Consider linear vector space c¹ ([a,b]) = {f: [a,b] → Rf and f are continous on [a, b]} ||f||0,1:= max If(x)| + max f(x)| Let f(x) = sin(x). Find fo.1 in C¹([0,1]).

Q: 2. Consider the floor function [.] whose output is the largest integer less than or equal to the…

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Q: Find the domain of the vector function r(t) = (t, √t — 4, √5 — t Domain: {t | <t< }

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Q: Find the limits of the following vector functions. 2t³-t² tan (2t)\ 21² (i) (ii) lim In 1-0 5t lim…

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Q: Show that the operator U is positive U: 1² ([0,1]) → L²([0,1]), Uƒ (x) = X[{0,1] [~*\f(t)\dt

A: To show that the operator U is positive, need to demonstrate that for any f∈L2([0,1]), the inner…

Q: Find the domain of the vector-valued function. r(t) = n(t) 0 (-∞, +∞0) O (1, + ∞ ) O (0, +∞ ) O [1,…

A: Find the domain for all three components and then take intersection of those.

Q: 1) Find the length of the function r = (1+ 2t, 3 – 2t, v2t²) on 0 <t < 1. %3D

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Q: Suppose f(x, y) = (a) ▼ f(x, y) = (b) ▼ f(-0.7, 7) = tan(x) + y and u is the unit vector in the…

A: Given that f(x,y)=tan(x)+y  and u is the unit vector in the direction of &lt;2,3&gt;now, ∇f(x,y)=fx,…

Q: Represent the following equations in R? by a vector-valued functions. Imet gaoglecomis shwing your n…

A: Given

Q: Let r(t) =[x(t), y(t), z(t)]. Provide a definition for the continuityof the vector function r at a…

A: Define continuity of the vector function.

Q: 2. Consider the vector-valued function R(t) = { Show that is continuous at t = 1. (el-¹, 1-², In (2-…

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Q: THEOREM 4.4 (LINEAR TRANSLATION) Let Y= aX + b; then: My(0) = eb" My(a0).

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Q: Consider the functions fi(x) = x and f2(x) = 5-9cx on the interval [0, 1]. (a) Find the value of the…

A: Consider the functions  and  on the interval .(a) Find the value of the constant  so that  and  are…

Q: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Verify that x − p is orthogonal to p.

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Q: Let A(t) = f(t)i + f₂(t)) and B(t) = g(t)i+g₂(t)), te[0, 1] where fi, f2, 81, 82 are continuous fun-…

A: Solution

Q: Suppose that f: R² → R² is a function such that f (1, 2) = the linear approximation. (Use syntax…

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Q: 5) Consider the vector function r = (9- 2t, 4+2t, √2t²) a. Find the length of the path on 0 ≤ t ≤1.…

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Q: 4: Which of the following functions has gradient vector [4x₁x₂ 2x² 3x²]¹? A: f (X) = 2x{+2x3+x3x3 B:…

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Q: Find the limits of the following vector functions. -2) 21³-t² tan (21) 21²" (i) (ii) lim In 1-0 5t…

A: Since you have posted multiple questions, we are supposed to answer only first  question. If you…

Q: show that the given vector functions are linearly independent on (−∞,∞).

A: To show that the given vector functions are linearly independent x1t=   et-et,        x2t=etet

Q: 5) Consider the vector function r = (9 – 2t, 4 + 2t, vZt²) a. Find the length of the path on 0<t<1.…

A: Given vector function The length of the path is found by the formula Given interval is 0 to 1

Q: Given the vector: j(x) sin(2x) X = X ,ex, 1-√1+x (2, 1,-2), x = 0 Determine its domain and show it…

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Q: 3. Use the Wronskian to determine whether the functions fi = e*,f2 sinx and f3 = x are linearly…

A: To find- Use the Wronskain to determine the functions f1 = ex, f2 = sinx and f3 = x3 are linearly…

Q: 1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2). Let f,…

A: We can just multiply h and k and integrate  Then after simplifying we are getting answer

Q: xy -) zi – xk -) xỉ + zk

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Q: a) Show that if u, v, and w are differentiable vector functions of t, then d du V X w +u - dt dv dw…

A: According to our guideline we are supposed to answer only one asked question.kindly repost other…

Q: Use the Wronskian to determine if the function vectors 7,= x? + e* and 72= e²x are linearly…

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Q: 2. Consider a function g(x, y, z) = 1 + ln(3x + 6y +9z) and a vector u = (0, -1, 1). (a) Find the…

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Q: Find the values of real numbers a, b, and c such that the vector-valued function - {(at?, a(t + 1),…

A: A function f is continuous at x=a if

Q: Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁,2, 8₁, 82 are continuous…

A: First we use the condition of parallel vectors and then by intermediate value theorem we will find…