Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 ln cos t k,     0 ≤ t ≤ π/4

Q: 1. Consider the curve C: r(t) = (t - sin t)i + (1– cos t)j, 0StS 2. %3| | (a) Find an equation of…

A: Since solve the first question for you. If you wanting any specific question to be solved then…

Q: A particle moves along the x-axis so that at any time t, t 2 0, its acceleration is a(t) = -4…

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Q: 15. Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = 3 sin t, y = 4t, z…

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Q: Q1. If f(x.y,z) = cosh²(\x² In y) + tan-(ey*NE) + sec*(sin Vz In x²) Find fx, fy, fz fyx

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Q: Suppose the position function of a particle is given r(t) = (√3 cos (t), t-4, √3 sin(t)). If the…

A: Given : r(t)=3cos(t), t-4, 3sin(t) is the position function of a particle. To find : If the particle…

Q: Find the point on the curve r(t) = (5 sin t)i + (5 cos t)j + 12t k at a distance 26p units along the…

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Q: Suppose the position function of a particle is given by r(t)=(√3 sin(t), t-2, √3 cos(t)). If the…

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Q: 4 .Using n=8 approximate the value of cos(1+Vx )dx using 0. (a) the Midpoint Rule, (b) the Trapezoid…

A: Formula used: Midpoint rule- The interval [a,b] into n subintervals of equal width,…

Q: 13. Find the value of t in the given interval at which the given curve is not smooth. r(t)=(5cost-…

A: Given curve: r(t)=5cos t-cos 5t, 5sin t-sin 5tt interval: π4,3π4

Q: A heat-seeking particle has the property that at any point (x, y) in the plane it moves in the…

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Q: Eliminate the arbitrary constant/s of this equation to obtain the DE. Express the DE in general…

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Q: 1. If H(y) = cos y – sin y, Find H (0), H (),H (1T), and also show that %3D + x) = H (T – x) = -H…

A: Given that H(y)=cosy-siny find H(0), Hπ2, Hπ Show that  Hπ2+x=Hπ-x=-H-x

Q: The slope of the tangent to the curve r = 3 sin 40 at 0 = 0 is %3D Şelect one: O a. -1 3D1 O b. 0 C.…

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Q: Q7) Set up the integral representing the length of the curve (sint, cos² t), cos"e). - – s1s. <t< 2

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Q: Find the point on the curve r(t) = (12 sin t)i - (12 cos t)j + 5t k at a distance 13π units along…

A: The smooth curve is rt=12sinti-12cost j+5t k at a distance L of 13π units a curve from point…

Q: 8. Find the length of the given curve over the indicated interval. r(t)= (r', 3 cos(t), 3t) [] 0, 2

A: Given r¯(t)=t2,3cost,3t in the interval [0,π2]

Q: 9. Using n= 8 approximate the value of cos(1+Vx )dx using (a) the Midpoint Rule, (b) the Trapezoid…

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Q: x = a(0 – sin 0), y = a(1 – cos 0). %3D %3D urve followed by the particle as a function of 0.

A: The given parametric equations are, x=a(θ-sinθ)...............(1) y=a(1-cosθ) .............(2)

Q: The function z = r' y is defined on the astroid r = cos 0, y = sin 0 where 0 < 0 < 2ñ. Use the dz…

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Q: Find the linear approximation to g(z) = cos(z + 1) at z = 0. Use the linear approximation to…

A: The formula to find the linear approximation Lz of a function fz at the point z=z0 Lz=fz0+f'z0z-z0

Q: Two particles start at the origin and move along the x-axis. For, 0 <t< 10 their respective position…

A: Given, x1=sint , x2=e-2t-1 ,0≤t≤10 To find the velocity, Differentiate with respect to t we get,…

Q: 7. Consider the function f: R → R³ given by: t+1 f(t) = sin(cos(t)), tet) t2 +4° Find f'(t) for all…

A: The derivative of the function is the measure of how fast the function is varying with respect to…

Q: а. z = t', y = ; 0 < t < 1 b. x = cos (2t), y = sin°(2t); 0 <t< 4

A: In the given question we have to find the exact length of the given curves.

