MATH 2ZZ3 Midterm #2 PDF

LAST (family) NAME: MIDTERM #2 FIRST (given) NAME: MATH 2ZZ3 ID # : Section # : Instructor: Subhajit Mishra...

LAST (family) NAME: MIDTERM #2 FIRST (given) NAME: MATH 2ZZ3 ID # : Section # : Instructor: Subhajit Mishra Version 1 Test Duration: 19:15 - 20:50 July 20, 2023 Instructions: This exam consists of 17 problems on 12 pages. Pages 3 - 9 contain all the questions. There is only one correct answer for each question. It should be entered into the Scantron sheet provided using only an HB pencil. The instructions for using the Scantron sheet are on page 2. Each question is worth of 1 mark. There is no penalty or negative marking for wrong answers. No books, notes, cheat sheets or electronic devices are allowed except McMaster-accredited calculator. Pages 10 - 11 of the test are for the scratch works. There is a formula sheet on the last page. Good Luck! Version 1 Page 1 of 12 Version 1 Page 2 of 12  1. Let f : R3 → R be a function given by f (x, y, z) = x2 + y2 − z2 + 4. Find the equation of level surface of f that passes through the point (0, 0, 0). (A) z2 = x2 + y2 − 4 (B) z2 = x2 + y2 + 4 (C) x2 = z2 + y2 (D) z2 = x2 + y2 (E) y2 = x2 + z2 2. Use the surface you have found in the previous question, to compute equation of the tangent plane to the surface at the point (1, 0, 1). (A) x = y + 1 (B) y = x + z (C) y + 1 = z (D) x = z (E) x = y + z 3. Let F : R3 → R3 be a vector field given by F(x, y, z) = 2xy2 z3 i + 2x2 yz3 j + 3x2 y2 z2 k. Let ϕ : R3 → R be the potential function such that F = ∇ϕ and ϕ(0, 0, 0) = 1. Find the value of ϕ(1, 1, 1). (A) 4 (B) 0 (C) 3 (D) 1 (E) 2 Version 1 Page 3 of 12  R 4. Use the vector field F from the previous question to find F · dr, where C is the curve
C parametrized by r(t) = t2 , t3 , sin πt2 , 0 ≤ t ≤ 1. (A) 2 (B) 1 (C) -1 (D) -2 (E) 0 5. The double integral Z 1Z 1 q 1 + y2 dy dx, 0 x can be written, after reversing the order of integration, as an iterated integral of the form Z 1Z h(y) q I= 1 + y2 dx dy. 0 g(y) Which of the following is TRUE for g(y) and h(y)? (A) g(y) = 1, h(y) = y (B) g(y) = 0, h(y) = 1 (C) g(y) = 0, h(y) = −y (D) g(y) = 0, h(y) = y (E) g(y) = −y, h(y) = y Version 1 Page 4 of 12  6. Find the value of the iterated integral I in the previous question. √ 2 2 (A) 3 (B) 1 1 √ (C) (2 2 − 1) 3 1 (D) 3 1 √ (E) ( 2 − 1) 3 7. Which of the following is the polar equation of y = a, for any a , 0? (A) r = a cos θ (B) r = a sec θ (C) r = a sin θ (D) r = a tan θ (E) r = a csc θ 8. Suppose that f (x, y, z) = xyz. Find the unit vector u such that Du f has maximum value at the point (1, 1, 1). (A) (1, 1, 1)   (B) − √1 , − √1 , − √1 3 3 3   (C) √1 , 0, √1 2 2   (D) − √1 , 0, − √1 2 2   (E) √1 , √1 , √1 3 3 3 Version 1 Page 5 of 12  9. Consider