 # 1. Find the equations of planes that just touch the sphere (x &minu...

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1. Find the equations of planes that just touch the sphere (x − 3) 2 + (y − 1) 2 + (z − 2) 2 = 25 and are parallel to (a) The xy-plane: and (b) The yz-plane: and (c) The xz-plane: and 2. Without a calculator or computer, match the func-tions with their graphs in the figures below. Note that two of the functions do not have a matching graph. (a) z = 2+x 2 +y 2 ? (b) z = 2−x 2 −y 2 ? (c) z = 2(x 2 +y 2 ) ? (d) z = 2+x−2y ? (e) z = −2 ? (f) z = 2−x ? 1. 2. 3. 4. 3. On a piece of paper, sketch each of the following surfaces: (i) z = x 2 +y 2 +4 (ii) 4 = x+y Use your graphs to fill in the following descriptions of crosssections of the surfaces. (a) For (i) (z = x 2 +y 2 +4): Cross sections with x fixed give ? Cross sections with y fixed give ? Cross sections with z fixed give ? (b) For (ii) (4 = x+y): Cross sections with x fixed give ? Cross sections with y fixed give ? Cross sections with z fixed give ? 4. By setting one variable constant, find a plane that intersects the graph of z = 2y 2 − 6x 2 + 3 in a: (a) Parabola opening upward: the plane = (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (b) Parabola opening downward: the plane = (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (c) Pair of intersecting straight lines: the plane = (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) 5. Find an equation for the contour of f(x,y) = 2x 2 y+ 6x+10 that goes through the point (4,3). Equation: 6. Find an equation for the plane containing the line in the xy-plane where x = 3, and the line in the yz-plane where z = 4.5. equation: 7. Find a possible equation for the linear function g(x,y) shown in the graph below: g(x,y) = 8. Find a formula for a function g(x,y,z) whose level surfaces are planes parallel to the plane z = 8x+4y−3. g(x,y,z) = 9. Find the limit of the function f(x,y) = sin(5 p x 2 +y 2) 5 p x 2 +y 2 as (x,y) → (0,0). Assume that polynomials, exponentials, logarithmic, and trigonometric functions are continuous. [Hint: limt→0 sint t = 1.] lim (x,y)→(0,0) sin(5 √ x 2+y 2) 5 √ x 2+y 2 = (Enter DNE if the limit does not exist.) 10. In this problem we show that the function f(x,y) = x 2 −y 2 x 2 +y 2 does not have a limit as (x,y) → (0,0). (a) Suppose that we consider (x,y) → (0,0) along the curve y = 4x. Find the limit in this case: lim (x,4x)→(0,0) x 2−y 2 x 2+y 2 = 1 (b) Now consider (x,y) → (0,0) along the curve y = 5x. Find the limit in this case: lim (x,5x)→(0,0) x 2−y 2 x 2+y 2 = (c) Note that the results from (a) and (b) indicate that f has no limit as (x,y) → (0,0) (be sure you can explain why!) . To show this more generally, consider (x,y) → (0,0) along the curve y = mx, for arbitrary m. Find the limit in this case: lim (x,mx)→(0,0) x 2−y 2 x 2+y 2 = (Be sure that you can explain how this result also indicates that f has no limit as (x,y) → (0,0). 11. Show that the function f(x,y) = x 2 y x 4 +y 2 . does not have a limit at (0,0) by examining the following limits. (a) Find the limit of f as (x,y) → (0,0) along the line y = x. lim (x,y)→(0,0) y=x f(x,y) = (b) Find the limit of f as (x,y) → (0,0) along the line y = x 2 . lim (x,y)→(0,0) y=x 2 f(x,y) = (Be sure that you are able to explain why the results in (a) and (b) indicate that f does not have a limit at (0,0)! 