3. (12 %) Find an equation of the tangent plane at the point (2,2,1) to the surface. u = t. Find z t. v = se t.

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1 EXAM - Math 17 NAME: I.D.: Instrction: Circle yor answers and show all yor work clearly. Messing arond may reslt in losing credits, since the grader may be forced to pick the worst to grade. Soltions with answer only and withot spporting procedres will hae little credit. Partial credit will only be gien to soltions that contain part of the procedre of a correct soltion. Yo may leae yor answers in fractions or radicals withot sing calclators to conert them into decimals. 1. (1 %) Let ρ (sin θ cos θ + cos φ) = 4 be the eqation of a srface. Conert it to rectanglar coordinates. xy. (10 %) Ealate the limit lim (x,y) (0,0) x + y. 3. (1 %) Find an eqation of the tangent plane at the point (,,1) to the srface z 3 + (x + y)z + (x + y ) = (14 %) Find all second-order deriaties of f(x, y) = sin(xy ). 5. (1 %) Let z = x tan 1 (xy), where = t = se t. Find t and s. 6. (1 %) The fnction f(x, y) = x 4 + y 4 4xy + 1 has two critical points at (0, 0) and (1, 1). Classify these critical points. 7. (14 %) Find the highest and the lowest point of the srface gien by z = f(x, y) = x + xy + 3y oer a sqare region with ertices ( 1, 0), ( 1, 1), (1, 1) and (1, 1). 8. (14 %) Let f(x, y) = x y. (8A) Find the maximm direcional deriatie of f at (6, ). (8B) Find the directional deriatie of f at the point (6, ) in the direction = i + 3 j. (8C) What is the direction in which the fnction has maximm rate of change at (6, ), and what is the maximm rate of change at (6, )? 1

2 EXAM - Math 17 NAME: I.D.: Instrction: Circle yor answers and show all yor work clearly. Messing arond may reslt in losing credits, since the grader may be forced to pick the worst to grade. Soltions with answer only and withot spporting procedres will hae little credit. Partial credit will only be gien to soltions that contain part of the procedre of a correct soltion. Yo may leae yor answers in fractions or radicals withot sing calclators to conert them into decimals. 1. (1 %) Let x + y + z = x + y be the eqation of a srface. (1A) Conert it to cylindrical coordinates. (1B) Conert it to spherical coordinates. x. (10 %) Show that the limit lim does not exist. (x,y) (0,0) x + y 3. (1 %) Find an eqation of the tangent plane at the point (,,1) to the srface z 3 + (x + y)z + (x + y ) = (14 %) Find all second-order deriaties of f(x, y) = sin(xy ). 5. (1 %) Let z = xy ln( + ), where = x + y = x 3 + y 3. Find x and y. 6. (1 %) The fnction f(x, y) = x 3 + 6xy + 3y 9x has two critical points at (3, 3) and ( 1, 1). Classify these critical points. 7. (14 %) Find the highest and the lowest point of the srface gien by z = f(x, y) = x + xy + 3y oer a trianglar region with ertices (0,0), (1,0) and (1,1). 8. (14 %) Let f(x, y) = xe xy. (8A) Find the maximm direcional deriatie of f at (1, 0). (8B) Find the directional deriatie of f at the point (1, 0) in the direction = 3 i+4 j. (8C) What is the direction in which the fnction has maximm rate of change at (1,0), and what is the maximm rate of change at (1,0)?

3 Reiew for Exam Fnctions of Seeral Variables 1. Cylindrical and Spherical Coordinates (x, y, z) rectanglar coordinates (r, θ, z) cylindrical coordinates (ρ, φ, θ) spherical coordinates Conersion Fomlas: x = r cos θ, y = sin θ, r = x + y, tan θ = y if x 0. x x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, ρ = x + y + z.. Limits of f(x, y) at (a, b). lim (x,y) (a,b) f(x, y) = f(a, b), if f(x, y) is continos at (a, b). If f(a, b) 0 0 or, try to se other coordinates. lim (x,y) (a,b) f(x, y) exists only if the ale of lim (x,y) (a,b) f(x, y) remains the same when (x, y) approaches (a, b) along all possible cres. 3. Tangent Plane The tangent plane to a srface at (x 0, y 0, z 0 ) is a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0, where a, b, c = n is a normal ector of the tangent plane. fx (x n = 0, y 0 ), f y (x 0, y 0 ), 1 if the srface is z = f(x, y) F x (x 0, y 0, z 0 ), F y (x 0, y 0, z 0 ), F z (x 0, y 0, z 0 ) if the srface is F (x, y, z) = 0. The srface z = f(x, y) has a horizontal tangent plane at (x 0, y 0 ) if (x 0, y 0 ) satifies f x (x 0, y 0 ) = 0 and f y (x 0, y 0 ) = Extrema of a fnction z = f(x, y) Absolte extrema oer a region R: First find all the candidates (interior critical points of R, critical points on the bondaries of R, cornor points of R, if there are any), then compare their z-ales to pick the highest and the lowest ales. Local maxima and local minima: fx = 0 First sole to get all the critical points. Then classify each critical point (a, b) f y = 0 by the discreminant (a, b) = f xx (a, b)f yy (a, b) fxy(a, b). If (a, b) > 0 and f xx (a, b) > 0, then f(x, y) has a local min at (a, b). If (a, b) > 0 and f xx (a, b) < 0, then f(x, y) has a local max at (a, b). 3

