3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+ t4x lim dt = 2 1+ xt

Q: Use Laplace transforms to solve the initial-value problem j – 9y = f(t), y(0) = 2, ý(0) = 0, where…

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Q: Let P(z) = | tin(t) dt. Use part 1 of the fundamental theorem of calculus to determine if P(z) is…

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Q: Find g'(x) if g(x) = t² cos 6t dt %3D 2

A: We can apply fundamental theorem of calculus here.  The second fundamental theorem of calculus…

Q: Let f(x, y,z) = x² sin y + y e², where x =t + 3,y = net-1 and z = In 2t. dL at t = 1 is: 1-2n O -12n…

A: Given, fx,y,z=x2siny+yez, where x=t+3,y=πet-1 and z=ln 2t.   The objective is to determine dfdt at…

Q: 30. Find the continuous function f(t) satisfying csin z for all r. 2+ f(t)dt = sin2015 /3

A: According to Leibnitz rule:

Q: Use Laplace transform to solve f(t) if y" + 3y' + 2y = t + 1 with y(0) = 1, y'(0) = 0

A: The Laplace transform of the function f(t) is represented as F(s) or L[f(t)], it is defined as…

Q: Let y(t) be the solution of dy/dt = 0.3y(2 − y) with y(0) = 1. Determine limt→∞ y(t) without solving…

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Q: (a) express dw/ dt as a function of t, both by usingthe Chain Rule and by expressing w in terms of t…

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Q: Let y(t) be the solution of dy/dt= 0.3y(2 − y) with y(0) = 1. Determine lim t→∞y(t) without solving…

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Q: t² + 1 t 1 %3D lim→1- h(t) = ?

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Q: Assume that y is a continuously differentiable furction in t Evaluate a) tP (+),A (+)) d tant sect…

A: Given: y is a continuously differentiable function in t. (a) To find: ∫xexπy(t)y'(t)dt

Q: .3 Suppose that w = x³yz + xy + z + 3 and x = 3 cos t ,y = 3 sin t , dw z = 2t. Evaluate dt t= ドIN

A: Ans:  we will use product rule of differentiation to solve this question.

Q: Assume x and y are functions of t. Evaluate dy,dt for each of the following.:…

A: Given:         8y3+x2=1dxdt=2x=3y=-1 Differentiate 8y3+x2=1 with respect to t,…

Q: For the relation siny – y² =x²y– x+x , is dx 3x²y+1 А. cos y – 2y –x T – 1 3x² + 2y – cos y cos y –…

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Q: Use the Laplace transform to solve the given initial-value problem. y" + 4y = sin tu(t - 2π), y(0) =…

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Q: integral f t^3 In t dt

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Q: If a and ß are positive constants, show that the particular solution of dy %3D У (В — ау) dt that…

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Q: Assume x and y are functions of t. Evaluate dy,dt for each of the following.:…

A: We have the given function in x and y as 2xy-5x+3y3=-51 On differentiating both of the side with…

Q: b) f(x) = In(1 – 3x) %3D c) f(x) = cos(t³) dt

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Q: 4- Calculate the Laplacian of the following functions: (a) T e* sin4y cos 3z. (b) V = x° x^+3xz y-…

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Q: Let A and B be constants. Prove that ifA > 0, then all solutions of dy/dt+ Ay = B approach the…

A: On some rearrangement you will get,

Q: Let y(t) satisfies dy dt 3ty-t-t 1+t2 for t2 0, and y(t = 0) = 1. Show that y(t) S 1+t for t2 0.

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Q: dr Let r = cos f(t) where f(0) = 0 and f'(0) = 3/2. Evaluate at dt t = 0. O None of these 3 00 O1

A: Given r=cosf(t),f(0)=0 and f'(0)=3π/2

Q: 1. Evaluate the limits. e' cos (t – 4) dt - 4 (а) lim x2 x2 – 4 e2 + 2 (b) lim x00 e2x + 100

A: Solve the following

Q: Let f (x,y, z) = xyz2 + 1 and suppose that x(t) = 2 – t, y(t) = In(t), and z(t) = 3t. df Use the…

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Q: (b) Prove that sint -dt > 0 t+1 for all a > 0.

