 + 13x - 7) = 0 2x^2 + 13x - 7 = 0
2x^2 + 14x - x - 7 2x(x+7) - (x+7) (2x-1)(x+7)
-7 and 1/2
6(2x^2 + 13x - 7)
-8 6(2(-8)^2 + 13(-8) -7) 102
-7 local maximum
0 6(2(0)^2 + 13(0) -7) -42
1/2 local minimum
1 6(2(1)^2 + 13(1) -7) 48
4(-7)^3 + 39(-7)^2 -42(-7) + 9 842
4(1/2)^3 + 39(1/2)^2 -42(1/2) + 9 -1.75 -7/4 max: (-7,842) min: (1/2, -7/4)
f’g - fg’ / g
f 9x^2 + 4 g x
f’ 18x g’ 1
18x^2 - 9x^2 - 4 9x^2 - 4
9x^2 -4 = 0 9x^2 = 4 x^2 = 4/9 x = sqrt(4/9) x = 2/3 or -2/3
9x
-1 9(-1)^2-4 9-4 5
-2/3
0 9(0)^2-4 -4
2/3
1 9(1)^2-4 9-4 5
max at -2/3 min at 2/3
(9(-2/3)^2 + 4) / (-2/3) -12
(9(2/3)^2 + 4) / (2/3) 12
5x^2-20x 10x - 20
10x - 20 = 0 10x = 20 x = 2
0 5(0)^2-20(0) 0
7 5(7)^2 -20(7) 5(49)-140 245-140 105
2 5(2)^2-20(2) 5(4) - 40 20 - 40 -20
absolute maximum at 7,105 absolute minimum at 2,-20
problem 17 step 1 attempt 2
problem 17 step 2 attempt 2
4x^3 + 24x^2 - 252x
12x^2 + 48x - 252 12(x^2+4x-21)
12(x^2+4x-21) = 0 x^2+4x-21 = 0 (x+7)(x-3) = 0
-7 and 3 are critical points
-9, 1296
-7, 1568
3, -432
5, -160
absolute maximum -7, 1568 absolute minimum 3, -432
156x - 6x^2 156 - 12x
156 -12x = 0 -12x = -156 12x = 156 x = 13
0,0
13, 1014
30, -720
172x-2x^2 - 2x^2 - 148x - 6 24x - 4x^2 -6 -4x^2 + 24x -6
-8x + 24
-8x = -24 8x = 24 x = 3
0, -6
3, 30
10, -166
maximum 3,30
product rule f(x) * g(x) f’g + fg’
chain rule f(g(x)) (x+5)^4 f = x^4 g = x+5
f’ = 4x^3 g’ = 1
f’(g(x)) * g’ 4(x+5)^3 * 1 4(x+5)
quotient rule f/g
(f’g - fg’) / (g^2)
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