
5. (5 points) Let Cº(-0,0) = {f(x) in C(-00,00)" () exists for all a} be the...
(5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) = -f(L). That is, W= {f() in C | F"(x) = -f(x)} In the previous worksheet, we showed that: • W is a subspace of Cº. . For all a and b, a sin(2) + b cos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b)...
5. Let V be the subset of Cº(R) consisting of all functions that can be expressed in the form a sin 2x + be 4x + cos2r for a, b, c ER. (a) (4 points) Prove that V is a subspace of C(R). (b) (3 points) Let fix) = sin 2x + e4r f2(x) = sin 2.c + cos 2.0 f3(x) = 4x + cos2r. The set B = (f1, f2, f3) is an ordered basis for V. (You do...
Let f(x, 2) Va r], (x1, X2) E R2. Determine all directions E R2 along (0,0) exists. which
Let f(x, 2) Va r], (x1, X2) E R2. Determine all directions E R2 along (0,0) exists. which
Let W denote the set of smooth functions f(x) in CⓇ such that f"(x) = -f(x). That is, W = {f(x) in "S"(t) = -f(x)} . W is a subspace of C . For all a and b, a sin(x) + bcos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b) Let's say we knew that dim(W)=2. Show that (sin(x),cos(x)} is...
3. Find lim f(,y) if it exists, and determine if f is continuous at (0,0. (x,y)--(0,0) (a) f(1,y) = (b) f(x,y) = { 0 1-y if(x, y) + (0,0) if(x,y) = (0,0) 4. Find y (a) 3.c- 5xy + tan xy = 0. (b) In y + sin(x - y) = 1.
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1 11-1, but that its value %(0.0) (Vf(0,0).u), fr at least one sucli u. (b) Show that the partial derivatives of f are not continuous at (0,0)
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
2. Let f:R2 + R be defined by gry, if (x, y) + (0,0) f(x,y) := { x2 + y2 + 1 0 if (x, y) = (0,0). Show that OL (0, y) = 0 for all y E R and f(x,0) = x for all x E R. Prove that bebu (0,0) + (0,0).
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed