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MATH 2720 MIDTERM 2 QUESTIONS AND ANSWERS ALL GRADED A 2020

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MATH 2720 MIDTERM 2 QUESTIONS AND ANSWERS Most of you did well on the course questions. These are “easy” questions that furthermore are important. 1. Course questions. [5] (a) State the definition of a function continuous at a ∈ Rn. [5] (b) State the definition of the arc length function. [10] 2. Consider the function f (x, y) = y sin 1 , x = 0 x 0, x = 0. 3. Consider the cycloid with equation c(t) = (t − sin t)i + (1 − cos t)j. [5] (a) What are the domain and range of c? [5] (b) Is c smooth? [5] (c) Show that the (orthogonal) projection of c on j is 2π-periodic, i.e., that projjc(t + 2π) = projjc(t) for all t ∈ R. [5] (d) Sketch c for t ∈ [−3π, 3π]. [5] (e) Compute the arc length function of c for t ≥ 0. [5] (f) Compute the curvature of c. Is this curvature defined for all t? Figure 1: The cycloid c(t) = (t sin t)i + (1 cos t)j when t [ 3π, 3π]. (Note that the coordinate system is not orthonormal (axes are not to proportion.) Plotted using the command plot([t-sin(t),1-cos(t),t=-3Pi..3Pi]) in Maple. Figure 2: The function t − sin t. [10] 4. Consider the system 3x + 2y + z2 + u + v2 = 0 4x + 3y + u2 + v + w + 2 = 0 x + z + w + u2 + 2 = 0. Can this system be solved for u, v, w in terms of x, y, z near x = y = z = 0, u = v = 0 and w = −2? [10] 5. Sketch the region of integration and evaluate both as indicated and as dy dx the integral ∫ 2 ∫ y2 x2 + y dx dy. Figure 3: The domain for the integral in Exercise 5. As an x-simple region, this is D = {(x, y) ∈ R2; −3 ≤ y ≤ 2, 0 ≤ x ≤ y2}. As a y-simple reg√ion, it is the union of two regions: the orange region, D1 = {(x, y) ∈ R2; 0 ≤ x√≤ 4, x ≤ y ≤ 2}, and the green region, D2 = {(x, y) ∈ R2; 0 ≤ x ≤ 9, −3 ≤ y ≤ − x}. [5] 6. (a) Show that the change of variables formula for cylindrical coordinates is given by ∫∫∫ T (W ∗) f (x, y, z) dV = ∫∫∫ f (r cos θ, r sin θ, z) r dr dθ dz, where W ∗ is an elementary region in rθz-space and T : (r, θ, z) (x, y, z) = (r cos θ, r sin θ, z). [10] (b) Use part (a) to integrate (x2 + y2)z2 over the part of the cylinder x2 + y2 1 inside the sphere x2 + y2 + z2 = 4. 7. Bonus question (5 marks). Recall that a function f : X → Y is injective (or one-to-one) if x1, x2 X, f (x1) = f (x2) x1 = x2, i.e., no two points in the domain of f have the same image. Consider the linear transformation T : R2 → R2 u ›→ Au, where u R2 and A a 2 2 matrix with entries aij, i, j = 1, 2. Show that T is injective if and only if det(A) = 0. [Hint: This is easy with MATH 1300-type ideas.] 8. Bonus question (5 marks). Show that if f : R3 → R is C2, then ∇ × ∇f = 0. ∂ ∂ ∂ ∂x ∂f ∂y ∂f ∂z ∂f

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