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(defn powerset [X] | |
(let [n (count X)] | |
(->> (range (inc n)) | |
(map #(clojure.math.combinatorics/combinations (vec X) %))/ | |
(apply concat) | |
(map set) | |
(into #{})))) |
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Surface Area | |
S is a surface | |
z=f(x y) | |
grad(f) exists | |
Divide D into small rectangles Rij with area dA | |
Pij(xi,yi,f(xi,yi)) is a point in S. |
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14.6 5 | |
Find the directional derivative of f at the given point in the direction indicated by the angle θ. | |
f(x, y) = y cos(xy), (0, 1), θ = π/6 | |
fx=-y^2sin(xy) | |
fx(0 1)= -1^2sin(0*1) =0 | |
fy= cos(xy)-sin(xy)y^2 | |
fy= 1 |
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14.5 39 | |
The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 8 m and | |
w = h = 1 m, and ℓ and w are increasing at a rate of 2 m/s while h is decreasing at a rate of 4 m/s. | |
At that instant find the rates at which the following quantities are changing. | |
l = 8 w = 1 h =1 | |
dl = 2 dw = 2 dh = -4 | |
v=lwh | |
dv = wh dl + lh dw + lw dh |
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r <9t t^2 t^3> (9,1,1) => t=1 | |
r' <9 2t 3t^2> | |
r'(1) <9 2 3> | |
|r'(1)| sqrt(81+4+9) = sqrt(94) | |
|r'(1)|^3= 94sqrt(94) | |
r" <0 2 6t> | |
r"(1) <0 2 6> | |
i j k i j | |
9 2 3 9 2 |
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13.1 19 | |
Find a vector equation and parametric equations for the line segment that joins P to Q. | |
P(0, −1, 4), Q(1/2,1/3,1/4) | |
r(t) = (1-t)r0+tr1 | |
r(t) = (1-t)<0,-1,4>+t<1/2,1/3,1/4> | |
r(t) = <0,-1,4> -t<0,-1,4>+ t<1/2,1/3,1/4> | |
r(t) = <0 -0t+1/2t, -1 +t+1/3t, 4-4t+1/4t> |
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12.4 35 | |
P(-2,1,0) Q(2,2,2) R(1,4,-1) S(3,6,4) | |
PQ = <2-(-2), 2-1, 2-0> = <4, 1, 2> | |
PR =<1-(-2), 4-1, -1-0> = <3, 3, -1> | |
PS =<3-(-2), 6-1, 4-0> = <5, 5, 4> | |
Triple Product | |
30 -20 12 | |
4 1 2 4 1 |