* Let X be the vector space of continuous real valued functions in [-1, 1]. Define (·, ·) : X×X → R as (x, y) = | x(1) y(1) dt. (a) Show that (-, ·) is a inner product. (b) Let Y c X be the set of even continuous real valued functions in [-1, 1]. That is, y € Y y(1) = y(-1) Vt e [-1,1]. Let Zc X be the set of odd continuous real valued functions in [-1,1]. That is, ze Z z(t) = -z(-1) VtE [-1,1]. Show that Y IZ.

Q: (d) Let f(x, y) = 2x² + xy-y². Prove that the directional derivative of f(x, y) at point x = (3,-2)…

A:

Q: Example// Find the centroid enclosed function Y=6x-x². y = x

A: Extremely sorry for the graphing mistake.

Q: Prove f-1(E U F) = f-\(E) U f-1(F)

A: We have to prove that f−1E∪F=f−1E∪f−1F. Let x∈f−1E∪F hence, x∈f−1E∪F⇔fx∈E∪F…

Q: Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every…

A:

Q: Q2) homeomorphism. Prove or disprove: The function f: (R, T1) - (R, 13) defined by f(x) = 3x is a…

A: Clearly the given function f(x)=3x is continuous. Now we want to check whether it is bijective,

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

A:

Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Let f be a function of R sucht hat Ә f(x, y) = f(x,y); for all (x, y) = R then prove that - y^ f(x,…

A: Since given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation for…

Q: Let f(x, y) = and let g(x, y) O O rºy6 26+y6 O 0 0 6x³y³ 6x6 + 6y6 Which of the above functions are…

A: fx,y=x6y6x6+y6if x,y≠0,00if x,y=0,0 gx,y=6x3y36x6+6y6if x,y≠0,00if x,y=0,0

Q: Assume that f(x, y) is a function given by if (r, y) # (0,0), f(x, y) 0 if (r, y) = (0, 0). Page 1…

A:

Q: { 2xy? 9x2 + y4 (x, y) # (0,0) (x, y) = (0,0) f (x, y) ·

A: Topic = Limits and  Continuity

Q: le can define an operator * : 2'(R?) n'(R?) by defining *dx = dy and *dy = -dx, and extending…

A:

Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

A:

Q: Use the inner product (f, 9) = | f(x)g(x) dr %3D in the vector space CU[0, 1] of continuous…

A:

Q: Let /: X →Y be a function, A, BC X and C, DCY. Show that: 1. f(AUB) = f(A) uf(B) 2. If AC B, then…

A:

Q: i+j+k t+ 0 Q2/B/ f(x){ (ti +t?j+t3k t =0 * Prove the continuity of function using definition of…

A:

Q: find the orthogonal trjectories of x=c^y^2

A: We need to find orthogonal trajectories.

Q: Determine the set of points at which the function is continuous. if (x, y) # (0, 0) 5x? + y? f(x, y)…

A: We know that given function is continuous in its domain so we need to check only at point where…

Q: Let f(x, y) and let g(x, y) = O O 8xy8 828+8y8 O 3xy6 3x² + 3y¹2 Which of the above functions are…

A:

Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: Let F(x , y , z) = g(y) + h(z) + 5 In(x - 2) , where g(y) and h(z) are differentiable functions of…

A: If a function is a function of more than one independent variable then the derivative of this…

Q: Let C[0, 1] be the vector space of continuous functions that are defined on the interval [0, 1].…

A:

Q: Let f be a function with continuous second derivatives, defined by: z = f (xy, x² + y") If we take u…

A: Solution:-

Q: Let u and v be defined implicitly as functions of x and y by the equations xv + yu 5 x² – y3 + uv² =…

A:

Q: what local linearizatoin of the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1)  (show all steps…

A:

Q: (b) 22). Compute the partial derivative frryz for f(x, y, z) = cos(cos(x³ + z²)) + x² In(y² +

A: Given f(x,y,z)=cos(cos(x3+z2))+x2ln(y2+z2).We need to find the partial derivative fxxyz for the…

Q: Use the limit to find partial derivative of…

A:

Q: 3. Let V denote the vector space of all functions f: Rn R, equipped with addition + : V × V → V…

A:

Q: Find the direction of maximum change of f(x, y, z) = xey + z² at (1, In 2,2).

A: Introduction: A function's maximum directional derivatives at a particular point P are produced in…

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = (x, y) = (0,0) y+2xy² x²+y2 Let…

A: