e3 - Exam 3 SHOW WORK!! 1)(20) Let f (x, y ) = xy + y 2 ; x...

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Unformatted text preview: Exam 3 SHOW WORK!! 1)(20) Let f (x, y ) = xy + y 2 ; x = cos(θ) cos(ψ ); y = cos(θ) sin(ψ ). a) Draw the tree diagram for these variables ∂f b) State the chain rule for ∂ψ c) If θ = t, ψ = t2 , draw the tree diagram for these new variables d) State the chain rule for df dt 2)(20) Let f (x, y ) = ln(x2 −y 2 ). Does f satisfy the partial differential equation y ∂f ∂f −x =0 ∂x ∂y 3)(60) Let z = f (x, y ) = 1 − 1 1 − (x2 + y 2 ); P = P (0, √2 ) a) Sketch the surface; locate P and f (P ). b) Find f ; f (P ). c) Sketch f (P ) on the graph in a), with its tail at P . d) In the xy plane, draw the level curve z = f (x, y ) = f (P ). e) On the graph in d), locate P and draw f (P ) with its tail at P . f) Find a parametric representation ¯(t) = (x(t), y(t)), that lies on the curve r ¯ in d), and goes through the point P . Find a t0 with r(t0 ) = P . r g) Compute ¯ (t0 ) and compute that f (P ) is perpendicular to ¯ (t0 ). r Exam 3c Show Work!! 3)(20 points) Let r = ti + tj 1)(10) Does the curveln(x2 0 ≥2 ) satisfy the partial differential equation a) Sketch f (x, y ) = for − y t < ∞ ¯ below? Does the line ¯ = (1+ t)¯− t¯− 2tk intersect the plane x − y + z = 1? 2)(20) r i b) Find the velocity vectorfatj t∂= 1. 2 2 ∂ f a) If yes, find the angle in b), with its = 0 at r(1) − c) Sketch the vector the two make.2 tail ∂x2 ∂y b) If not, show the two are parallel. − 1)(10) Let f (x, y ) = xx2 yy . Does f satisfy the partial differential equaQ(2, 0, −1) and R(1, 12, 1) tion: ∂2f Exam 3makeup = Show Work!! 0 √ ∂x∂y 2)(20) Find the area 2of the triangle with vertices P (1, 1, 2), 2 √ ¯) = xi − ¯ − let P = P be two planes. 3)(20 points) Let yz= 3t¯ y 2t+ x; 3y + 2z = 1(1, 1). In what direction r= 2 j 2)(20) Let x f (y − and x 4)(20) Let + x, a) Do the planes intersect? t < ∞ Sketch the curve for 0 ≤ a) should I move away from P , so that the slope is greatest? What b) Find answer to a) is yes, t = intersect in a line; find the direction b) If the the velocity vector atthey 1. is that slope? c) Sketch the the answer with its tail at ¯ that the planes are parallel. r Exam 3makeup Show of that line. If vector in b),to a) is no, show (1) Work!! 4)(20) Let x, y r = 22t i 2 2 j xy , let r 1)(10)points)((x, yf=x,ln(¯ −−ty 2 ) =yr cos θ sinpartial(0=1)sin θ sin φ. 3)(20 Doesffz = )¯ ( xey− = e ;¯x satisfy thePφ= P ydifferential equation Let ) 5)(30) Let tree diagram for 2 + 2 . Let )x xthese quantities , a) Draw below? the a) Without plotting points, sketch the curve for −∞ < t < ∞ a) Sketch the surface ∂f ∂2 b) State the chain rule for at2t = ∂f f b) Find the velocity vector finding 0. /∂φ 0 b) Sketch the rule compute− both in (x, ∂f /∂φ= c) Use the chaintrace to = f∂x2 1),∂y 2 r the c) Sketch the vector inzb), with its tail at ¯(0) xz plane and on the d) Compute fx (P ). Referring back to b), explain why it has the value it does. 5)(30)points)= f (x, =) e2t¯ − et¯ P = P (1, 0) x2 j 3)(20 Let z Let ¯ y = i . Let r a) Sketch the surface a) Without plotting points, sketch the curve for −∞ < t < ∞ b) Sketch the trace = f (x, 0), on the b) Find the velocityzvector at t = 0. surface. c) Sketch the level curve z = 1, its tail at ¯(0) c) Sketch the vector in b), with on the surface. r d) Compute fx (P ); fy (P ). Referring back to b), c) explain why they have the value they do. 4)(20) Let f (x, y ) = xy ; x = u2 − v 2 , let y = u2 + v 2 ; u = r + s; v = r − s. a) Draw the tree diagram for these quantities b) State the chain rule for finding ∂f /∂s c) Use the chain rule to compute ∂f /∂s surface. c) Sketch = y − x2 Let = , y ), y = 5)(30) Let (x, y 2 = xy . z = P2f (02 ,(0, 1) =the v 2 plane s; v on the 4)(20) Let fz the ) trace;3xandux− vPy let 2in=u2 +yz; u = r + and = r − s. 2)(20) Let the surface x+y−z = a) Draw the tree diagram for − 3 + z 1 be two planes. Sketch surface. a) Do the planes intersect? these quantities a) Sketch the trace z = f (x, 1), in the xz plane and on the surface. b) d) the answer f P yes, they intersect in a b) IfCompute to(a)).for finding ∂f /∂s to b) and c), the direction b) State the chain rule isfReferring back plane. line; find explain why c) Sketch the direction it does. the yz c) Use line. If traceanswer(0, ya) in no, show that the planes are parallel. it points the the z = to ), is of that the chain rule to compute ∂f /∂s 5)(30) Let z = f (x, y ) = x2 . Let P = P (1, 0) a) Sketch the surface b) Sketch the trace z = f (x, 0), on the surface. c) Sketch the level curve z = 1, on the surface. d) Compute fx (P ); fy (P ). Referring back to b), c) explain why they ∂f ∂f + 2x =0 ∂x ∂y Exam 3a Show Work!! 1)(10) A line that f (x,P (11, (x + y )2 satisfies the − z = 2 differentialz = 0. 