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MATH 226 SAMPLE MIDTERM 1-SOLUTIONS Fall 2014 1. Let f (x, y) = p y − x2 . Find the domain of f , and draw several level curves. The domain is {(x, y) ∈ R2 : y ≥ x2 }. The level curves f (x, y) = c are parabolas y = x2 + c2 . 2. Decide whether each of the sets below is open, closed, or neither. (a) {(x, y) ∈ R2 : 0 ≤ x < 1, 0 ≤ y < 2} – neither (b) {(x, y) ∈ R2 : x + y < 2} – open 3. Find the area of the triangle in R3 with vertices (1, 0, 0), (4, 0, 1), (1, 2, −1). Denote the vertices (1, 0, 0), (4, 0, 1), (1, 2, −1) by P, Q, R. Then the area is 12 kP~Q × P~Rk. We have: i j k P~Q × P~R = h3, 0, 1i × h0, 2, −1i = 3 0 1 = −2i + 3j + 6k. 0 2 −1 √ √ Hence the area is 21 4 + 9 + 36 = 12 49 = 72 . 4. Find the scalar parametric equations of the line which passes through the point (4, 5, −2) and is perpendicular to the plane through the three points (1, 0, 1), (3, 2, 0), (−1, 1, 2). The line should be parallel to the normal vector to the plane in the plane by P, Q, R. Then the normal vector is i n = P~Q × P~R = h2, 1, −1i × h−2, 1, 1i = 2 −2 question. Denote the three points in j k 1 −1 = 3i + 6k. 1 1 Hence the vector parametric equation is r = (4 + 3t)i + 5j + (6t − 2)k, and the scalar parametric equations are x = 4 + 3t, y = 5, z = 6t − 2. 5. Let (a) Find 3 x + xy − y 3 f (x, y) = x2 + y 2 0 if (x, y) 6= (0, 0) if (x, y) = (0, 0) ∂f ∂f (0, 0) and (0, 0). ∂x ∂y We have f (x, 0) = −y 3 ∂f x3 ∂f = x and f (0, y) = = −y, hence (0, 0) = 1 and (0, 0) = −1. 2 2 x y ∂x ∂y (b) Does f have a limit at (0, 0)? Explain your answer. From part (a), limx→0 f (x, 0) = limy→0 f (0, y) = 0. But on the other hand, limx→0 f (x, x) = x2 1 limx→0 2x 2 = 2 . Since f approaches different values as (x, y) approaches (0, 0) along different trajectories, the limit does not exist. 6. Find all values of (a, b) such that the tangent plane to the surface z = 4x2 − y 2 at (a, b, 4a2 − b2 ) is parallel to the line with parametric equations x = t + 1, y = 2t, z = −4t + 9. Let f (x, y) = 4x2 −y 2 , then fx (a, b) = 8a and fy (a, b) = −2b. Hence the tangent plane at (a, b, 4a2 −b2 ) is perpendicular to the vector n = −8ai + 2bj + k. We want n to be perpendicular to the direction vector v of the given line. From the equations of the line, v = i + 2j − 4k. We therefore want n · v = 0, so that −8a + 4b − 4 = 0, or simplifying, −2a + b = 1. All values (a, b) with b = 1 + 2a satisfy this condition.