Let T: P' → R, T(p) S p(t)dt be a linear functional acting on polynomials of degree no more -1 than 1. Find an element in ker(T) that has 1 as a fixed point (i.e. p(1) = 1). Return the value of p(-1).

Q: Develop Laplace transform of the piecewise continuous function t, 0 ≤ t < 1 f(t): 2-t, 1st < 2 t≥2…

A: The solution is given as

Q: For the given functions, (a) express dt as a function of t, both by using the chain rule and by…

A: Topic = Derivative

Q: In Exercises 1-6, (a) express dw/dt as a function of 1, both by using the Chain Rule and by…

A: As per our company guidelines we are supposed to answer ?️only first question. Kindly repost other…

Q: Use the Fourier transform to solve the BVP Urr + Uyy = 0, -0 0 u (x, 0) = f (x) lim u (æ, y) = 0,…

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Q: Show that the two linearly independent solutions of xy" – y + xy = 0, x > 0 can %3D be obtained as…

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Q: b) Use Laplace transform to solve the system of differential equations UTM dr UTM x = 4y, dt y+Bet…

A:

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

A:

Q: a. Express dw/ dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

A:

Q: Determine the Laplace Transform of the given function 1-t, 0<t<1 1-1, 1. f(t) = [0, 1st

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: y/ + 2ry = 0, • State the order of the DE (1.1). • Classify the DE (1.1) in terms of linearity. •…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Use the Laplace Transform operator to find the Solution of the IVP, 3y + 5y- 2y = 4x°. y(0) = 0 У…

A: Given 3y''+5y'-2y=4x3  y(0)=0  y'(0)=0Taking Laplace Transform on both sideL(3y''+5y'-2y)=L(4x3)…

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

A: Consider the following equation: w=x2+y2 Substitute x=cost+sint and y=cost-sint in the above…

Q: f(t) = u(t− 3)(t - 3)e-4t

A: In this question, the concept of Fourier Transform is applied. Fourier Transform The Fourier…

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

A: Given, 1. w = x2 + y2 , x = cos t,  y = sin t;  t = π  2. w = x2 + y2 , x = cos t + sin t,  y = cost…

Q: Derive the Fourier transform for f(t) = te2t for t>0 and 0 for t<0.

A: The Fourier transform of the function f(t) is represented as F(k), it is defined as ∫-∞∞e-iktf(t)dt,…

Q: Using the second fundamental theorem of calculus and chain rule, compute dz t dt. t+1 2x

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Q: Let f(x, y, z) = x'y³ + z³ and st, y = st?, x = and (a) Calculate the primary derivatives af af dy…

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Q: Solve f'(t) + f(t)=3 where f(0)=0 using laplace transformation

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Q: Let z = g (x, y) = x³ + xy³ + y4 - 6x - 2y + 4, where x = r cos (0) and y дz Use the chain rule to…

A: We will use chain rule to find the derivative

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

A: A) The function x is differentiated as, x=cost⇒dxdt=-sint The function y is differentiated as,…

Q: Let f(x, y) xy, x = 1- Vt, and y = 1+ Vt. Find df using the chain rule. dt

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Q: =) Use the Laplace Transform operator to solve the IVP 3y" + 10 y' +7y = 4 y(0) = 0 %3D y'(0) = 0

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Q: Determine continuous time Fourier transform for the given signals. x(t) 1 1, t20 0, t < 0

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Q: C. Determine the Laplace transform of 1. F(x)= x2 2 f(x) = e®, a €R 2.

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Q: 3. Construct a suitable Liaponov function of the form ax + by and use it to determine the type of…

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: . Verify that the differential operator defined by…

A:

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

A: We find dw/dt using chain rule. Here plug x = cos t, y = sin t, z = 4sqrt(t) and simplify.

Q: Use the Laplace Transform method to solve the given system d²x (b) +- y = 0, dt2 dt dx dz 2 - x + -…

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Q: Use the Laplace transform to solve the system of the DE + 2y = 0, y'- 2: = 2, Yo = 0 = 0.

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Q: Determine if the following functions are linearly dependent or independent by determining their…

A: The Wronskian of a set of function y1, y2, y3 is defined as, If the Wronskian W is non-zero then…

Q: Exercise 11.3 Prove the equality [* | f(x) dx = fm({x : [ f(x)| ≥ t}) dt, 0 where m is Lebesgue…

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Q: Find Z transform for 3 F(t) = t * * F(t) = u(t-2)

A: Given that F(t)=t3 F(t)=u(t-2) We have to find the Z transformation of f(t).

Q: Find the Laplace Transform. f (t) = {1-3t for 03

A: The solution is given as

Q: Find the Laplace transform of the given function: f(1) = (1-6)u4 (1) - (1-4)u6 (1), where u. (t)…

A: We find Laplace transform of given function

Q: Consider the following. d²x = t² dt² d²x - = 6t dt² x(0) = 4, x'(0) = 0, y(0) = 0, y'(0) = 0 Give…

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Q: Let f1(x, y) - e cos(5y), f2(x, y) = x² – 25 y². Find Vf, and Vf2, and show that %3D (fv) = 0 for…

A:

Q: Suppose that w = f(x,y, z), x = h(s), y= g(s, t), z= k(s,t) and all the conditions required for the…

A: Given that w=fx,y,z, x=hs, y=gs,t and z=ks,t. We have to compute ∂w∂t by using chain rule. Since, by…

Q: Find a linear differential operator that annihilates the given function. f(x) = 3x2 - x sin 3x +…

A: Given function is fx=3x2−xsin3x+x2e−xcos4x We have to find linear differential operator that…

Q: Consider the following. w = In(x2 + y), x = 2t, y = 8 - t dw by using the appropriate Chain Rule. dt…

A: Solution is given below..

Q: Let T: C* (R) → C (R), T(f) = f' – 2f be a (linear) differential operator on the space of infinitely…

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Q: Using the second fundamental theorem of calculus and chain rule, compute t dt. t+1 dx 2x

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Q: Let w f(t,0) where t and 0 are functions ofr and s, %3D dw is equal to Or then according to the…

A:

Q: Suppose z = x² sin y, x = -s² – 21² , y = 8st. dz A. Use the chain rule to find and * as functions…

A:

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

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Show that the functions In x , In(x2) linearly dependent (0, co) are on

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Q: f (t) = tcos3 3t dt 0, e-4t

A: Given: f(t)=∫0∞tcos33te-4tdt To find: Value of integral using Laplace transform.

Q: 1. Find the linearization of r – z at a = 2.

A: Find the linearization of x3-x at a=2.

Q: Find the charge and current at time t by using Laplace transformation, in L-R-C circuit i.e., d²Q dQ…

A: As we know that laplace transform is given byLy''=s2Ys-sy0-y'0and L(y')=sYs-y(0)and Lδt=1

Q: Q2/ Express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

A:

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.