Q: The value of f(z) = e² at z = 3 + 4i in terms of x + iy is O a. 1 O b. e cos 4+ie sin 4 O c. e*…

A: Given, We have to find the value of  fz = ez  at  z = 3+4i  in terms of  x+iy  .

Q: Find the speed (as a function of t ) of a particle whose position at time t seconds is c(t) = (sin t…

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Q: Q4: Find an equation for the line tangent to the curve 2y cos x = 4x sin y , at the point where x =…

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Q: d the area of the surface generated by revolving the curve x =cos (2t), y = 7+ sin ( 2° - xis. .....…

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Q: 9. Find the length of the curve: r = 1+ cos(20),0 < 0 s 2n

A: given equation of curve r=1+cos(2θ)   , 0≤θ≤2π as per our guideline, we are supposed to answer the…

Q: Find the length of the curve. x = cos t y = sint 0 <t< 2n

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Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t = 2n. %3D 2. Find the length…

A: As we know that, The expression for the length of vector-valued curve is: L=∫abx't2+y't2dt   1).…

Q: A heat-seeking particle has the property that at any point (x, y) in the plane, it moves in the…

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Q: Q1: Find the length of the cardioid r= 1 + cos 0 0 < 0 <T

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

Q: The function z = cos 0, y = sin³ 0 where 0 < 0 < 27. Use the x2 Vy is defined on the astroid x = dz…

A: The chain rule is stated as follows:

Q: 5. Express the curve by an equation in x and y. x(t) = 2- sin(t) y(t) = cos(t) a. (x-2)² + y² =1 b.…

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Q: 11)Consider the curve C in R3 parameterized by: r(t) = sin(t) î +t² î+3 cos(t) k, con 0 <t < T An…

A: Given that, r(t) = sin(t)i + t^2j + 3cos(t)k Let x(t) = Sin(t) y(t) = t^2 z(t) = 3cos(t)

Q: Suppose a particle begins to move from the origin(t = 0) following the curve in space. r(t) =…

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Q: Find T, Ñ and B for the curve 7 (t) = (3 cos(3t), 3 sin(3t), 5t) at the point t = 0 Give your…

A: Space Curves and T, N, B vectors The unit tangent vector T, the unit normal vector N and the unit…

Q: Question 9

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Q: f cos(mx) cos(nx) dx is equal to {A} [sin(m + n) + sin(m − n)] dx {B} [sin(m + n) - sin(m - n)] dx A…

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Q: Find the root for the function f(z) = r(cos(r) – 1) at 5. steps by using the Secant Method for the…

A: The secant method is  xn+2=xnfxn+1-xn+1fxnfxn+1-fxn            → i And let the initial conditions…

Q: Find the points on the interval -n s0s n at which the cardioid r = 1- cos(0) has a vertical or…

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Q: f(t) is given by: 3t for 0 = 2.

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Q: 7. Find the length of the curve f(1) = In(sin r), 0 < r < a/6.

A: Topic = Application of integration

Q: 7. Find the length of the curve: r = 1 + cos 0

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Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

A: Differentiate x with respect t. dxdt=ddtcos2t=-2sin t cost  Differentiate y with respect to t.…

Q: 3 Evaluate F. dr along the curve C if F(x, y) = (x² + y²) ²(xi+yj), and C : r(t) = e' sin ti+ e' cos…

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Q: Find the length of the curve defined by the following function for 0 <t < 2n. 7(t) = (– cos 2t)i +…

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Q: For the smooth curve, find any points where the tangent line is either horizontal or vertical. x(t)…

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Q: 2 tan-1(y). x In(xyz), q ƏV when x = y = % = 1. ду Let V = eP cos(qr), with p = z cos(TY) and r = By…

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.