Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 =
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1 Test Review Find te determinant of te matrix below using (a cofactor expansion and (b row reduction Answer: (a det + = (b Observe R R R R R R R R R Ten det B = (((det Hence det Use Cramer s rule to solve: Answer: x = = 9, y = x + y + = x + y z = x y + z = = 9, z = Find te inverse of te matrix using (a row reduction and (b te adjoint formula = B Answer: (a Reduce [A I] to [I A ] (b Use A = det A adj det A [C ij] T In bot cases, A = Let C = F T F were F is an n n real matrix Sow det (C Answer: det C = det (F T F = (det F (det F T = (det F (det F = (det F Sow tat if A exists, ten det ( A = det A 6 Let = 9 Wic of te following are eigenvectors? If so, wat is te associated eigenvalue If not, justify your answer (a [ ] T (b [ ] T (c [ ] T (a (b = = 8 7 Yes, an eigenvector wit eigenvalue No, not an eigenvector
2 (c x = is never an eigenvector Tis is part of te definition of eigenvector 7 Is λ = and eigenvalue for te matrix below? If not, explain If so, wat is te associated eigenspace? (a det(a I = (b [A I ] row reduces to = Tus, λ = is an eigenvalue 8 Te matrix A below as eigenvalues λ = and λ = (a Determine te tird eigenvalue Hence, te eigenvectors are of te form x = t (b For eac distinct eigenvalue, find a basis for te associated eigenspace (a It follows from eiter tr(a = λ + λ + λ or det(a = λ λ λ tat λ = (b [A I ] row reduces to Hence E = Span (c [A I ] row reduces to Hence E = Span 9 Diagonalize te matrix or explain wy it is not possible D =, S = Diagonalize te matrix or explain wy it is not possible D =, S = Find A given ( [ Te eigenvalues of A are λ =, wit corresponding eigenvectors ] [ and ] Terefore A = ( ( ( ( = +
3 Let f(t = t i + e t j + sin(tk Evaluate (a f ( (b f ( (c f(t dt (a j + k (b i + j (c i + (e j + ( cos + k Sketc te curve r(t = e t i + e t k, t R Since x = e t and y = e t, te curve satisfies te relation y = x Note tat x = e t > Terefore it is te part of te curve y = x satisfying x > Find a parametrization for te curve of intersection of te cylinder x + y = and te plane x + y + z = r(t = cos ti + sin tj + ( cos t sin tk Using te rules of differentiation, simplify (( f f f (f(t f (t f (t 6 Suppose f (t is a vector valued function and f (t = for all t Sow f (t f (t = Starting wit f(t f(t =, differentiate to obtain f(t f (t = Hence f(t f (t = 7 Find te line tangent to r(t = (t + t + i + (t + j + (t + t + k at (,, ( + ti + ( + tj + ( + tk 8 Tere are two surfaces, ( x + y = and x + y + z = wic intersect in a curve Find an equation of te tangent line to tis curve at,, Te curve of intersection is given by x + y =, z = ± Te point of interest is on te( curve x + y =, z = ± Tis curve can be parameterized by f(t = cos ti + sin tj + k Te tangent line at,, is r(t = ( ( t i + t j + k 9 Find te arc lengt of te curve r(t = cos ti + sin tj + cos tk from t = to t = ln ln ln + sin t dt = cos t dt = sin t ln = Let r (t = t i + (t + j + t k Find te following at t = : (a T (b N (c κ (d a T (e a N (a T = (b N = v = i + j + k T (t T (t = κ κ = a a T T a a T T = 6 i 6 j + 6 k (Note tat κ = (v va (v av (v v (c κ = v a = 6 6 (d a T = v a = (e a N = v a = 6 Let r (t = cos ti + sin tj + tk Find a, a T, a N, T, and N Verify tat a = a T T + a N N a = cos ti sin tj, a T =, a N =, T = sin t i + cos t j + k, and N = cos ti sin tj Note tat a = cos ti sin tj = ( sin t i + cos t j + k + ( cos ti sin tj = a T T + a N N Calculate te curvature of y = x / at (, Let r(t = ti + t / j Ten κ = v a = 6 Let f (x, y = x y Find te domain of f {(x, y x + y }