12. What value of c makes the following function continuous at (0,0)? f(x,y) = x 2 +y 4 +4,(x,y) =6 (0,0)c,(x,y) = (0,0) c = 13. Find the value(s) of a making~v = 7a~i −3~j parallel to ~w = a 2~i + 6~j. a = (If there is more than one value of a, enter the values as a comma-separated list.) 14. (a) Find a unit vector from the point P = (2,2) and toward the point Q = (6,5). ~u = (b) Find a vector of length 15 pointing in the same direction. ~v = 15. Find all vectors ~v in 2 dimensions having ||~v|| = 5 where the ˜i-component of~v is 3˜i. vectors: (If you find more than one vector, enter them in a commaseparated list.) 16. A plane is heading due west and climbing at the rate of 100 km/hr. If its airspeed is 500 km/hr and there is a wind blowing 90 km/hr to the northwest, what is the ground speed of the plane? ground speed = 17. Let ~a,~b,~c and~y be the three dimensional vectors ~a = 4˜j+4 ˜k, ~b = 5˜i+6˜j+5 ˜k, ~c = 4˜i+5˜j, ~y = −5˜i−3˜j Perform the following operations on these vectors: (a)~c ·~a+~a ·~y = (b) (~a ·~b)~a = (c) ((~c ·~c)~a)·~a = 18. The force on an object is ~F = −25˜j. For the vector ~v = ˜i + 5˜j, find: (a) The component of ~F parallel to~v: (b) The component of ~F perpendicular to~v: The work, W, done by force ~F through displacement ~v: ˜ 19. Compute the angle between the vectors ˜i + ˜j + and −˜i k − ˜j − ˜k. angle = radians (Give your answer in radians, not degrees.) 20. Find the equation of a plane that is perpendicular to the vector −2˜i + ˜j − 2 ˜k and passing through the point (−1,5,−1). The plane is given by: 21. Use the algebraic definition to find ~v × ~w if ~v =˜i−˜j+3˜k and ~w = −3˜i−3˜j−˜k. ~v×~w = 22. Find an equation for the plane through the points (5,3,5),(−1,2,0),(−1,0,−2). The plane is 23. Consider the planes given by the equations x+y−4z = 3, 4x+y+z = 8. (a) Find a vector ~v parallel to the line of intersection of the planes. ~v = (b) Find the equation of a plane through the origin which is perpendicular to the line of intersection of these two planes. This plane is 24. Find a vector ~v parallel to the intersection of the planes 4z − (3x + 5y) = 4 and 3x + 2y − z = 7. ~v = 25. Let P = (0,0,1),Q = (1,−1,2),R = (−2,1,1). Find (a) The area of the triangle PQR. area = (b) The equation for a plane that contains P, Q, and R. This plane is 26. Suppose ~v · ~w = 7 and ||~v×~w|| = 4, and the angle between~v and ~w is θ. Find (a) tanθ = (b) θ = 2 27. Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. z = 2x 2 y 6 −y 6 12xy−5 . zy = 28. Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. ∂ ∂x (x 3 e √ 5xy) = z = sin 4x 5 y−4xy3 29. Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined.  . zx = 30. Find the partial derivatives indicated Assume the variables are restricted to a domain on which the function is defined. z = x 5 +3 y +x y . zx = zy = 31. Find the equation of the tangent plane to the surface determined by x 3 y 4 +z−20 = 0 at x = 2, y = 3. z = 32. Find the equation of the sphere centered at (10,10,8) with radius 10. = 0. Give an equation which describes the intersection of this sphere with the plane z = 9. = 0. 33. Find the area of the parallelogram with vertices: P(0,0,0), Q(2,-1,4), R(2,-2,5), S(4,-3,9). 34. Find the distance the point P(3, -2, -4), is to the plane through the three points Q(-1, 0, -2), R(1, -1, -5), and S(1, -4, -5). 35. Find the distance from the point (-1, 5, -4) to the plane −4x + 2y+2z = 6. 36. Find the limits, if they exist, or type DNE for any which do not exist. lim (x,y)→(0,0) 2x 2 3x 2 +4y 2 1) Along the x-axis: 2) Along the y-axis: 3) Along the line y = mx : 4) The limit is: 37. Find the first partial derivatives o f f(x,y,z) = z arctan( y x ) at the point (2, 2, -2). A. ∂ f ∂x (2,2,−2) = B. ∂ f ∂y (2,2,−2) = C. ∂ f ∂z (2,2,−2) = 38. Find the partial derivatives of the function f(x,y) = −4x+2y 2x+1y fx(x,y) = fy(x,y) = 39. Find the partial derivatives of the function w = p 1r 2 +1s 2 +2t 2 ∂w ∂r = ∂w ∂s = ∂w ∂t = 40. p Find the linearization of the function f(x,y) = 54−1x 2 −4y 2 at the point (3, 3). L(x,y) = Use the linear approximation to estimate the value of f(2.9,3.1) = 3

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