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Additional Exercises

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University of Toronto St. George

Math 237Y1 Jeremy Wong Section L0201, 2006-07 Additional Exercises (last updated 3/12/07) Here are additional exercises to supplement the acknowledged paucity of problems in the textbook, roughly grouped according to the chapter material they pertain to. Within each chapter heading, the exercises are not placed in any particular order. The majority of the exercises are computational, rather than conceptual in nature. Starred problems are ones which are more involved or dicult, and therefore are ones which are not likely to be on a Term Test. Additional exercises and short answers will be posted in the future. In this handout, not all material from all chapters is represented among the types of questions below. Also, as mentioned, some parts of this handout-in-progress (such as questions on integration) need to be updated in the future. 1 Chapter 1 n 1. Show that the closure S is connected if S R is connected 2. True or false: If f : S ! R where S R , and all partial derivatives @ f exist and are bounded on j S, then f is continuous on S. m n n 1 3. Suppose f : R ! R is a map such that for any compact set K R , the preimage set f (K) = fx 2 Rm : f(x) 2 Kg is compact. Is f necessarily continuous? n 4. Show that if f : S ! R is uniformly continuous and S is bounded then f(S) is bounded. x y3 5. Determine whether lim exists, and if so, its value. (x;y)!(0;0)2 + y4 xy x y + 7x y 6 6. (*) Prove or disprove that lim exists. (x;y)!(0;0) x + y 5 2 7. (*) Give an example of a set S R , a point (a;b) 2 S, and a dierentiable function f : S ! R such that @f(x;y) and @f (x;y) ! 0 as (x;y) ! (a;b), but for which f(x;y) ! 1 as (x;y) ! (a;b). @x @y q 2 1 12 8. (i) Let i denote the line segment in R from the origin (0;0) to the pointi; 1 i ) on the unit p S 1 circle f(x) = 1 x2. Is the union S := i=1L iompact? q 2 1 1 (ii) Let i denote the line segment in R from the origin (0;0) to the point ( i; i) on the curve f(x) = p x. Is the union S :=S 1 L compact? i=1 i 9. Dene a sequence recursively by x1= 1 and x k+1 = 2xk+ p1 . Show that lim x kxists and nd its k k!1 value. 2 Chapter 2 2 1. Find the global minimum and maximum, if any, of (x +4xy)e y on the set S = f(x;y) 2 R : x 1g. 2. Find and classify the critical points of (i) f(x;y) = 3xe x e 3y (ii) f(x;y) = (x 1) (x y x 1)2 2 4 (iii) f(x;y) = x y 1 R 3. (#4, sect. 2.7) Use a Taylor series approximation to e x 2 to compute 1et2 dt to three decimal 0 places, and prove the accuracy of your answer. 4. (*) Instead of expanding a real-valued function f(x;y) of two variables by a Taylor polynomial in several variables, suppose one rsts expands f(x;y) as a polynomial in x, treating y as constant, so that x 2 x3 xn1 xn f(x;y) = g0(y) + g1(y)x + g2(y) + g 3y) + + gn1(y) + gn(x;y) 2! 3! (n 1)! n! where the g (y) (i = 1;:::;n 1) are functions of y, and the last term g (x;y) possibly involves both i 2 2 n x and y. If n = 2 and f is C , then is 2 (x;y) necessarily C as well? 0 1 2 0 0 5. Suppose f is a C function whose Hessian matrix at some point a for which rf(a) = 0 [email protected] 0 3 0 . 0 0 7 Is this point a local maximum, local minimum, or neither? 2 2 6. Find and classify all possible local extrema of the function f(x;y) = (2x x )(2y y ). 7. Suppose the post oce adopts a new regulation that only boxes whose sum of length, width and height is less than 2 meters can be mailed through their overnight delivery service. Under this regulation, could you mail a package overnight which contained 0:3m worth of goods? 3 Chapter 3 1. (i) Explain why the z-axis in R cannot be represented as the level set of a C function F : R ! R. 2 1 (ii) Explain why the upper half-plane f(x;y) 2 R : y 0g cannot be represented as a level set of a C function F : R ! R. (It can, however, be represented a sublevel or superlevel set, e.g., F([0;1)), where F(x;y) = y.) 2. Let f(x;y) = (x + 6y;3xy;x 3y ), where x;y 2 R. Describe the image of f. 2 2 3. Show that if F : R ! R is a continuous function and the set f(x;y) 2 R : F(x;y) = 0g is disconnected, then there exists a factorization F(x;y) = G(x;y)H(x;y), for some functions G and H. (This is a converse to exercise #6, sect. 3.2) 4. Find the tangent plane to the surface 2 (i) given by f(u;v) = (u + v;4u + 1;uv) when u = 1;v = 2 (ii) given by F(x;y;z) = xy z + y = 0, at (x;y;z) = (3;2;2) 5. Are the functions f(x;y;z) = x + y 2 g(x;y;z) = y + z2 h(x;y;z) = z + x2 functionally dependent on some open set U R ? If so, determine the rank of Df at each point of U, where f = (f;g;h). 6. (# 8, sect. 3.1) Near which points (x;y;z;u;v) = (x0;y0;z0;u 0v 0, if any, can the equations xy + xzu + yv = 3 u yz + 2xv u v = 2 be solved for u and v as functions of x;y;z? 2 7. Sketch by hand the curve or surface determined by each of the following equations 3 3 2 (i) x + y 6xy = 0 (rectangular coordinates) in R (ii) r 2r sin = 3 (cylindrical coordinates) in R . 8. (Conversion between implicit (level set) and parametric descriptions) (a) Find a parametric description of the intersection of the following two equations: 2 2 2 x + 5(y 1) + 4(z + 2) = 1 2x + y z 3 = 0: (b) Find either a real-valued function F(x;y;z) or a vector-valued function F(x;y;z) = (F(x;y;z);G(x;y;z)) whose 0-level set is the image of the map f(u;v) = (2uv;v ;u + v). 1 9. (#5, sect. 3.4) Find a one-to-one C map f from the rst quadrant of the x-y plane to the rst quadrant of the u-v plane such that the region where x y 2x and 1 xy 3 is mapped to a 1 rectangle. Compute the inverse map f . 10. (i) Give a full statement of the Implicit Function Theorem and the Inverse Function Theorem. (ii) (#7, sect. 3.4) Assuming the Inverse Function Theorem, prove the Implicit Function Theorem. (note: it is possible to prove the Inverse Function Theorem directly, so this would not be a circular argument.) 1 2 11. Does there exist a C map f : R ! R which is one-to-one? (recall that a map f is one-to-one or injective if the images of two distinct points are distinct. Equivalently, if f(x) = f(y) then x = y.) Hint: examine the derivative of f. 12. (*) Does there exist a C map f : R ! R which is onto? (recall that a map f is onto or surjective 1 if its image is the entire target space. Equivalently, the question is: can there exist a C function f such that f(R) = R holds?) Hint: the notion of zero content (from chapter 4) could be helpful in answering this. See Proposition 4.19(c), for example. 4 Chapter 4 (In problems 1-5 below, integrability refers to Riemann integrability, instead of improper Riemann integra- bility, although several of the results do extend to the latter situatio
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