Math 243 Spring 2015 (Practice) Final Exam 5/11/2015 Time Limit: 2 hours Name: • No calculators or notes are allowed. • One side of each sheet is blank and may be used as scratch paper. • Check your answers whenever possible. • Show your work clearly. Grade Table (for instructor use only) Question Points Score 1 15 2 20 3 20 4 15 5 15 6 15 7 15 8 20 9 10 10 15 11 20 12 20 Total: 200 Math 243 (Practice) Final Exam - Page 2 of 16 Common trig function values x sin x cos x tan x 0 0 1 0 π 6 1 2 π 4 √ 5/11/2015 The cross product If ~u = ha, b, ci and ~v = hp, q, ri, then ~u × ~v = hbr − cq, cp − ar, aq − bpi. 3 2 √1 3 √1 2 π 3 √1 2 √ 3 2 1 2 1 √ 3 π 2 1 0 undefined Differential geometry d~r ~v • Tˆ = = ds |~v | The 2nd derivative test 2 H = fxx fyy − fxy H<0 saddle H > 0, fxx (or fyy ) < 0 local maximum H > 0, fxx (or fyy ) > 0 local minimum H=0 inconclusive ˆ ˆ ˆ = 1 dT = 1 dT • N dTˆ dt κ ds dt dTˆ dTˆ 1 • κ= = ds |~v | dt Z • arclength = b |~v (t)| dt a Math 243 (Practice) Final Exam - Page 3 of 16 5/11/2015 1. (15 points) Let l be line through (3, 2, 1) that is normal to the plane 2x − y + 2z = −2. Where does l meet the plane? Solution: The vector h2, −1, 2i is normal to the plane, therefore parallel to l. So l can be parametrized as follows: ~r(t) = h3, 2, 1i + th2, −1, 2i. To see where l passes through the plane, plug the parametrization into the equation for the plane: 2(3 + 2t) − (2 − t) + 2(1 + 2t) = −2, and solve it to get t = −8/9. Plugging this value of t back into the parametrization yields the point h11/9, 26/9, −7/9i. Math 243 (Practice) Final Exam - Page 4 of 16 5/11/2015 2. For each of the following limits, find it if it exists, otherwise show why it does not. x3 − y 2 (a) (10 points) lim (x,y)→(0,0) x2 + y 2 Solution: When x = 0, the limit becomes −y 2 = 1. y→0 y 2 lim On the other hand, when y = 0, the limit becomes x3 = lim x = 0, x→0 x→0 x2 lim and so the limit doesn’t exist. (b) (10 points) lim (x,y)→(0,0) x2 xy + y2 + 1 Solution: Plug in to get 0. Math 243 (Practice) Final Exam - Page 5 of 16 3. (a) (10 points) Sketch the level curves of z = g(x, y) = z = 2. 5/11/2015 p x2 − y at z = 0, z = 1, and y 4 z=0 2 x −4 −2 2 z=1 −2 −4 4 z=2 Math 243 (Practice) Final Exam - Page 6 of 16 5/11/2015 (b) (10 points) Sketch the level curves of z = f (x, y) = ex+y at z = 1, z = e, and z = 1/e. y 4 2 x −4 −2 2 4 −2 z=e z=1 −4 z = 1/e Math 243 (Practice) Final Exam - Page 7 of 16 5/11/2015 4. (15 points) Suppose that the acceleration of a particle is given by ~a(t) = ht, 0, t2 i and we know that its initial velocity ~v (0) = h1, 1, 0i and its initial position ~r(0) = h0, 1, 0i. Find ~r(t). Solution: ~r(t) = t3 t4 + t, t + 1, 6 12 Math 243 (Practice) Final Exam - Page 8 of 16 5. (15 points) Find the curvature of the curve ~r(t) = ht, t2 , 2t + 1i at the point (2, 4, 5). Solution: ~v (t) = h1, 2t, 2i, so |~v (t)| = √ 5 + 4t2 . Then 1 Tˆ(t) = √ h1, 2t, 2i, 5 + 4t2 and dTˆ 1 4t =√ h0, 2, 0i − h1, 2t, 2i, 2 dt (5 + 4t2 )3/2 5 + 4t which when t = 2 is equal to the vector 2 h−4, 5, −8i. 213/2 The curvature is √ 2 1 2 5 h−4, 5, −8i = 3/2 . κ= 3/2 |~v (2)| 21 21 5/11/2015 Math 243 (Practice) Final Exam - Page 9 of 16 5/11/2015 6. (15 points) Find a vector equation for the line that is normal to the surface z = ln(x2 + y 2 ) at the point (1, 0, 0). Solution: The given surface is the level surface where g(x, y, z) = ln(x2 +y 2 )−z = 0, so we take its gradient: 2x 2y ~ ∇g = , , −1 = h2, 0, −1i. x2 + y 2 x2 + y 2 (1,0,0) (1,0,0) So an equation for the normal line is ~r(t) = h1, 0, 0i + th2, 0, −1i. Math 243 (Practice) Final Exam - Page 10 of 16 5/11/2015 7. (15 points) The equation x − y 2 + z 3 + sin(xyz) = 3 defines a surface in 3-dimensional space. Find an equation for the tangent plane to this surface at the point (2, 0, 1). Solution: Let g(x, y, z) = x − y 2 + z 3 + sin(xyz). Then ~ ∇g = h1 + yz cos(xyz), −2y + xz cos(xyz), 3z 2 + xy cos(xyz)i(2,0,1) (2,0,1) = h1, 2, 3i, so an equation for the tangent plane is x + 2y + 3z = 5. Math 243 (Practice) Final Exam - Page 11 of 16 5/11/2015 8. Let f (x, y) = 2x2 + y 2 − 8x − 20. (a) (10 points) Classify all local extrema and saddle points (if