12. A linear map T: P (R) → R² satisfies T(x) = (1,-1) and T(1) (1, 2). Find a formula for T(a1x + ao), where a1x + ao E P1(R).

Q: Find R(t) if R′(t)=⟨1+2t,2e2t⟩ and R(0)=⟨1,3⟩.

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Q: Let r(t): = 1 t-1' +-1/2). Find the domain of r(t). et+1, t-1,

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Q: Let X {1,2, 3} and Y {3,4}. Define f : X → Y as follows: = f(2) = 3 and f(3) = 4. Draw an arrow…

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Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

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Q: C: 1. Is there a linear map from R2 to R³ such that T(1, 3) = (0, 0, 0) and T(-2,0) = (0, 0, 0) and…

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Q: Consider the function d: R² × R² → R defined by {} d(x, y) = min{|y2x₂, 1} if x₁ = y₁ otherwise, 1…

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Q: Find the linear approximation of the function f (x, y) x² + y² at the point (2,-1, 1)

A: Given function:- fx,y=5x2+y2

Q: Consider the function d: R² × R² → R$ defined by [min{\32 - x₂, 1} if x₁ = y₁ 1 otherwise, d(x, y) =…

A: A metric on a set X is defined as a function d: X×X→ℝ which satisfies the following conditions:…

Q: 1. Determine the linear dependence of the following functions: fi(t) = 2t – 3, f2(t) = t² + 1, f3(t)…

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Q: Let A = T(u) = T(v) = 30 03 4 and define T: R²R² by T(x) = Ax. Find the images under T of u = -7 a…

A: To solve the following

Q: xercise II.2. let F(): R² R³ and G(): R² R² be given by (fi(x)) f2(x) f3(x)) F(x) = (iii) f3 and ere…

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Q: Find a path c(t) that traces the line y = 2x + 1 from (1, 3) to (3, 7) for 0 ≤ t ≤ 1.

A: Given:

Q: A function is defined as f(x,y) = 25xy + 3x³ - 7x² sin(y) aj ay a2f (ii) Find and 1325 3²x (iii)…

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Q: 2. Let g (x, y) = 2x² – y² . Compute 9(1, 2), g (2, 1), g (1, 1), g (–1, 1) and g (2, –1).

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Q: 3. Consider the functions, m(x) = x + 3x and p(x) = 2x + 5, Find m(p(1)) b. Find m(p(x)) a. c. Find…

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Q: 1. Use the Chain Rule to find u and ou for w = x²y3 where x = dw at s2 +t2 and y = ds

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Q: 18. Let f(t) = t2 |t| and g(t) = t3. a. Show that the functions f(t) and g(t) are linearly dependent…

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Q: Find the functionr that satisfies the following conditions. r'(t) = (1,4t ,5t4); r(1) = (7, – 6, –…

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Q: F . dr, (c) Let f(x, y, z) = x³ + 2yª + 325 and F consists of three line segments from (1,0, 0) to…

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Q: 4. Let L: R² → R? be a linear operator. If L((1, 2)") = (-2, 3)" and L((1, – 1)") = (5, 2)" find the…

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Q: 1. Let f (-1,1) R, f(x) = (1+x³3)-3. Compute g': (0,1) → R where : (a) g(x) = få ƒ. 0 (b) g(x) = [ x…

A: To compute g′(x) using the formula for Differentiation Under the Integral Sign, we'll need to apply…