Review Problems. Calculus IIIA: page 1 of??

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1 Review Problems The final is comprehensive exam (although the material from the last third of the course will be emphasized). You are encouraged to work carefully through this review package, and to revisit all homework assignments and tests. True or False? Explain! Determine if each of the following is true or false, and give a complete written argument for your response. 1. There is exactly one unit vector parallel to a given nonzero vector u. 2. If v and w are any two vectors, then v + w = v + w. 3. If u v < 0 then the angle between u and v is greater than π/2. 4. If u = 1 then the vector v ( v u) u is perpendicular to u. 5. u v is a vector and u v is a vector. 6. ( i j) k = i ( j k) and ( i j) i = i ( j i). 7. ( i j) k = i ( j k). 8. If v, u, w are all nonzero vectors and v u = v w, then u = w. 9. If v is a nonzero vector and v u = v w, then u = w. 10. It is never true that v w = w v. 11. u v has direction parallel to both u and v. 12. The square of any complex number is a real number. 13. If z = x + iy where x and y are positive, then z 2 = a + ib has a and b positive. 14. For all k R, the equation y +ky = 0 has trigonometric functions as solutions. 15. Polynomials are never solutions to differential equations. 16. A function f(x, y) can be an increasing function of x with y held fixed and be a decreasing function of y with x held fixed. 17. The point (1, 2, 3) lies above the plane z = There is only one point in the yz-plane that is distance 3 from the point (3, 0, 0). 19. There is only one point in the yz-plane that is distance 5 from the point (3, 0, 0). 20. The graphs of f(x, y) = x 2 + y 2 and g(x, y) = 1 x 2 y 2 intersect in a circle. 21. A line parallel to the z-axis can intersect the graph of f(x, y) at most once. 22. A line parallel to the y-axis can intersect the graph of f(x, y) at most once. 23. The linear function f(x, y) = 2x + 3y 5 has exactly one point (a, b) satisfying f(a, b) = The level surfaces of g(x, y, z) = x 2 + y + z 2 are cylinders with axis along the y-axis. 25. A level surface of a function g(x, y, z) cannot be a single point. Calculus IIIA: page 1 of??

2 Review Problems Calculus IIIA: page 2 of?? 26. If all the contours of f(x, y) are parallel lines, then the graph of f is a plane. 27. If f(x, y) = x 2 + y 2 then f y (1, 1) < 0. sin(v) z u 28. The function z(u, v) = u cos(v) satisfies the equation cos(v) z = 1. u v 29. If f(x, y) has f y (x, y) = 0 then f must be a constant. 30. An equation for the tangent plane to the surface z = x 2 + y 2 at the point (1, 1) is z = 2 + 2x(x 1) + 2y 2 (y 1). 31. The tangent plane approximation of the function f(x, y) = ye x2 at the point (1, 0) is f(x, y) y. 32. The local linearization of f(x, y) = x 2 + y 2 at (1, 1) gives an overestimate of the value of f(x, y) at the point (1.04, 0.95). 33. If two function f and g have the same tangent plane at a point (1, 1), then f = g. 34. The gradient vector f(a, b) is tangent to the contour of f at (a, b). 35. If you know the gradient vector of f at (a, b) then you can find the directional derivative f u (a, b) for any unit vector u. 36. The directional derivative f u (a, b) is parallel to u. 37. At the point (3, 0), the function g(x, y) = x 2 + y 2 has the same maximal rate of increase as that of the function h(x, y) = 2xy. 38. If P 0 is a critical point of f, then P 0 is either a local maximum or local minimum. 39. If P 0 is a local maximum or local minimum of f and not on the boundary of the domain of f, then P 0 is a critical point of f. 40. The function f(x, y) = x 2 + y 2 has a global minimum on the region x 2 +y 2 < The function f(x, y) = x 2 +y 2 has a global maximum on the region x 2 +y 2 < If P and Q are two distinct points in 2-space and f has a global maximum at P, then f cannot have a global maximum at Q. 43. The function f(x, y) = sin(1 + e xy ) must have a global minimum in the square region 0 x 1, 0 y The point (2, 1) is a local minimum of f(x, y) = x 2 +y 2 subject to the constraint x + 2y = The parametric curve x = 3t + 2, y = 2t for 0 t < 5 passes through the origin. 46. The parametric curve x = (3t + 2) 2, y = (3t + 2) 2 1 for 0 t 3 is a line. 47. Both x = t + 1, y = 2t and x = 2s, y = 4s + 2 describe the same line. 48. A particle whose motion in the plane is given by r(t) = t 2 i + (1 t) j has the same speed at t = 1 and t = If a particle moves with constant speed, then the path of the particle must be a line. 50. If r(t) for a t b is a parameterized curve, then r( t) for a t b is the same curve traced backward.

