3- vt) = 3 sin 21 3 2 2

Q: Find the points on the cardioid r = 1−cosθ at which there is a horizontal tangent line, a vertical…

A: Consider the given: r=1−cosθWrite the cartesian…

Q: Find the length L of the curve R(t) = e* cos(3t) i + e* sin(3t) j + 3e*k over the interval [1, 5]. L…

A: given: Rt=etcos3ti+etsin3tj+3etk interval 1,5

Q: The linear approximation at z = 0 to sin(8x) is A + Bx where A is: %3D and where Bis:

A: Given : The linear approximation at x=0 to sin(8x) is Ax + B   To Find : Value of A and B

Q: What is the maximum value of f(t) = sin t + cos t in the interval 0<t< =?

A: Given; f(t) = Sin t + Cos t.

Q: Consider the graph of f(x, y) = excos(xy²). Find an equation of its tangent plane at (0,0, 1) and…

A:

Q: 4 - Find the direction A in which the function f(x, y) = x cos y + y sin x decreases most rapidly at…

A: We have to find the direction A in which the given function decreases most rapidly.

Q: The velocity of a particle moving back and forth on a line isy = ds/dt = 6 sin 2t m/sec for all t.…

A:

Q: Let the velocity of a particle be given by v(t) = –sin(t). What is the displacement of the particle…

A: We need to find displacement and total distance.

Q: Two masses hanging side by side from springs have positionss1 = 2 sin t and s2 = sin 2t,…

A:

Q: - Sva cos 2t dt from x= 0 to x= Find the length of the curve y =

A:

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…

A:

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

A:

Q: Express in form y=f(x) by eleminating the parameter x = t , y = tan−1(t3 + et )

A: It is given that,

Q: Find the mass of a thin wire shaped in the form of a helix, r(t) = ( 3 cos t, 3 sin t, 4t) with 0 <…

A: The curve is given by  rt=⟨3cost,3sint,4t⟩ , 0≤t≤3π2                   (i) Then,…

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: The velocity of a particle moving back and forth on a line is v = ds/dt = 6 sin(2t) m/sec for all t.…

A: Given:

Q: Find the length L of the curve R(t) = e" cos(2t) i + e* sin(2t) j+2e"k over the interval [1, 4]. L =

A:

Q: Find the tangent to y = cos x at x = 5- O A. y=x+7 O B. y= -x+7 O C. y= -x-7 O D. y= 1

A:

Q: Suppose that a particle following the given path c(t) flies off on a tangent at t = to. Compute the…

A: Consider the path of the particle           c(t) = (2t2, t3-4t, 0) The velocity of vector is…

Q: A particle moves in the xy-plane in such a way that its path is defined by x = e^t cos t and y = e^t…

A: Given that, x = e^t cos t y = e^t sin 2t Differentiating x with respect to t we get, dx/dt = e^t cos…

Q: Find the unit tangent of the following curve at the specified point. r(t) = 2ti + 1 · sin(2t)j+3 ·…

A: We have to find the unit tangent vector at t=pi/3

Q: 3. Given that the acceleration function of an object moving along a line is a(t) = 3 cos(t)-2 sin(t)…

A: Proceed as shown

Q: Find the linear approximation of the function at the given point: f(x, y) = 2 sin x – 3 cos y at (0,…

A:

Q: Suppose that the position of a body moving along a coordinate line at time t is a. s = a cos kt + b…

A: Part (a):Part (a):Given,Given,

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

A:

Q: 4. Use the arc length formula to show that the length of a linear function f(x) = mx +c on the…

A:

Q: Given a particle is moving in a curve r(t) = cos 7t i + sin 7t j + 6t k. Find the tangential…

A: Given a particle is moving in a curve rt=cos 7t i^+sin 7t j^+6t k^to find the tangential and normal…

Q: The slope of the tangent line to the curve parametrized by the equations x (t) = -2(cos(t) + t…

A: Slope of tangent is equal to dy/dx.

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

A: 1)- Given: r→t=sin 2t, cos 2t from t=0 to t=2π. The objective is to determine the length of the…

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

A: Since you have posted multiple question in this question which are not interlinked so i have solved…

Q: Find the linear approximation at x = 0 to each of f(x) (x + 1)?, g(x) = 1+ sin(2x) and h(x)= e2*.…

A:

Q: Find an equation of the tangent to the curve at the point corresponding to the given value of the…

A:

Q: Please solve

A: Parametrization of tangent lines is given by

Q: Let y(t) be the position function of a particle in rectlinear motion. Suppose at time t = 0 hr, that…

A:

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

A: I am going to solve the given problem by finding the arc length of the given parametric curve to get…

Q: Given a particle is moving in a curve r(t) = cos  7t i + sin 7t j +  8t k. Find the tangential…

A: Given a particle is moving in a curve, r(t) = cos  7t i + sin 7t j +  8t k. To Find the tangential…

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

A: Given : x = cos2ty = cos t0 ≤ t ≤ 4π

Q: The graph of the function f(x) = sin(x+ 8) is shown along with the points (– 8, f( – 8)) and (– 7,…

A: (1) f(x)=sin(x+8)f'(x)=cos(x+8)slope =f'(-8)=cos(0)=1equation of tangent at…

Q: The velocity of a particle moving back and forth on a line is y = ds/dt = 6 sin 2t m/sec for all t.…

A: given the velocity of a particle moving back and forth on a line is y = ds/dt = 6 sin 2t m/sec for…

Q: Find an equation of the tangent llne to the về at the given poln 6x y = (x + 1)2' (0, 0)

A:

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

A:

Q: The parametric form of the tangent line to the image of f(t) = (5t², 2,t – 2) at t = -1 is L(t) =

A: Topic = Application of derivative

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

A: Topic = Application of integration

Q: 1. If an object moves with velocity v(t) = sin(t) + 3t and begins at position p(0) = 0, find the…

A: integration of velocity funct

Q: A rod with density p(x)=4+sin(x) (in grams per cm) lies on the x-axis between x=0 and x=pi/3. Find…

A: Please see the explanation below

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

A:

Q: use Euler's method to obtain an approximation to four decimal places of the indicated value by first…

A:

Q: The position of a particle in the xy-plane at time t is given by r(t) = (cos 2t) i+ (2 sin 2t) j…

A: Given: rt=cos2ti+2sin2tj For finding equation in x and y whose graph is the path of the particle, we…

Q: Find F. dř for F = 5yi – (sin y)j on the curve counterclockwise around the unit circle C starting at…

A:

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.