A: Given:- ∫0x sintt+1dt >0

Q: Solve the following simultaneous DE using Laplace transforms. 2 dy fut dx 5 sin t = 0 dt −y+x+ dt dy…

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Q: If y(t) solves the IVP dy = y2021 y2019 y(2021)= -2019, dt then lim y(t) is: t- 00 Oa. 1 O b. 00 O.O…

A: Given differential equation is dydt=y2021-y2019.       ........1 It is an autonomous differential…

Q: of Let f (x, y) = tanh-¹(x-y) with x=e" and y= usinh (t). Then the value of dt (u,t) =(4, In 2) is…

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Q: 2. Let L:= D2 - x-D+4x2. Verify that Ly 0 where y(x) = sin(x). %3D %3D %3D

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Q: 2. Suppose that z = f (x, y) and both x and y are functions of a single variable t such that dy t2 –…

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Q: Suppose that r = x(t) and y = y(t) are both functions of t. If %3D y = 2x/2 + 3, and dx/dt = 5 when…

A: The given function is: y=2x12+3

Q: Give the function z f(x, y)=4x²+ 3y, x(t)=sin tand y (t)= cos t. zp =2sin t.cos t 2.1 Show that dt…

A: The derivative of the function can be calculated by differentiating the function with respect to the…

Q: 1. Evaluate the integral ab- a + b à) + a(b a)-2(à ab - bla dt in which a, b are the derivatives of…

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Q: Which of the following graphs is the slope field for 15/1 1~/1ST ~\~ in 13t-ii-i ~~~ -1 1 2 3 4 5…

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Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: Let T = x²y - xys + 6x, where x = r cos 0 and y = r sin 6. Use the chain rule to find dT/00.

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Q: Evaluate dt dW at t=0 for the function W = e' – In x , x = t², y= In t

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Q: Assume x and y are functions of t. Evaluate dy / dt for the following. y3 = 2x 4 + 6 ;        dx /…

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Q: ди ди Solve the following PDE by using Laplace transform: ax %3D at ди x > 0,t > 0 with boundary and…

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Q: Let w(t) = ((g(t))^2 + 1))^1/3, where g(t) is a continuous function. If g(t) = √7 and g'(0) = 2,…

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Q: 2t – 2 t2- 1, Let r(t) = - î - tan (t) ĵ+ In(2t) k. %3D (a) Determine the domain of r. (b) Compute…

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Q: Assume x and y are functions of t. Evaluate dy / dt for 4xey = 8 - ln 64 + 6ln x, with the…

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Q: 32 2 f is continuous and | f(t) dt = 6, find the integral / t* f(t) dt.

A: Given problem:-

Q: assume x and y are functions of t.  Evaluate dy/dt for the following: y3=2x4+6; dx/dt=2, x=1, y=2

A: The given function is  y3 = 2x4 + 6    ...... (1) dx/dt = 2, x = 1,   y = 2

Q: Assume x and y are functions of t. Evaluate DY/DT for 4xy-4x + 6y^3 = -258, with the conditions…

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Q: Use the Laplace transform to solve the initial value problem. y''+ 36y = sin(6t), y(0) = 1, y'(0) =…

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Q: If a and ß are positive constants, show that the particular solution of dy %3D У (В — ау) dt that…

A: Given that dydt=yβ-αy and y=y0 at t=0 We have to find the solution of the above equation.

Q: Assume x and y are functions of t. Evaluate dy,dt for each of the following .xey = 2 - ln 2 + ln x;…

A: It is given that, therefore the time derivative of the given function is, But it is also given…

Q: Assume x and y are functions of t. Evaluate dy over dt for the following. y = 2x 2 + 6 ; dx over dt…

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.