Calculus Volume 3

# Review Exercises

Calculus Volume 3Review 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 the 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.

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