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Review Exercises

Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

394.

The domain of f(x,y)=x3sin−1(y)f(x,y)=x3sin−1(y) is x=x= all real numbers, and −π≤y≤π.−π≤y≤π.

395.

If the function f(x,y)f(x,y) and all its second derivatives are continuous everywhere, then fxy=fyx.fxy=fyx.

396.

The linear approximation to the function of f(x,y)=5x2+xtan(y)f(x,y)=5x2+xtan(y) at (2,π)(2,π) is given by L(x,y)=22+21(x−2)+(y−π).L(x,y)=22+21(x−2)+(y−π).

397.

(34,916)(34,916) is a critical point of g(x,y)=4x3−2x2y+y2−2.g(x,y)=4x3−2x2y+y2−2.

For the following exercises, sketch the function in one graph and, in a second, sketch several level curves.

398.

f ( x , y ) = e − ( x 2 + 2 y 2 ) . f ( x , y ) = e − ( x 2 + 2 y 2 ) .

399.

f ( x , y ) = x + 4 y 2 . f ( x , y ) = x + 4 y 2 .

For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it.

400.

lim ( x , y ) → ( 1 , 1 ) 4 x y x − 2 y 2 lim ( x , y ) → ( 1 , 1 ) 4 x y x − 2 y 2

401.

lim ( x , y ) → ( 0 , 0 ) 4 x y x − 2 y 2 lim ( x , y ) → ( 0 , 0 ) 4 x y x − 2 y 2

For the following exercises, find the largest region of continuity for the function.

402.

f ( x , y ) = x 3 sin −1 ( y ) f ( x , y ) = x 3 sin −1 ( y )

403.

g ( x , y ) = ln ( 4 − x 2 − y 2 ) g ( x , y ) = ln ( 4 − x 2 − y 2 )

For the following exercises, find all first derivatives, full or partial, as appropriate.

404.

f ( x , y ) = x 2 − y 2 f ( x , y ) = x 2 − y 2

405.

u ( x , y ) = x 4 − 3 x y + 1 , x = 2 t , y = t 3 u ( x , y ) = x 4 − 3 x y + 1 , x = 2 t , y = t 3

For the following exercises, find all second partial derivatives.

406.

g ( t , x ) = 3 t 2 − sin ( x + t ) g ( t , x ) = 3 t 2 − sin ( x + t )

407.

h ( x , y , z ) = x 3 e 2 y z h ( x , y , z ) = x 3 e 2 y z

For the following exercises, find an equation of the tangent plane to the specified surface at the given point.

408.

z=x3−2y2+y−1z=x3−2y2+y−1 at point (1,1,−1)(1,1,−1)

409.

z=ex+2yz=ex+2y at point (0,1,3)(0,1,3)

410.

Approximate f(x,y)=ex2+yf(x,y)=ex2+y at (0.1,9.1).(0.1,9.1). Write down your linear approximation function L(x,y).L(x,y). How accurate is the approximation to the exact answer, rounded to four digits?

411.

Find the differential dzdz of h(x,y)=4x2+2xy−3yh(x,y)=4x2+2xy−3y and approximate ΔzΔz at the point (1,−2).(1,−2). Let Δx=0.1Δx=0.1 and Δy=0.01.Δy=0.01.

412.

Find the directional derivative of f(x,y)=x2+6xy−y2f(x,y)=x2+6xy−y2 in the direction v=i+4j.v=i+4j.

413.

Find the maximal directional derivative magnitude and direction for the function f(x,y)=x3+2xy−cos(πy)f(x,y)=x3+2xy−cos(πy) at point (3,0).(3,0).

For the following exercises, find the gradient.

414.

c ( x , t ) = e ( t − x ) 2 + 3 cos ( t ) c ( x , t ) = e ( t − x ) 2 + 3 cos ( t )

415.

f ( x , y ) = x + y 2 x y f ( x , y ) = x + y 2 x y

For the following exercises, find and classify the critical points.

416.

z = x 3 − x y + y 2 − 1 z = x 3 − x y + y 2 − 1

For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints.

417.

f ( x , y ) = x 2 y , x 2 + y 2 = 4 f ( x , y ) = x 2 y , x 2 + y 2 = 4

418.

f ( x , y ) = x 2 − y 2 , x + 6 y = 4 f ( x , y ) = x 2 − y 2 , x + 6 y = 4

419.

A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of 5%5% in height and 2%2% in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height 66 cm and radius 22 cm.

420.

A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of 22 ft/sec and 33 ft/sec, respectively. Find the rate at which the liquid level is rising when the length is 1414 ft, the width is 1010 ft, and the height is 44 ft.