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Mathematics LibreTexts

5.10: Chapter 5 Review Exercises

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

1. The domain of f(x,y)=x3arcsin(y) is {(x,y)|xR and πyπ}.

2. If the function f(x,y) is continuous everywhere, then fxy(x,y)=fyx(x,y).

Answer
True, by Clairaut’s theorem

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

4. (34,916) is a critical point of g(x,y)=4x32x2y+y22.

Answer
False

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

5. f(x,y)=e(x2+2y2)

6. f(x,y)=x+4y2

Answer
Contour Plot for function z = x + 4y^2

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

7. lim(x,y)(1,1)4xyx2y2

8. lim(x,y)(0,0)4xyx2y2

Answer
Does not exist.

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

9. f(x,y)=x3arcsiny

10. g(x,y)=ln(4x2y2)

Answer
Continuous at all points on the xy-plane, except where x2+y2>4.

For the following exercises, find all first partial derivatives.

11. f(x,y)=x2y2

12. u(x,y)=x43xy+1, with x=2t and y=t3

Answer
ux=4x33y,

uy=3x,

dxdt=2 and dydt=3t2

dudt=uxdxdt+uydydt=8x36y9xt2=8(2t)36(t3)9(2t)t2=64t36t318t3=40t3

For the following exercises, find all second partial derivatives.

13. g(t,x)=3t2sin(x+t)

14. h(x,y,z)=x3e2yz

Answer
hxx(x,y,z)=6xe2yz,
hxy(x,y,z)=6x2e2yz,
hxz(x,y,z)=3x2e2yz2,
hyx(x,y,z)=6x2e2yz,
hyy(x,y,z)=4x3e2yz,
hyz(x,y,z)=2x3e2yz2,
hzx(x,y,z)=3x2e2yz2,
hzy(x,y,z)=2x3e2yz2,
hzz(x,y,z)=2x3e2yz3

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

15. z=x32y2+y1 at point (1,1,1)

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

Answer
z=x2y+5

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

18. Find the differential dz of h(x,y)=4x2+2xy3y and approximate Δz at the point (1,2). Let Δx=0.1 and Δy=0.01.

Answer
dz=4dxdy,dz(0.1,0.01)=0.39,Δz=0.432

19. Find the directional derivative of f(x,y)=x2+6xyy2 in the direction v=ˆi+4ˆj.

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

Answer
38527,6

For the following exercises, find the gradient.

21. c(x,t)=e(tx)2+3cost

22. f(x,y)=x+y2xy

Answer
f(x,y)=x+2y22x2yˆi+(1x+1xy2)ˆj

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

23. z=x3xy+y21

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

24. f(x,y)=x2y, subject to the constraint: x2+y2=4

Answer
maximum: 1633, minimum: 1633,

25. f(x,y)=x2y2, subject to the constraint: x+6y=4

26. A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of 5% in height and 2% in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height 6 cm and radius 2 cm.

Answer
2.3228 cm3

27. 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 2 ft/sec and 3 ft/sec, respectively. Find the rate at which the liquid level is rising when the length is 14 ft, the width is 10 ft, and the height is 4 ft.


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