3 Calculus IIIA: page 3 of?? Review Problems Exercises 1. Let a and b be the vectors a = < 1, 1 > and b = < 1, 1 >. (a) Find numbers r and s such that v = r a + s b if v = < 2, 1 > (b) Describe the set of vectors { w = s u + t v 1 s 1, 2 t 3} geometrically. (c) Describe the set of vectors { w = s u + t v 2 s 1, 1 t 2} geometrically. (d) Describe the set of vectors { w = 2 u + t v 1 t 1} geometrically. 2. For what values of t are the following pairs of vectors parallel? (a) 2 i + ( t t + 1) j + t k, 6 i + 8 j + 3 k (b) t i + j + (t 1) k, 2 i 4 j + k (c) 2t i + t j + t k, 6 i + 3 j + 3 k. 3. Consider the plane 5x y + 7z = 21. (a) Find a point on the x-axis on this plane. (b) Find two other points on the plane. (c) Find a vector perpendicular to the plane. (d) Find a vector parallel to the plane. 4. Describe the following regions in complex plane. (a) {z = r(cos(θ) + i sin(θ)) 0 r 2, π 6 θ π 3 }. (b) {z = r(cos(θ) + i sin(θ)) 1 r 2, π 6 θ π 4 }. 5. A plane is heading due east and climbing at the rate of 80 km/h. If its airspeed is 480 km/h and there is a wind blowing 100 km/h to the northeast, what is the ground speed of the plane? 6. A large ship is being towed by two tugs. The larger tug exerts a force which is 25% greater than the smaller tug and at an angle of 30 degrees north of east. Which direction must the smaller tug pull to ensure that the ship travels due east? 7. A man wishes to row the shortest possible distance from north to south across a river which is flowing at 4 km/h from the east. He can row at 5 km/h. (a) In which direction should he steer? (b) If there is a wind of 10 km/h from the southwest, in which direction should he steer to try and go directly across the river? What happens? 8. Find the points where the plane z = 5x 4y+3 intersects each of the coordinate axes. Find the lengths of the sides and the angles of the triangle formed by these points.

4 Review Problems Calculus IIIA: page 4 of?? 9. Given the points P = (1, 2, 3), Q = (3, 5, 7) and R = (2, 5, 3) find: (a) A unit vector perpendicular to a plane containing P, Q, R. (b) The angle between PQ and PR. (c) The area of the triangle PQR. (d) The distance from R to the line through P and Q. (e) Find the equation of the plane spanned by P, Q and R. (f) Find the distance between the origin the plane spanned by P, Q and R. 10. (a) A vector v makes an angle α with the positive x-axis, angle β with the positive y-axis, and angle γ with the positive z-axis. Show that v = v cos(α) i + v cos(β) j + v cos(γ) k. (b) cos(α), cos(β), cos(γ) are called direction cosines. Show that cos 2 (α) + cos 2 (β) + cos 2 (γ) = Perform the following calculations. Give your answer in the form z = x + iy. (a) (2 + 3i)(5 + 7i) (b) (1 + i) 4 + (1 i) 4 (c) (e iπ/3 ) 2 (d) 4 10e iπ/2 12. If the roots of the equation x 2 + 2bx + c = 0 are the complex numbers p ± iq, find expressions for p and q in terms of b and c. 13. Find the general solution to the following differential equations. (a) z + 2z = 0 (b) y + 3y + 2y = 0 (c) 4s + 8s + 3s = 0 (d) s + 4s + 4s = Solve the following initial value problems. (a) y + 5y + 6y = 0, y(0) = 5, y (0) = 1. (b) y 3y 4y = 0, y(0) = 1, y (0) = 0. (c) y + 6y + 10y = 0, y(0) = 5, y (0) = Let f be a differentiable function with f x (2, 1) = 3, f y (2, 1) = 4 and f(2, 1) = 7. (a) Give an equation for the tangent plane to the graph of f at x = 2, y = 1. (b) Give an equation for the tangent line to the contour for f at x = 2, y = 1. (c) Near x = 2 and y = 1, how far apart are the contours f(x, y) = 7 and f(x, y) = 7.3?

5 Calculus IIIA: page 5 of?? Review Problems 16. Match the following equations with their graphs. Given reasons for your choice. (a) z = x 3 sin(y) (b) z = x + 2y + 3 (c) z = e x2 y 2 (d) x 2 + y 2 z 2 = 1 (e) x 2 + y 2 z 2 = 1

6 Review Problems Calculus IIIA: page 6 of?? 17. Find a function f(x, y, z) whose level surface f = 1 is the graph of the function g(x, y) = x + 2y. 18. What do the level surfaces of f(x, y, z) = x 2 y 2 + z 2 look like? 19. An experiment to measure the toxicity of formaldehyde yielded the following table. The values show the percent, P = f(t, c), of rats surviving an exposure to formaldehyde at a concentration of c (in parts per million, ppm) after t months. Estimate f t (18, 6) and f c (18, 6). Interpret your answers in terms of formaldehyde toxicity. Time t (months) Conc. c (ppm) Find the linear approximation of the function f(x, y) = x 2 + y 2 at (3, 4) and use it to estimate (3.1) 2 + (3.9) Dieterici s equation of state for a gas is P(V b)e a/rv T = RT, where a, b and R are constants. Regard volume V as a function of temperature T and pressure P and prove that V T = R + a TV RT. a V b V The temperature at (x, y) is T(x, y) = 100 x 2 y 2. In which direction should a heat-seeking bug move from the point (x, y) to increase its temperature fastest? 23. (a) If w = f( x+y ), show that xy x 2 w x y2 w y = 0. (b) Let z = g(x, y), x = e r cos(θ) and y = e r sin(θ). Show that u xx + u yy = e 2r (u rr + u θθ ). (c) Suppose F is a function satisfies F(tx, ty, tz) = t d F(x, y, z) for some integer d. Show that x F x + y F y + z F z = df. 24. Suppose f(x, y) = cos(x + 2y) sin(x y).