2)(20) Show through y ) = 2) is parallel to x − y partial and x + y − equation vector and scalar parametric equations for the line. Find ∂2f ∂2f − 2 =0 ∂x2x2 y 2 + y ; x = cos θ + cos φ, ∂y 3)(20 points) Let f (x, y ) = x + b) If the answer to a) is yes, find the plane containing both lines. If the y 2 . Let P to both 12 ) 4)(30) to z is no, yExam13a x perpendicular = P )= − answerLet a) = f (x,find a vector 2 − Show Work!! (0, √lines. a) Sketch the surface; locate P . b) Sketch the Lety¯ z sin((x,j 1y ) in the xz partial differential equation 3)(20 Does f x, r t2 if x t − 1)(10)points) (trace = = ¯ − 2 ¯√2 ), satisfy the plane; locate P ) 1 a) Sketch the trace z = −∞ < t)< ∞ surface. curve for f (x, √ on the c) 2 ∂f b) Sketch the level curve z = ft(= ), in the xy plane; locate P Find the velocity vector at P 1. ∂f d) + 2x =0 c) Sketch the vector in b), with its tail P r ∂x ∂y . e) Find the normal to the level curve atat ¯(1) let y = cos θ − sin φ. a) Draw the tree diagram t¯ +theseand ¯ = (2, −1, 2) + t(1, 1, 1) be two ¯ 2)(20) Let ¯ = (1 + t)¯ − for 2tk quantities r i j r b) State the chain rule for finding ∂f /∂φ lines. c) Do the chain rule to compute ∂f /∂φ a) Use the lines intersect? 4)(20) Let f (x, y ) = x + x2 y 2 + y ; x = cos θ cos φ, let y = cos θ sin φ. 2 5)(20 A the You are (11for −1) quantities y − = (1 2)(20) points)tree diagramP 2),these on the surfacez z = (y −x +)3− z = 0. a) Draw line through Pat , 3bis parallel to xWork!! 2 and x y . What Exam Show − direction should scalar for finding equations for the line. Find vector and you move from P , so that your height remains the same? b) State the chain rule parametric ∂f /∂φ c) Use the chain rule to compute ∂f /∂φ 3)(20 points) Find an = x + x2 2 + the plane containing the line 1)(20 points) Let f (x, y ) equationy for y ; x = cos θ + cos φ, 1 let −cos θ − 2t − 2, = 1 − t − y . Let axis. 5)(30) 1, z = f ( φ x =yt=Let y = sinx,.y ) z = 1 − x,2 and2the y P = P (0, √2 ) a) Draw the tree diagram for these quantities a) Sketch the surface 1 b) State the chain rule for x, √ ), in the xz plane and on the surface. ∂f /∂φ b) Sketch the Let = f ( finding sin(xy 2)(10 points) tracezz= f (x, y )2 =∂f /∂φ ); let P = P (1, π ). Find the c) Use the chain rule to compute c) Sketch the trace z = f f y ), in in yz direction i − j. directional derivative of (0, at P , thethe plane. Referring to b), Work!! d) Compute fx (P ). Exam 3c back Show explain why it has the value 1 4)(30) Let z = f (x, y ) = 1 − x2 − y 2 . Let P = P (0, √2 ) it does. 2 3)(30 points)x, y ) = =22 y2 . Does s2 satisfy the. partial = x; t = −x. f2 1)(10) Let f ( Let q locate P a) Sketch the surface;xx − (s, t) .= f + t − st Let s differential equay 1 a) Draw athe trace z = f (functions the xz plane; locate P tree for these x, √ ), in tion: b) Sketch 22 b) Sketchthe chain z = ffor these f the surface. State the trace rule (x, √ )∂on functions 1 c) =0 dq 2 ∂x∂y c) Compute dx using the chain rule. d) Sketch the level curve z = f (P ), in the xy plane; locate P e) Find the normal to the level curve at P . 2 5)(50 points) Let z ¯= f(1+ t)) − t¯− 2tx2intersect the plane x − y + z = 1? 2)(20) Does the line r = (x, y ¯ = j − k − y . i1¯ 1 LetIf yes, find2the 1angle the two make. a) P = P ( √ , √2 ) 5)(20 not, show theare at P (1, −1) on the surface z = (y − x2 )3 . What b) If points) You two are parallel. direction the surface; locate P , point your a) Sketchshould you move fromthe so that P . height remains the same? √ ¯) i 3)(20 points) Let (P=. t¯ − t¯ r j b) Find f ; f CODE NAME FOR ONLINE GRADES: ...
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