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Unformatted text preview: ESP 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=t3-t2. 8.) Find awe» and aw/as where w=f(3t2-s) 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 x-zX - y.zy = z. 12.) Assume that f and g are twice differentiable functions. Show that u =f(x+at)+g(x-at) 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)=x3-3xy2+3y°" b.) f(x,y)=3x2-6xy+y2+12x-16y+1 c.) f(x,y)=x2-ln(xy)+y2 14.) Find the shortest distance between the planes 2 x + 3 y - z = 2 and 2x+3y-z=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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