HABIB UNIVERSITY
Math 102 Test 2
Spring Semester 2019
INSTRUCTIONS:
Please show all your work wherever possible and attempt all questions. You may use a calculator, but do show
the work and explain your thinking wherever possible / applicable. For MCQs, circle only ONE answer. You
have 60 minutes. Good luck!
1. A plane is drawn through the points P = (2, 1, 0), Q = (0, 1, 3) and R = (1, 0, 1).
(a) Which of the following is a normal vector to the plane? [2]
! ! ! ! ! ! ! ! ! ! ! !
A. ! 3i + j + 2k B. ! 3i − j + 2k C. ! 2i − j + 3k D. ! −3i − j − 2k
(b) Which of the following is the equation of the plane containing P, Q, and R? [2]
5 − 3x + y
A. ! 3x + y + 2z = 5 B. ! z = C. ! 2x − y + 3z = 5
2
2. The contour plot of a function f is given.
(a) Use the contour plot to determine the signs
(negative or positive) of! f x (2,1) and f y (2,1). [2]
f x (2,1) is ___________
f y (2,1) is ___________
(b) Use the plot to estimate the value of ! f y (2,1) . [2]
(c) Mark an X on the contour for where you think there is a local minimum on the function f. [1]
, 3. Given a particular function, f (x, y), which of the following is the expression for the derivative at f (2, 1)
! !
in the direction of vector! i − j ? [2]
f y (2,1)
( f (2,1)) ( f (2,1)) f x (2,1)
2 2
A. ! f x (2,1) − f y (2,1) B. ! x
+ y
C. ! −
2 2
4. Which of the following is the normal vector to the curve ! x 2 − y 2 = 3 at the point (2, 1)? [2]
! ! ! ! ! ! ! !
A. ! i − j B. ! 4i − j C. ! 4i + 2 j D. ! 4i − 2 j
1
5. Find an equation of the tangent plane to the surface ! z − = 0 at the point (1, 1, 1). [4]
xy
Math 102 Test 2
Spring Semester 2019
INSTRUCTIONS:
Please show all your work wherever possible and attempt all questions. You may use a calculator, but do show
the work and explain your thinking wherever possible / applicable. For MCQs, circle only ONE answer. You
have 60 minutes. Good luck!
1. A plane is drawn through the points P = (2, 1, 0), Q = (0, 1, 3) and R = (1, 0, 1).
(a) Which of the following is a normal vector to the plane? [2]
! ! ! ! ! ! ! ! ! ! ! !
A. ! 3i + j + 2k B. ! 3i − j + 2k C. ! 2i − j + 3k D. ! −3i − j − 2k
(b) Which of the following is the equation of the plane containing P, Q, and R? [2]
5 − 3x + y
A. ! 3x + y + 2z = 5 B. ! z = C. ! 2x − y + 3z = 5
2
2. The contour plot of a function f is given.
(a) Use the contour plot to determine the signs
(negative or positive) of! f x (2,1) and f y (2,1). [2]
f x (2,1) is ___________
f y (2,1) is ___________
(b) Use the plot to estimate the value of ! f y (2,1) . [2]
(c) Mark an X on the contour for where you think there is a local minimum on the function f. [1]
, 3. Given a particular function, f (x, y), which of the following is the expression for the derivative at f (2, 1)
! !
in the direction of vector! i − j ? [2]
f y (2,1)
( f (2,1)) ( f (2,1)) f x (2,1)
2 2
A. ! f x (2,1) − f y (2,1) B. ! x
+ y
C. ! −
2 2
4. Which of the following is the normal vector to the curve ! x 2 − y 2 = 3 at the point (2, 1)? [2]
! ! ! ! ! ! ! !
A. ! i − j B. ! 4i − j C. ! 4i + 2 j D. ! 4i − 2 j
1
5. Find an equation of the tangent plane to the surface ! z − = 0 at the point (1, 1, 1). [4]
xy
