Let D be a disk in R². This exercise shows that if V f(x, y) = 0 for all (x, y) in D, then f is constant. Consider points P = (a, b), Q = (c, d), and R = (c, b) as in Figure 18. (a) Use single-variable calculus to show that f is constant along the segments PR and RQ. (b) Conclude that f (P) = f(Q) for any two points P, Q e D. Q = (c, d) P = (a, b) R = (c, b) Disk D FIGURE

Q: Let f be a function such that Vf(x, y) = (2x, 1) g(x, y) = x² + y². Find all the points on the curve…

A: Let f be a function such that ∇f(x, y)=2x, 1 and let g(x, y)=x2+y2 We have to find all the points on…

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

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Q: Show that the into R² defined by T₁ = ( 9₁ ₂1₂) — 172,9₁) and T2: (F₁, F₂) → (Y¶₁₂ Y {₂) : are…

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Q: 1. Determine the domain and range of the vector-valued function r(t) = (√t+3, In(2 – t), t²)

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Q: 3. Find the fixed point(s) of f : [1, 0) → [2, 0) defined by f(x) = vx² + x.

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Q: 1. Let f: R" → R be the function f(x) = cx+b for some fixed vector c ER" and scalar b. Show that f…

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Q: a) f : R? → R, f (x, y) = Vy sin x 5) f : R2 → R. f (r. u) = GGrul + In (a2 + 1? – 1). /4 – 72 -

A: We will find the domain

Q: Draw a picture on the z-plane

A: In this question, we draw a picture of the given curve in C(t). here, first, we write the real and…

Q: Let A = (1,1) and B = (1,5) be in the Poincar ́e plane. Find C = (xC,yC) for which xC > 1 and…

A: The natural point that comes in our mind is (2,6).

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: (c) If f is a C² scalar function, then V × (Vƒ) = 0. True False (d) The lines 1₁ (t) = (1 − t, 1+ t,…

A: (C) If c is scalar function Means      curl(grad ƒ)  is = 0   Therefore correct ans is  True

Q: – x²y? x2 + y? f(x, y)

A: fx, y=-x2y2x2+y2 Now, z0=fa, b=f1, -2=-12-2212+-22=-45 Now partially differentiating fx,…

Q: (b) Prove that Vx, y ≤R : x² + xy + y² ≥ 0, and that x² + xy + y² = 0 if and only if x = y = 0.…

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Q: Find the plane through P = (4, −1, 9) containing the line r(t) =〈1, 4, −3〉+ t〈2, 1, 1〉.

A: The plane contains the line r(t)=1, 4, -3+ t2, 1, 1, so a vector in the plane is 2, 1, 1. The line…

Q: Let T: R2 R2 be a linear trasformation such that T (x1, 22) T (x) = (3, 1). (x1 +x2, 2x1 + 3x2).…

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Q: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

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Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Find B and t for the space curve r(t) = N 11 ਕਰਜ |·+|· 1 +1 + 2 3 ()+( 3 +1 j, t> 0.

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Q: Let F(x, y, z) = (14xyz + 7sin(x))i + 7x²zj + 7x²yk. Find a function f such that F = Vf. f(x, y, z)…

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Q: Suppose f: X→Y, and A, B CY. Prove that (a) f-¹(AUB) = ƒ−¹(A) U ƒ−¹(B). (b) f-¹(ANB) = f¯¹(A) n…

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Q: Consider the following function. T: R2 → R3, T(x, y) = (Vx, 4xy, vy) Find the following images for…

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Q: Use the function to find the image of v and the preimage of w. T(V1, V2) = (V1 + V2, V1 - V½), v =…

A: To solve this problem we follow the following steps : Finding image :  Write down the given…

Q: Show that r|(1) and r2(1) define the same line, where rị(t) = (3, –1, 4) + t (8, 12, –6) r2(1) =…

A: Two lines have the same direction if their direction vectors v1→ and v2→ are directly proportional.…

Q: 1 (b) Find the span of vectors i = |0 and j = |1 in R3

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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: Consider the following vector functions: r(t)= tcost i + 2t sint j + 2 tk for t ≥ 0 t≥ 0. a. The…

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Q: Let r(t) = (-5t³ – 1, t² + 5t³, 3t²). Find the line (L) tangent to (t) at the point (39, -36, 12)

A: r→(t)=⟨−5t3−1,t2+5t3,3t2⟩Find the line (L) tangent to r→(t) at the point (39,−36,12)