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REVIEW QUESTIONS FOR MIDTERM EXAM
Question 1. Which number is in the range of the function f px, yq “ 5.25 ` sinpx3 ` y2 ` 1.23q? A 2.16 B 4.21 C 6.19 D 8.23 E None of the others
Question 2. Which function whose domain is enclosed by an ellipse? A
f px, yq “ coshp2 ´ x2 ´ 3y2q B
f px, yq “ lnp7 ´ 5x2 ´ 2y2q ? C
f px, yq “ sinp5 ´ 2x2 ´ y2q D None of the others E
f px, yq “ 4 ` x2 ´ y2 Bf Bf
Question 3. Let f px, yq be a function so that
p5, 7q “ 2.15 and
p5, 7q “ 3.67. Consider the Bx By Bg
function gps, tq “ fpt2 ´ 2st, 3t ` s2q. Evaluate p´2, 1q. Bs Bg Bg A None of the others B
p´2, 1q “ ´18.88 C
p´2, 1q “ ´18.92 Bs Bs Bg Bg D
p´2, 1q “ ´18.96 E
p´2, 1q “ ´18.98 Bs Bs
Question 4. Let zpx, yq be a function defined implicitly from the equation 5xz3 `4xyz “ z5 `xz `1. Bz Evaluate p0, 1q. Bx Bz Bz A p0, 1q “ ´2.6 B None of the others C p0, 1q “ ´1.6 Bx Bx Bz Bz D p0, 1q “ ´2.4 E p0, 1q “ ´1.8 Bx Bx
x3 sinpy ´ zq
Question 5. Let f px, y, zq “ . Evaluate f , q 1 ` cospx2 ` yzq xp1, 2 3 . A
fxp1, 2, 3q “ ´1.7987 B
fxp1, 2, 3q “ ´1.7887 C None of the others D
fxp1, 2, 3q “ ´1.7997 E
fxp1, 2, 3q “ ´1.7897
Question 6. What is the equation of the tangent plane to the paraboloid z “ x2 ` y2 ` xy at the point M p1, 2q. A
z “ 5x ` 4y ´ 7 B
z “ 5x ` 4y ´ 14 C
z “ 4x ` 5y ´ 14 D
z “ 4x ` 5y ´ 7 E None of the others
Question 7. Let pP q be the tangent plane to the paraboloid z “ 4x2`5y2´3 at the point Mp1, 2, 21q.
Let pCq be the intersection between pP q and the cylinder 4x2 ` 4y2 “ 1. The projection of pCq onto
the plane Oxy is a circle. Evaluate its radius. A 0.5 B 0.25 C 2 D 4 E None of the others
Question 8. Let f px, yq “ 2x3 ´ 3xy2 and Mp3, 2q. Let Ý Ñ
u be the unit direction so that the rate of
change of the function f at M in this direction is maximal among all unit directions. Find Ý Ñ u . A None of the others B Ý Ñ
u “ p0.7518, ´0.6578q C Ý Ñ
u “ p0.7535, ´0.6528q D Ý Ñ
u “ p0.7576, ´0.6580q E Ý Ñ
u “ p0.7593, ´0.6508q 1
Question 9. Find the directional derivative of the function f px, y, zq “ x3y ` z2 ` 2xyz at the point
p1, ´1, 2q in the direction of the vector Ý Ñ
u “ p3, 4, ´5q. Bf Bf Bf A “ ´1.4752 B “ ´1.4983 C “ ´1.5318 BÝ Ñ u BÝ Ñ u BÝ Ñ u Bf D None of the others E “ ´1.5556 BÝ Ñ u
Question 10. Let f px, yq be a function such that fp4, 2q “ 0.3, fxp4, 2q “ ´0.6, fyp4, 2q “ 1.4. Use
the linear approximation to obtain an approximation for f p4.01, 1.98q. A 0.266 B 0.288 C 0.334 D 0.322 E None of the others
Question 11. Let f px, yq “ expy2 ´ x2q. Which of the following sentences is correct? A None of the others B
f has a local maximum at p´2, 0q and p0, 0q is a saddle point C
f has a local minimum at p´2, 0q and p0, 0q is a saddle point D
f has a local maximum at p2, 0q and p0, 0q is a saddle point E
f has a local minimum at p2, 0q and p0, 0q is a saddle point
Question 12. Find the point on the plane x ´ y ` z “ 3 that is closest to the point p1, 0, ´1q. A p2, 1, 2q B p2, ´1, 0q C p1, ´1, 1q D p´1, 2, 6q E None of the others
Question 13. Find the maximum value of the function f px, yq “ x3 ´ 6y2 on the disk x2 ` y2 ď 1. A None of the others B ´5 C ´6 D 0 E 1 ij
Question 14. Evaluate the double integral I “
px2`2yq dA where D is the region bounded by D
the curve y “ 4 ´ x2 and the x-axis. 64 128 128 64 A I “ B I “ C I “ D None of the others E I “ 3 3 15 15 ij
Question 15. Evaluate the double integral I “
px2`y2`xyq dA where D is the region bounded D
by the curve x2 ` y2 “ 2x and the lines y “ 0 and y “ x. 5π 19 5π 19 3π 19 A I “ ` B I “ ` C I “ ` 8 24 8 12 8 12 3π 19 D I “ ` E None of the others 8 24
Question 16. Let D be the lamina bounded by the triangle with vertices p0, 0q, p5, 0q and p0, 5q and
has the density function ρpx, yq “ xy. Evaluate the mass of the lamina. 325 425 A M “ B None of the others C M “ 24 24 625 725 D M “ E M “ 24 24 2 ANSWER: 01. C 02. B 03. E 04. C 05. A 06. D 07. A 08. D 09. E 10. A 11. C 12. B 13. E 14. B 15. C 16. D 3