7 Calculus IIIA: page 7 of?? Review Problems (a) Find the first-order Taylor polynomial about (0, 0). (b) Find the second-order Taylor polynomial about the point (0, 0). (c) Find a 2-vector perpendicular to the level curve through (0, 0). (d) Find a 3-vector perpendicular to the surface z = f(x, y) at the point (0, 0). 25. At the point (1, 3), suppose f x = f y = 0 and f xx < 0, f yy < 0, f xy = 0. Draw a possible contour diagram. 26. Find the local maxima, local minima and saddle points of the following functions. Decide if the local maxima or minima is global maxima or minima. Explain. (a) f(x, y) = x + 6y 3x 2 y 2 (b) f(x, y) = x 2 + y 3 3xy (c) f(x, y) = xy + ln(x) + y 2 10, x > Find rigorously the global maximum/minimum and global maximizer/minimizer of the following functions subject to the given constraint. (a) f(x, y) = x 2 y 2 2x 2y, 0 x 1 and 0 y 1. (b) f(x, y, z) = x 2 y 2 z, x + y + z = 1, x 0, y 0 and z Determine whether the following functions are strictly convex or strictly concave? Do they achieve its global maximum or global minimum? (a) f(x, y) = x 2 + 3xy + 4y 2 + x y + 1. (b) f(x, y) = 2x 2 + xy y 2 + x y + 1 (c) f(x, y) = e x+y 2x 3y Use Lagrange multipliers to find the maximum or minimum values of f subject to the given constraint. (a) f(x, y) = xy, (1 + x 2 )(1 + y 2 ) = 4. (b) f(x, y, z) = x 2 y 2 2z, x 2 + y 2 = z (c) f(x, y, z) = x + y + z, x 2 + y 2 + z 2 = 1 and x + y z = (a) Find the points on the ellipsoid x 2 + 4y 2 + z 2 = 1 where the tangent plane is parallel to the plane 2x + 2y 3z = 2. (b) Find the distance between the ellipsoid x 2 +4y 2 +z 2 = 1 and the the plane 2x + 2y 3z = What is the shortest distance from the surface x2 2 xy+4x+z2 = 9 to the origin? 32. Find the point on the plane 3x + 2y + z = 1 that is closest to the origin by minimizing the square of the distance.

8 Review Problems Calculus IIIA: page 8 of?? 33. A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities q 1 and q 2 supplied by each factory and is expressed by the joint cost function, C(q 1, q 2 ) = 2q q 1q 2 + q The company s objective is to produce 200 units, while minimizing production costs. How many units should be supplied by each factory? 34. A company sells two products which are partial substitutes for each other, such as coffee and tea. If the price of one product rises, then the demand for the other product rises. The quantities demanded, q 1 and q 2, are given as a function of the prices, p 1 and p 2, by q 1 = p p 2, q 2 = p p 1. What prices should the company charge in order to maximize the total sales revenue? 35. Let f be differentiable and f(2, 1) = 3 i + 4 j. You want to see if (2, 1) is a candidate for the maximum and minimum values of f subject to a constraint satisfied by the point (2, 1). (a) Show (2, 1) is not a candidate if the constraint is x 2 + y 2 = 5. (b) Show (2, 1) is a candidate if the constraint is (x 5) 2 + (y + 3) 2 = 25. From a sketch of the contours for f near (2, 1) and the constraint, decide whether (2, 1) is a candidate for a maximum or minimum. 36. Consider the line x = 5 2t, y = 3 + 7t, z = 4t and the plane ax + by + cz = d. All the following questions have many possible answers. Find values of a, b, c, d such that: (a) The plane is perpendicular to the line and passes through the point (5, 3, 0). (b) The line lies in the plane. 37. Find the length of the curve x = cos(e t ), y = sin(e t ) for 0 t Find the length of the curve: x = 3 + 5t, y = 1 + 4t, z = 3 t for 1 t 2. Explain your answer. 39. A particle moves at a constant speed along a line from the point P = (2, 1, 5) to the point Q = (5, 3, 1). Find a parametric equation of the line if: (a) The particle takes 5 seconds to move from P to Q. (b) The speed of the particle is 5 units per second. 40. Suppose r(t) = cos(t) i + sin(t) j + 2t k represents the position of a particle on a helix, where z is the height of the particle above the ground. (a) Is the particle ever moving downwards? When? (b) When does the particle reach a point 10 units above the ground? (c) What is the velocity of the particle when it is 10 units above the ground? (d) When it is 10 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line.

Practice problems from old exams for math 233

Practice problems from old exams for math 233 William H. Meeks III January 14, 2010 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These