Use Fundamental Theorem of Calculus to evaluate the following: d a) sin t dt dx

Q: Let c (t) = (e", sin(-21), t'), and c2 (t) = (e", cos(t), -t) d [c (t) c2 (t)] = [C (t) x c2 (t)] =…

A: To find d/dt(c1(t).c(t)) d/dt(c1(t)*c2(t))

Q: A particle moving along R(t) = r cos tî + r sin tj, r > 0, has an instantaneous speed that varies…

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Q: Find the differential of each fuction. a) %3D dy= ? b) U COS U dy= ?

A: Given,  (a) y = s7+8s  (b) y = u cosu  We have to find the differential of each function.

Q: Let f(x, y, z) = ln () Vxy – sin (cos (zy)) , evaluate ху %3D дх ду

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Q: Use symmetry to evaluate the followingintegrals.

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Q: Apply runge kutta method of order two on du/dx = sin(xu), u•=0.1,h=0.1 to find u(0,3)

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Q: Show that the following function satisfies the two-dimensional Laplace's equation. z = In(x? + y?) +…

A: Solution:Given   z=ln (x2+y2)+2 tan-1yxTo prove :∂2z∂x2+∂2z∂y2=0Formula:∂∂xln…

Q: Obtain the following Laplace ti 1) L{cos3t + 5e-t} 2) L{e¬3t (2t² – 5t + 1)} 3) L{sin7t – cosh3t} -

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Q: Let z = e* sin(y+a), show that %3D ôx? oy?

A: Given, z=e-xsiny+a, then we have to show that ∂2z∂x2+∂2z∂y2=0.

Q: Let y = (sin z+ cos r)e", x E [0, 27]. %3D

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Q: Exercise 3. Verify the following equalities: a) -st sin t dt = (1 – e 28) L[sin t](s). (1-e

A: Given, Verify the following

Q: Let R(t) = cos ti+ sin t j+ 2k, 0<t < • Compute - D;||R'(t)||-

A: We are asked to compute ||R'(t)||

Q: Prove that J seccu) du - Ln/Sec (u)+ tan (w) + C

A: Integrated in general

Q: 9) f(t)=tu,(1) 10) f(t)=u,(1)sin(t)

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Q: Use Part 1 of the Fundamental Theorem to simplify the following expressions.

A: Let us consider a function ddx∫0xftdt. The fundamental theorem of the calculus states that,…

Q: Show that the following function satisfies the two-dimensional Laplace's equation. z = In(x² + y²) +…

A: The given function is -  z = ln(x2 + y2) + 2tan-1yx We have to test whether the above function…

Q: r(t) = (tan(5t))i+t cos(t)j – Vtk

A: Given: rt=tan-15ti+tcostj-tk

Q: Differentiate with respect to t. y = a cos(t) + e sin(t) y' =

A: Given : y = acost+t2sintt

Q: Differentiate with respect to t. y = a cos t + t2 sin t

A: Differentiate the function with respect to t.

Q: (b) Show that, the function u = In(x² + y²) + 2 tan" satisfies Laplace's equation.

A: A Laplace equation is a 2nd-order partial differential equation. This is defined by,…

Q: find dy/dx

A: According to Leibnitz  rule of differentiation:

Q: Let r(t) = t²i + t cos(πt)j + sin(ät) k, find ſª r(t) dt.

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Q: du Q. Show that du and dy for u = In Vx2 + y? %3D dy and v = tan

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Q: Show that the function z = e* sin y satisfies Laplace's equation a²z a?z = 0 дх? ду?

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Q: Consider the following position function. r(t) = (2 sin t,2 cos t) for 0sts 2x |v(t)| =]

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Q: Using a suitable substitution, the integral T/4 9 (13+tan t)8 sec? t dt of

A: Given,          ∫0π4913+tant8sec2tdt

Q: Show that the following function satisfies the two-dimensional Laplace's equation. z = In(x² + y?) +…

A: Solution:Givenz=ln (x2+y2)+2 tan-1 yxFormula:ddxln x=1xddxtan-1x=11+x2ddx(xn)=nxn-1To prove:z…

Q: Differentiate with respect to t. y = a cos t + 2 sin t

A: The given equation is y=acost+t2sint.

Q: Let / = x tan(x) dx. The approximation of I using the two-point Gaussian quadrature formula is:

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Q: Prove that ∫ cot u du = ln∣sin u∣ + C.

A: Prove that ∫ cot u du = ln∣sin u∣ + C

Q: ha)= Jo So Sin't dt

A: We have to find the given integral .

Q: If r (t) = (cos(t), sin(t), 2) , then fr (t) dt %3D

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Q: Show that the following function satisfies the two-dimensional Laplace's equation. z = In(x? + y²) +…

A: Solution:Given   z=ln (x2+y2)+2 tan-1yxTo prove :∂2z∂x2+∂2z∂y2=0Formula:∂∂xln…

Q: If r (t) = (cos(t), sin(t), 2) , then ſr (t) dt =

A: r(t)=&lt; cos(t), sin(t) ,2&gt;

Q: sin (at

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Q: Use the convolution theorem to show that cos T sin (t – T) dr 1 sin t. %3D 2

A: Convolution theorem use integration of product of 2 functions and transforms them to pointwise…

Q: If u = x3 + y3 where x = a cos t, y = b sin t, then du dt

A: Solution

Q: Evaluate the integral. 7/2 cos(t) dt V1+ sin?(t)

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Q: Prove the following hyperbolic trig identity. cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y).

A: cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y)

Q: Use Part 1 of the Fundamental Theorem to simplify the following expressions.

A: To find the derivative of the integral, one need to use fundamental theorem of calculus, by…

Q: If r (t) = (sin(t), cos(t), 2). then ſr (t) dt =

A: we have to find the following

Q: Let f(x,y) = x sin ry. Find: (a) fr(T,y) (b) fy(x, y)

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Q: Use laplace trasfarng fo Solve dt² 2e cosi2t +e Sin at

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Q: Does the fundamental theorem of calculus imply that e' dt = e*?

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Q: Prove that d/dz sin(z) = cos(z) using Cauchy-Riemann equations

A: Cauchy-Riemann Equation- If f(z)=u(x, y)+i v(x,  y) is analytic, then In particular, These…

Q: Integrate the following (a) / tan(z)dx (b) Ju 1 dr /ez – 1 In 2

A: Evaluate the Integrals: (a) ∫tan x dx=∫sin xcos x dx Let cos x =t implies -sin x dx =dt , substitute…

Q: Verify the rule by differentiating ∫ du/(a2 + u2) = (1/a) arctan (u/a) + C . Let a &gt; 0.

A: Consider the given integral identity. ∫dua2+u2=1aarctanua+C Differentiate the LHS of the equation…

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = 5t cos(t) dt %3D F'(x) =

A: Simple use the second fundamental theorem of calculas   go Through the solution..