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Kouba
Worksheet 4 1.) Determine the limit as (x, y) approaches (0,0) for each of the
following. a) f(x,y)= £11:
x2+y2
2x b) f(X.y) ___Y___
5x4+3y4 c) f(X.y)= X_3’"’_Xy_
X3/2+y3 2.) The two shortest sides of a right triangle are measured as 3 cm. and
4 cm., resp., with a maximum absolute error of 0.02 cm. for each measurement. Use differentials to approximate the maximum absolute
error in measuring a.) the hypotenuse.
b.) the area. 3.) Use differentials to approximate the change in w = r 2 + 3 s v + 2 p 3 if r changes from 1 to 1.02, s from 2to 1.99, v from4to 4.01, and p
form 3 to 2.97 . 4.) The dimensions of a rectangular room are 9 x 12 x 8 ft. with
possible errors of i 0.01, i 0.02, and i 0.03 ft., resp. Calculate the length of the long diagonal across the room and the possible error in this
measurement. 5.) A rectangular solid has sides of length 1.02, 3.01, and 4.2 cm. a.) Compute the volume.
b.) Use a differential to estimate the volume. 6.) The specific gravity of an object is s = A/ (A — W) , where A and W
are the weights of the object in air and water, resp. If A = 12 lbs. and
W: 5 lbs. with maximum absolute errors of 1/2 oz. in air and 1 oz. in
water, what is the maximum absolute error in the calculated value of s ? 7.) Find dw/dt where w=ln(3u+v2), u=e‘2‘,andv=t3t2.
8.) Find awe» and aw/as where w=f(3t2s) and f'(x) =sin x. 9.) Find Zx where 2 satisfies xy2+22+cos(xyz) = 4. 10.) Assume that f is a differentiable function with w = f( a x + b y ),
where a and b are constants. Show that a(8w/ay) = b(aw/Bx).
11.) Assume that f is differentiable with z =x f(xy). Show that
xzX  y.zy = z. 12.) Assume that f and g are twice differentiable functions. Show
that u =f(x+at)+g(xat) satisfies a2 E32u/£)x2 = 32u/at2,
where a is aconstant. 13.) Find the critical points and classify each as a relative maximum,
relative minimum, or saddle point. a.) f(x,y)=x33xy2+3y°"
b.) f(x,y)=3x26xy+y2+12x16y+1
c.) f(x,y)=x2ln(xy)+y2 14.) Find the shortest distance between the planes 2 x + 3 y  z = 2 and
2x+3yz=4. 15.) Find the dimensions of the rectangular parallelepiped of maximum
volume that can be inscribed inside the ellipsoid 16X2+4y2+922 =144. 16.) Determine the minimum surface area of a closed rectangular box
with volume 8 ft.3 17.) Determine the maximum and minimum values of f on the given
region. a.) f(x, y) = (x  1) 2 + (y  2) 2 on the triangle with vertices
(0,0), (0,4), and (5, 0) b.) f(x, y) = x y on the unit circle ...
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 Spring '09
 Kouba
 Calculus

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