4 If (a, b) < 0, then f(a, b) is neither a local max nor a local min. If (a, b) = 0, then no conclsion may be made. The test fails. 5. Partial Deriaties (A) If w = f(x, y, z), then there are three first order partial dertiaties x, y,, and nine second order partial dertiaties. = (t) (B) If w = f(, ) and Then w(t) and w = (t) (t) exist and dw dt = f d dt + f d dt. = (x, y) (C) If w = f(, ) and Then w(x, y) exists and = (x, y) (D) If w = f(x, y,, ) and x = f x + f x, = (x, y) = (x, y) x = f x + f x + f x, y = f y + f y. Then w(x, y) exists and y = f y + f y + f y. 6. Directional Deriaties of f(x, y, z) The gradiaent ector f = f x i + fy j + fz k. The directional deriatie of f on the direction is D f = f, where mst be a nit ector. D f is the rate of change of the fnction in the direction. The maximm directional deriatie of f at (a, b) is f(a, b). f points to the direction in which f has the maximm rate of change. 4

5 Soltion of Exam 1. (1 %) Let x + y + z = x + y be the eqation of a srface. (1A) Conert it to cylindrical coordinates. (1B) Conert it to spherical coordinates. Soltion (1A) r + z = r, sing x + y = r. (1B) Use ρ = x + y + z to get Ths the answer is ρ = cos φ. ρ = ρ cos φ cos θ + ρ cos φ sin θ = ρ cos φ. x. (10 %) Show that the limit lim does not exist. (x,y) (0,0) x + y x Soltion lim (x,y) (0,0) x + y = lim r cos θ = cos θ. Therefore the answer aries (r,θ) (0,θ) r as θ changes. For example, if θ = 0, (this means (x, y) goes to (0,0) along the x-axis), then the answer is 1; and if θ = π, (this means (x, y) goes to (0,0) along the line y = x), 4 then the answer is. Since the answer aries as the limiting process takes different cres, the limit does not exists. 3. (1 %) Find an eqation of the tangent plane at the point (,,1) to the srface z 3 + (x + y)z + (x + y ) = 3. Soltion n = F x, F y, F z = z + x, z + y, 3z + (x + y)z, n(,, 1) = 5, 5, 11. The eqation of the tangent plane is 5(x ) + 5(y ) + 11(z 1) = (14 %) Find all second-order deriaties of f(x, y) = sin(xy ). Soltion f x = y cos(xy ), f y = xy cos(xy ). f xx = y 4 sin(xy ), f yy = x cos(xy ) 4x y sin(xy ), and f xy = f yx = y cos(xy ) xy 3 sin(xy ). 5. (1 %) Let z = xy ln( + ), where = x + y = x 3 + y 3. Find x and y. Soltion x = f x + f x + f x = xy + (x) + xy (3x) + y ln( + ), + 5

6 y = f y + f y + f y = xy + (y) + xy + (3y) + x ln( + ). 6. (1 %) The fnction f(x, y) = x 3 + 6xy + 3y 9x has two critical points at (3, 3) and ( 1, 1). Classify these critical points. Soltion f x = 3x + 6y 9, f xx = 6x, f xy = 6, f y = 6y + 6x, f yy = 6. (3, 3) = 18(6) 36 > 0 and f xx (3, 3) = 18 > 0, a local min at (3, 3) ( 1, 1) = ( 6)(6) 36 < 0, neither local max nor local min at ( 1, 1). 7. (14 %) Find the highest and the lowest point of the srface gien by z = f(x, y) = x + xy + 3y oer a trianglar region with ertices (0,0), (1,0) and (1,1). Soltion Interior points: Sole f x = x + y = 0 and f y = x + 6y = 0 to get x = 0, y = 0. (0,0) is a corner point. So there is no critical point inside the triangle. Critical Points on the Bondaries: On the edge y = x: f(x, y) = f(x, x) = x + x + 3x = 6x, f = 1x, x = 0. = (0,0). On the edge x = 1: f(x, y) = f(1, y) = 1 + y + 3y, f = + 6y = 0, y = 1. = Not in 3 domain. Discarded. On the edge y = 0: f(x, y) = f(x, 0) = x, f = x, x = 0. = (0,0). There are three corner points (0,0), (1,1), (1,0). All together there are for candidates. Compare their z-ales: f(0, 0) = 0 (minimm ale) f(1, 0) = 1 f(1, 1) = = 6 (maximm ale) f(1, 1) = = The highest point is at (1,1) and the lowest point at (0,0). 8. (14 %) Let f(x, y) = xe xy. (8A) Find the maximm direcional deriatie of f at (1, 0). (8B) Find the directional deriatie of f at the point (1, 0) in the direction = 3 i+4 j. (8C) What is the direction in which the fnction has maximm rate of change, and what is the maximm rate of change? Soltion (8A) f = f x i + fy j = (e xy + xye xy ) i + x e xy j. f(1, 0) = i + j, f =. (8B) = = 1 5 (3 i + 5 j). D f = D f = f(1, 0) = 1, 1 3 5, 4 5 = 7 5. (8C) The direction is f(1, 0) = 1 1, 1, the max. rate of change is f(1, 0) =. 6