Problem 4: Freely using the fact that if u(s,t) is a “nice enough" function of the variables s and t, then the following identity holds: d d u(s, t)dt = | ds"(s, t)dt, ds Prove that d. (L{tf(t)})(s) (L{f})(s). ds = --

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: Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, and the following applies.…

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Q: Tell whether inductive or deductive reasoning is involved. "The passing score is 40%. Out of 80…

A: Difference between Inductive and deductive reasoning Inductive Reasoning Deductive reasoning…

Q: Let B = {1, t − 8, (t − 8)²} : - and C = {1, t − 5, (t − 5)²} - be two bases for P2. The function…

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Q: If f(r, v)= +y-2y, find fa (2, 1), f (2, 1), fay(2, 1) and fy(2, 1).

A: Given a function f(x,y)= x3+x2y3-2y2 Now, fxx= ∂2f∂x2=∂∂x∂f∂x fyy= ∂2f∂y2=∂∂y∂f∂y…

Q: Let the invertible function Evaluate f(x) = 3x³+x+3 1 = ["] † ^¹ (2²) dar dx (Hint: You may use the…

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Q: d Calculateri(t) r2(t)] and [r₁(t) × r2(t)] first by differentiating the product directly and then…

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Q: = (15) Determine whether or not the functions y₁ 4x³, y2 = π, y3 = 12x², and y4 = 24x are linearly…

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Q: 2. Find a function f and a number a such that f(t) -dt = tanx. Give proper reasoning.

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Q: 3. Let f: A-→ B and g : B A be functions. Suppose that y = f (x) if and only if r =g (y). Prove that…

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Q: Suppose f: R → R is continuously differentiable. Show that if f'(x) > 0 for xo R, then there exists…

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Q: fi(x) 1 A = {fi, f2, f3} c F(R) and f(x) = f:(x) = |x+ 1| PROVE: A is linearly independent. (A L.I.)

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Q: If f(x)=x² -4, where x is an integer and 0<x<3, then which of the following sets of ordered pairs…

A: We have,

Q: uppose that w = f(x, y, z), where z = r(u, v), y = y(u, v), and z = z(u, v). 'hich of the following…

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Q: Define briefly u(t) and 8(t)

A: u(t): u(t) represents the unit step function, which is defined as u(t) = 0 for t &lt; 0 and u(t) = 1…

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: The functions f1 (z), f2 (z) and f3 (z) defined in the following equations (being EC) are given. i.…

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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: Find the inverse of laplace, f(t) if: 2s + 3 s2 – 6s + 8 1. F(s) =

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Q: 1. Find the following for the function f(x, y) = 3x³y² - 5x + 26y - 9. a. f(1,2)= b. fy(x, y) = c.…

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

A: Given that

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: If A%3 2x3 -Зх +2:х € [О, 2) then in (R, T,), A %3D а. [-3, 1] b. [-1, 1] с. [1, 3] С. d. [0, 4] e.…

A: Open sets and Closed sets: The sets in topology τ are called the open sets of the topological space…

Q: 6. Let f be a function such that f(1) = 1, ƒ′(1) = 4, ƒ”(1) = −2 and ² ƒ""(t)(2 – t)²dt = 3. Find…

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Q: 1. Determine whether the following pairs of operators commute dx х (a) (b) dx

A: The function f(x) is a rule that associates or fixes a number to the particular point x. When an…

Q: 1. For T € L(V,W), prove that null(T) = (range(T*)) and range(T) = (null(T*))+.

A: Given these, we can now prove the statements:  Proving (null(T)=(range(T∗))⊥):By definition, for any…

Q: 1. Find the domains and ranges of f, g, f + g, and f. g: f(x) = x, g(x) = Vx – 1

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Q: Problem 2. Now consider the relation D' on the domain z+ where rD'y if and only if r|y. Use the…

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Q: (5) Which of the following sets are linearly independent. Show all work. set 1:[1, e, ex] or set…

A: Step 1:Solution-Step 2:Step 3: Step 4:

Q: Let u be a root of ƒ = t³ − t² +t+ 2 € Q[t] and K : = (a) Show that f = mo(u). (b) Express…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.