3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane?

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1 Math 4 Practice Problems for Midterm. A unit vector that is perpendicular to both V =, 3, and W = 4,, is (a) V W (b) V W (c) 5 6 V W (d) 3 6 V W (e) 7 6 V W. In three dimensions, the graph of the equation x + y + z 4y + 8z + 6 = 0 is (a) A sphere centered at the origin of radius (b) A sphere centered at the point (0, 4, 8) of radius (c) A sphere centered at the point (0,, 4) of radius (d) A sphere centered at the point (0, 4, 8) of radius 4 (e) A sphere centered at the point (0,, 4) of radius 3. The three points (, 4, ), (,, ) and (5,, ) determine a plane. Which of the following points is in that plane? (a) (,, ) (b) (,, ) (c) (3, 0, 0) (d) (, 3, 0) (e) (0, 0, 5) 4. The slope of the line tangent to the graph of x = ln t, y = t at the point (0,) is (a) 0 (b) (c) (d) /e (e) /e 5. The line of intersection of the planes 3x + y z = 0 and x + y + z = 4 is given by the equations: (a) x = t +, y = t, z = t (b) x + (c) x = t, y = t + 6, z = t (d) x (e) x = t, y = t, z = t 6 = y 6 = z 3 6 = y + 6 = z 6. Let V =,, and W = 0,,. For which constants a and b is the vector av + bw orthogonal to the vector U = 4,,? (a) any value of a, but b must be zero (b) any value of b, but a must be zero (c) a must be equal to b (d) a must be equal to b (e) a must equal b 7. Draw a sketch of the part of the curve given parametrically by x = 6t, y = 8t 3 for t. Then find the length of the curve.

2 8. Consider the cycloid x = t sin t, y = cos t. Let L be the line tangent to the cycloid for t = π/ and let L be the line tangent to the cycloid for t = 3π/. Find the point where these two lines intersect. 9. Let f(x, y) be a (smooth) function and suppose f x = 3x y +. Which of the following could y be f y? (A) 3x y (D) x 3 x y + sin y (B) x 3 y + x y (E) x3 y + tan y y (C) 3x y + ln y The length of the curve given parametrically by x = 3t, y = 4t 3 for 0 < t < 0 is (A) 48 (B) 96 (C) 9 (D) 364 (E) 78. Let f(x, y) = (x +y +z ) α. For which value(s) of α is it true that f xx +f yy +f zz = 0 (except for possibly at the origin)? (A) Only for α = 0 (B) Only for α = (C) For α = 0 or (D) For α =. Calculate or show that it does not exist. 3. Calculate or show that it does not exist. or 0 (E) For α = or 0 x 3 y 3 x y lim (x,y) (0,0) x + y lim (x,y) (0,0) x y x 4 + y 4. Find the tangential and normal compoents of acceleration for the motion of a particle given by ln(t + ), t arctan t. Write the acceleration vector in the form a T T + a N N. 5. Calculate the curvature of the plane curve t + 3, 5 t. 6. Consider the conical helix given parametrically by r = at, θ = t, z = bt, where r, θ and z are the usual cylindrical coordinates (this means that r and θ are polar coordinates in the xy-plane and z is what it usually is). What is the length of the part of the curve traced between t = 0 and t = T?

3 7. The area of the triangle (not the parallelogram) whose vertices are (,,3), (,0,5) and (4,,) is (a) 37 (b) 5 (d) (e) 5 5 (c) What is the largest value of A that makes the following statement true? The sphere x + y + z x + 4y + 4z = a is contained entirely in the interior of the sphere x + y + z = 64 provided a < A. (a) A = (b) A = (c) A = 4 (d) A = 5 (e) A = 6 9. The set of points that are equidistant from the two points (,,3) and (5,, ) is a plane. Find the equation of this plane. (a) x y z = 4 (b) 4x y z = (c) x + y + 3z = 9 (d) 5x y + z = 5 (e) 6x + 4z = 8 0. The curve given parametrically by x = t 3, y = t 6 intersects the graph given parametrically by x = + cos t, y = + sin t at the point (, ). What is the angle between the tangent lines to the two graphs at that point? (a) 0 (b) π (c) π 3 (d) π 4 (e) π 6. Calculate the length of the curve given parametrically by x = e t cos t, y = e t sin t, z = e t for 0 t ln. (Work carefully, the integrand really does simplify!) (a) 3 (b) 3 (c) 6 (d) 6 ln (e) e 3. Which of the following statements is true about the curvature κ(x) of the graph of y = x 4 /4? (a) The maximum of κ occurs at x = 0 and the minimum is never achieved (except in the limit as x ±. (b) The minimum of κ occurs at x = 0 and the maximum is never achieved (except in the limit as x ± ). (c) The maximum of κ occurs at ± 6 7/3 and the minimum occurs at x = 0. 3

4 (d) The maximum of κ occurs at ± 6 3/7 and the minimum occurs at x = 0. (e) The maximum of κ occurs at ± 6 7/3 and the minimum is never achieved (except in the limit as x ± ). 3. Suppose f is a differentiable function of x and y and that Which of the following could be f/ y? f x = x cos y. (a) y cos x (b) x sin y (c) x sin y (d) x cos y (e) sin y 4. Suppose z = x + e xy. Which of the following equations is true? (a) x + y = (d) x y y = + x (b) x x + y y = (e) x x y y = x (c) x + x y = x 5. At time t = 0, a particle is at point r = (, 3, ), its velocity is v = (,, ) and its acceleration is a = (0, 0, ). What is the curvature of the particle s path at that point? 5 (a) (b) (c) (d) 5 4 (e) 7 7 4

5 Matching: match each equation with its graph A. B. C. D. E. () x y z = () x + y z = (3) x y z = (4) x + y z = (5) x + y + z = 5