any) of f . ~ = h4x − 8, 2yi, so the only critical point is (2, 0). fxx = 4 and Solution: ∇f fyy = 2, while fxy = 0, so the Hessian is always positive. Since fxx is also positive, we conclude that (2, 0) is a local minimum. Math 243 (Practice) Final Exam - Page 12 of 16 5/11/2015 (b) (10 points) Now consider f restricted to the domain x2 + y 2 ≤ 25. Find the maximum and minimum values of f on this restricted domain and where they occur. Solution: To find max and min values on the boundary, we can parametrize the circle or use the method of Lagrange multipliers. For the latter, the constraint ~ = h2x, 2yi, which gives us the system of is g(x, y) = x2 + y 2 − 25 = 0, so ∇g equations 4x − 8 = λ2x 2y = λ2y 2 x + y 2 = 25 Multiplying the first by y and the second by x gives us the equation 4xy − 8y = 2xy, and so y(x − 4) = 0, which means that either y = 0 or x = 4. From y = 0, we get the two points (5, 0) and (−5, 0) and from x = 4 we get the points (4, 3) and (4, −3). Finally compute the value of f at all the points we have found so far: x y f 2 0 -28 5 0 -10 -5 0 70 4 3 -11 4 -3 -11 from which we can see that the minimum value of f is −28, attained at (2, 0) while the maximum value is 70, attained at (−5, 0). Math 243 (Practice) Final Exam - Page 13 of 16 5/11/2015 9. (10 points) Find the direction of maximum increase of the function f (x, y, z) = yx2 + z 2 at the point with the coordinates (−1, 2, −2). Find the directional derivative of f at that point in this direction. Solution: ~ ∇f (−1,2,−2) = h2xy, x2 , 2zi(−1,2,−2) = h−4, 1, −4i. The directional derivative is the magnitude of this vector, which is √ 33. Math 243 (Practice) Final Exam - Page 14 of 16 5/11/2015 10. (15 points) Find a direction in which the derivative of f (x, y) = xy + y 2 at (3, 2) is 0. Then find a vector equation for the line that passes through (3, 2) that is parallel to this direction. Solution: ~ ∇f (3,2) = hy, x + 2yi|(3,2) = h2, 7i. f increases the fastest in the direction of this vector, and has a directional derivative of 0 in any direction perpendicular to this vector, for example, in the direction of h7, −2i. The line is then given by the equation ~r(t) = h3, 2i + th7, −2i. Math 243 (Practice) Final Exam - Page 15 of 16 5/11/2015 11. (20 points) You are required to construct a closed rectangular box that encloses a volume of 512 cm3 . What dimensions of the box would minimize its surface area? Solution: Let x, y, and z represent the dimensions of the box. Then g(x, y, z) = xyz − 512 = 0 is the constraint and f (x, y, z) = 2xy + 2xz + 2yz is the surface area we want to minimize. Then ~ = hyz, xz, xyi ∇g and ~ = h2y + 2z, 2x + 2z, 2x + 2yi. ∇f We get the system of equations 2y + 2z 2x + 2z 2x + 2y xyz = λyz = λzx = λxy = 512 Note that x, y, and z must be positive (they represent the lengths of the sides of the box, after all). Multiply the first equation by x and the second by y to get 2xy + 2xz = 2xy + 2yz, and so 2xz = 2yz, which means that x = y (we can cancel z, because we know it is nonzero). Similarly, we can conclude from the second and third equations that y = z, so x = y = z, which means they all equal 8 cm. Math 243 (Practice) Final Exam - Page 16 of 16 5/11/2015 12. (20 points) Use the Lagrange multiplier method to find the point (or points) on the curve 2y = x2 nearest to the point (0, 3). Clearly state the function f (x, y) whose extrema you wish to find, and the function g(x, y) in the constraint g(x, y) = 0. Explain why there is no point on the curve that is farthest from (0, 3). Solution: Let f (x, y, z) = x2 + (y − 3)2 , which the square of the distance between the point (x, y) and (0, 3), and let g(x, y) = 2y − x2 . Then ~ = h2x, 2y − 6i ∇f and ~ = h−2x, 2i, ∇g so we get the system of equations 2x = −λ2x 2y − 6 = λ2 2y = x2 Eliminating λ from the first and second equations gives us 2x = −x(2y − 6), which can be rearranged to give x(y − 2) = 0. From x = 0, we get the point (0, 0) and from y = 2, the points (2, √ 2) and (−2, 2). Of these (2, 2) and (−2, 2) are at the minimum distance, which is 5. There is no farthest point, because the graph of 2y = x2 is unbounded.
© Copyright 2024