the integral √ Z 3Z 9−y2 e−(x 2 +y2 ) dx dy. −3 0 Suppose the integral above is rewritten in terms of the polar coordinates that has the following form Z dZ 3 J= g(r, θ) dr dθ. c 0 Which of the following are the values of c, d (in that order)? (A) 0, 2π π π π 3π (D) 0, π (E) π, 2π (B) − , (C) , 2 2 2 2 10. Which of the following is correct expression for g(r, θ), in the previous question? (B) e−r (C) re−r 2 2 2 2 (A) rer (D) r (E) er
11. Let f : R → R be a function defined as f (t) = h x(t), y(t) , where h : R2 → R is differen- tiable. Which of the following is TRUE for any t ∈ R? ∂h   ∂h   (A) f ′ (t) = x(t), y(t) y′ (t) + x(t), y(t) x′ (t) ∂x ∂y ∂h   (B) f ′ (t) = x(t), y(t) y′ (t) ∂y ∂h   ∂h   (C) f ′ (t) = x(t), y(t) + x(t), y(t) ∂x ∂y ∂h   ∂h   (D) f ′ (t) = x(t), y(t) x′ (t) + x(t), y(t) y′ (t) ∂x ∂y ∂h   (E) f ′ (t) = x(t), y(t) x′ (t) ∂x Version 1 Page 6 of 12 12. Suppose, f (t) = h(x(t), y(t)), x(t) = t, y(t) = t2 and ∂h ∂h ∂h ∂h (1, 2) = 3, (1, 1) = 2, (1, 2) = −4, (1, 1) = 1. ∂x ∂x ∂y ∂y Evaluate f ′ (1). (A) 0 (B) 2 (C) 6 (D) 4 (E) 10 13. Find a function g(z) such that the vector field F(x, y, z) := (y, x + g(z), 7yz6 ) satisfies ∇ × F = (0, 0, 0). (A) 7z6 (B) 7z7 (C) z7 (D) 3z7 (E) z6 − z 14. Let f : R3 → R be a function given by f (x, y, z) = xy2 z + g(z), where g is a real-valued differentiable function. Find the directional derivative of f at the point (2, 2, 2) along the direction of the vector (2, 0, 0). (A) 2 (B) 16 (C) 8 (D) 0 (E) 4 Version 1 Page 7 of 12 15. Let f (x, y, z) = xyz be a scalar function and F(x, y, z) = (x, y, z) be a vector field of a fluid flow. Which of the following statement(s) is(are) TRUE at the point (1, 1, 1)? (i) ∇ × ∇ f = 1 (ii) ∇ · (∇ × F) = 0 (iii) F is incompressible. (A) (iii) only (B) (ii) only (C) (ii) and (iii) only (D) None of the choices (E) (i) only Version 1 Page 8 of 12 16. Let C be a curve given by r(t) = (4 sin t, 4 cos t), for t ∈ [0, π]. Which of the following describes the curve C with the correct orientation? y y 4 4 2 2 (A) −6 −4 −2 2 4 6 x (D) −6 −4 −2 2 4 6 x −2 −2 −4 −4 y y 4 4 2 2 (B) −6 −4 −2 x (E) x 2 4 6 −6 −4 −2 2 4 6 −2 −2 −4 −4 y 4 2 (C) −6 −4 −2 2 4 6 x −2 −4 R 17. Find C xdy − ydx, where C is the curve in the previous question. (A) −64π (B) −8π (C) −32π (D) −16π (E) −4π Version 1 Page 9 of 12  SCRATCH Version 1 Page 10 of 12  SCRATCH Version 1 Page 11 of 12  Some Formulas You may find Useful i j k ∇×F = ∂ ∂ ∂ , for F = (P, Q, R). ∂x ∂y ∂z P Q R ∂P ∂Q ∂R ∇·F = + + ∂x ∂y ∂z Z Z b F · dr = F(r(t)) · r′ (t) dt, a ≤ t ≤ b C a 1 1 sec θ = , csc θ = cos θ sin θ sin θ 1 tan θ = , cot θ = cos θ tan θ sin2 θ + cos2 θ = 1 Z Z b f ds = f (r(t)) ||(r′ (t))|| dt, a ≤ t ≤ b C a Z t s(t) = ||(r′ (u))|| du, a ≤ t ≤ b a Version 1 Page 12 of 12