4 Here is te grap of a quadratic surface (a Wic of te following equations could possibly represent tis surface? (i z ( x + y = (ii z = x + y (iii x + y = z (iv x + y z = (v z = x y (vi x + y + z = (b Wat is te name of tis surface? (a z = x y (b yperbolic paraboloid Find te c-level surface of f(x, y, z = x y + z tat passes troug te point (,, x y + z = 6 Let f(x, y = x sin ( x + y Find (a f x (b f y (c f xy (d f yx (a f x = sin(x + y + x cos(x + y (b f y = xy cos(x + y (c f xy = y cos(x + y x y sin(x + y (d f yx = y cos(x + y x y sin(x + y 7 Under wat conditions is it true tat f xy = f yx? If f x, f y, f xy, and f yx are continuous, ten f xy = f yx 8 If possible, find f x (, were f (x, y = f(, f(, f x (, = { yx+x +xy x +y if (x, y (, if (x, y = (, 9 Specify te interior and boundary of te set State weter te set is open, closed, bot or neiter (a {(x, y : x, y } (b {(x, y, z : x + (y + (z = } (a Te boundary is {(x, y : x = or, y } {(x, y : x, y = or } Te set is closed (b Te boundary is {(x, y, z : x + (y + (z = } Te set is closed Find lim not exist (x,y (, lim (x,y (, y=mx x y x +y, if it exists If te limit does not exist, explain wy x y x +y x m x x x +m x = m +m Since different values of m give different limits, lim x y (x,y (, x +y does
5 x Find lim y+x +xy +x +y (x,y (, x +y, if it exists If te limit does not exist, explain wy x lim y+x +xy +x +y r (x,y (, x +y cos θ sin θ+r cos θ+r cos θ sin θ+r r r = Find f (x, y were f (x, y = x y + sin (xy f (x, y = (xy + y cos(xy i + ( x + x cos(xy j Using te definition of te directional derivative find f v (, were v = i + j and f v f (, (, f(, f (x, y = { x y +x +xy x +y if (x, y (, if (x, y = (, + + = Find f v (, were f (x, y = x y + sin (xy and v is te unit vector in te direction of i + j f v (, = f(, v + cos = (( + cos i + ( + cos j (i + j = Let f (x, y = x y+sin (xy Find te largest value of f v (, over all unit vectors v Tat is, find te largest directional derivative of f(x, y f = + 6 cos + cos 6 Prove tat te direction of maximal rate increase of a function f(x, y is in te direction of f and tat te maximal rate of decrease is in te direction of - f Wat are te rates of increase in tese directions? Note tat f v = f v = f cos θ = f cos θ Since cos θ, ten f f cos θ f Hence f f v f Note tat equality olds on te left only if θ = π, ie f and v are in te opposite directions Likewise, equality olds on te rigt only if θ =, ie f and v are in te same directions 7 Let z = xy and let x = ts + ps wile y = s + t Find z s wen (s, t, p = (,, Note tat x(,, = and y(,, = Ten z s = z x x s + z y y s = y (t + p + xy(s = 66 8 Let z = x sin (y and let x = t s + r wile y = t s Find z t wen (s, t, r = (,, Note tat x(,, = and y(,, = Ten z cos t = z x x t + z y y t = x sin y(st + x cos y(t = sin + 9 A triangle as sides x and y, and included angle θ Given tat x and y increase at te rate of inces per second, but te area of te triangle is kept constant, at wat rate is θ canging wen x = inces, y = inces, and θ = π/ radians? Note tat xy sin θ = (( sin π = Tus θ = sin ( xy dx x x y dt + dy y x y dt = 9 radians/sec Hence dθ dt = θ dx x dt + θ dy y dt =
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