FINAL REVIEW Answers and hints Math 311 Fall 2017

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1 FINAL RVIW Answers and hints Math 3 Fall 7. Let R be a Jordan region and let f : R be integrable. Prove that the graph of f, as a subset of R 3, has zero volume. Let R be a rectangle with R. Since f is integrable, given ε >, there is a partition P such that Uf, P Lf, P < ε. This gives a finite family of boxes B,..., B n with n n Gf B j such that V B j < ε. j= j=. Let fx, y = y if < x < y <, fx, y = x if < y < x <, and fx, y = otherwise, and let = [, ]. a Show that f is not integrable on, but x fx, y is integrable on [, ] for each fixed y, and y fx, y is integrable on [, ] for each fixed x. b Show that the iterated integrals are not equal. fx, ydxdy and fx, ydydx exist, but Since f is unbounded, it can not be integrable. For a fixed y, we get and for a fixed x we get fx, ydy =. fx, ydx = 3. Let fx, y = x + y, e x y. Prove that there is a differentiable function f defined in a neighborhood of f,, and compute the differential df f,. df f, =. 4. Suppose that F x, y, z = can be solved to yield any of the three variables as a differentiable function of the other two. Show that x y z y z x =. Use x y = F y F x etc. 5. The plane x + y + z = intersects the paraboloid z = x + y in an ellipse. Find the points on this ellipse that are closest to and farthest from the origin.

2 /, /, / and,,. 6. Use the change of variables x = u + v, y = u + v to evaluate where is the triangular region with vertices,,, and,. x 3ydV x, y = u u 3v3dvdu = 3. x 3ydV x, y 7. Find the volume of the solid ellipsoid x + y + x y + z + 3z. V = π Let be the region in the first quadrant bounded by the curves xy =, xy = 3, x y =, and x y = 4. Use a change of variables to compute x + y dv x, y. Let u = xy, v = x y. Then 9. Prove that the system x + y dv x, y = 3. e x + yz 4t = 3, y cos x 6x + z u = can be solved for x, y around the point,, 3,, 7. Let x = gz, t, u, y = hz, t, u be the solution. Compute g 3,, 7. z g z 3,, 7 = 4.. Find and classify the critical points of fx, y = xy 3x 4y.,, 4,,, 3 saddle points; 4/3, local max.. For the equation xy + yz + xz = find all points on the solution set S where locally this is a smooth surface. At such a point a, b, c find an equation of the tangent plane. a, b, c,, ; b + cx a + a + cy b + a + bz c =.. valuate R y 3 x + y x dx y 3 x dv x, y where R = [, ] [, ]. + y x dy = ln. 4

3 3 3. Let A, B R d be open with A B =. Prove that A B = A B =. Since B c is closed and A B c, it follows that A B c. 4. Prove that Q R is not connected. Find its connected components. It can be separated by, α, α, with α irrational. The connected components are points. 5. Let fx, y = e x cos y, e x sin y. Prove that every point in R has a neighborhood in which f is one-to-one, but f is not one-to-one on R. Find an explicit formula for f defined in a neighborhood of f, π and compute df f, π. 3 3 For example, f, = f, π. We get f u, v = ln u + v, arctan v, df 3 u, = Show that the system of equations 3x + y z + u =, x y + z + u =, x + y 3z + u = can be solved for x, y, u in terms of z; for x, z, u in terms of y; for y, z, u in terms of x; but not for x, y, z in terms of u. We have f, f, f 3 x, y, z =. 7. Let f : R 3 R, fx, y, z = x 3 y z. Determine its affine approximation at,,. T x, y, z = + 6x 4y + + z. 8. Let fx, y = xy + 3 x + 4. Show that f has a minimum but no maximum on y {x, y : x, y > }. Find the minimum. f min = Find the extrema of g : D R, gx, y, z = x y + 3z where D = {x, y, z R 3 : x + y + 3z 9}.

4 4 g min = 6 5, g max = Find a compact infinite subset of R such that its connected components are single points. For example, let {/n : n } {}.. Does the series x k y k converge uniformly on the open square,,? k= No.. Let fx, y, z = x y + e x + z. Show that there is a differentiable function g in a neighborhood of, such that g, = and fgy, z, y, z =. Compute g y, and g z,. g y, =, g z, =. 3. Let gx, y = fe x cos y, e x sin y, where f is of class C. Compute g xx and g xy in terms of f uu, f uv and f vv. g xx = e x f uu cos y + f uv sin y cos y + f vv sin y + e x f u cos y + f v sin y, g xy = e x f vv sin y cos y f uu sin y cos y + f uv cos y + e x f v cos y f u sin y. 4. Find the Taylor polynomial of degree 4 at, for gx, y = e xy cosx + y 3 without computing any derivatives. T 4 x, y = + xy + x y x4. 5. Find the point on the line through,,, and,,, that is closest to the line through,,, and,,,. /, /,,.

5 5 6. Find the domain of continuity for the functions xy if xy > afx, y =, if xy y x y if y x y x bgx, y =, if y = x { x y 3 if x, y, x chx, y = +y if x, y =,. a R, b y x, c R \ {, }. 7. Suppose A R is connected and contains the points, 3 and 4,. Show that A contains at least one point on the line y = x. Otherwise, the sets {x, y R : y > x} and {x, y R : y < x} will separate A. 8. Calculate z 3 x + y + z dv x, y, z where H is the solid hemisphere given by H x + y + z, y.. 9. Show that the series x n e nx converges uniformly on [, A] for each A >. Does it n= converge uniformly on [,? x n n Yes. Use the inequality. e x e 3. Suppose A R p and B R q are Jordan regions. Find a formula for A B and prove that A B is a Jordan region in R p+q. A B = A B A B, so V A B =.