valuate (F. dr along the path r(t) = (t. sint.

Q: dy Differentiate parametrically to find dx in terms of t: x = 4t?,y= 2t3

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Q: Estimate ∆ y using differentials equ ∆y ≈ dy = f ' (a)dx y = cosx, a= π/6, dx = 0.014

A: Given y = cos x  a= π/6, dx = 0.014

Q: Find the equation of the tangent plane to z = ey + x + x2 + 11 at the point (5, 0, 42).

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Q: What is the point of intersection of the tangent lines to the curve r(t) = sin(πt)i + 8sin(πt)j +…

A: Let us consider the given curve r(t) the tangent line is obtained as Find the derivative of the…

Q: 5) Sketch the curve X = e', Y = e?t + 1

A: Given, The curve  X = et  and  Y = e2t+1 We have to sketch the curve.

Q: calculate dy/dx at the point indicated. c(t) = (t3 + t, t2 − 1), t = 3

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Q: Find the rightward-pointing (positive x-direction) unit tangent to the curve P = {(x, y) E R² |y =…

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Q: Plot the curve c(t) = (cos t, esin t ) for 0 ≤ t ≤ 2π

A: The plot of the curve for t within 0 to 2pi is shown in figure below.

Q: find y'' for the curve cosx-siny=4

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Q: Cvaluate (xy) s ds where C is the curve y = x², for 0 < x < 2.

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Q: a - Find h (Stet sinh t sint dt)

A: The objective is to find ∫te-3tsintsinhtdt It is known that, sint=et-e-t2isinht=eit-e-it2…

Q: Consider the curve defined by xy + y2 = 21 Determine dy/dx for this curve at the point (4, 3)

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Q: Find dy/dx and dy/dx. x = e°, y = te-t dy %3D dx %3D For which values of t is the curve concave…

A: Given: x=ety=te-t Find dydx and d2ydx2 as shown below.

Q: Find an equation for the line tangent to the curve x= 2sin(t), y= 4cos(t) at the point defined by…

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Q: dy +y de : 2 et dx Arowar 1- ye" (c -x*) 1= yet (c -x*) %3D

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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: The line y = mx + 1 is a tangent to the curve y2 = 4x .Find the value of m

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Q: Evaluate y dx + x dy along the curve y x² from (0,0) to (1, 1).

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Q: h(x) = S*(8t³ + 2t)dt, use the FTC Part I to find h'(x). %3D

A: Given :  hx = ∫1x8t3 + 2tdt

Q: dy Sketch the direction field for dx y II

A: The direction field is as follows

Q: %3D Find the equation of the tangent line at (1, 1) of 6x²y- ncos(y) = 7m y= -2nx +3T B y=2nx+3n C.…

A: Solving the problem by doing the derivative and then calculate the equation of tangent line.

Q: Find the condition that the line x cos a + y ain a =p may touch the curve yh =1. am bm

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Q: dy -from the function y= sint, x= cost as a parameter. da? Obtain

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Q: Find the moment of inertia, with respect to the line x=1, of the area enclosed by the loop of the…

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Q: Find the differential of the function. y = V6 - t8 dy %3D

A: Solution: The objective is to find the derivative of the given function

Q: tiny, no bug lands on a piece of graph paper and walks along a path so that t minutes after it…

A: Increasing function

Q: Sketch the curve. And indicate the direction of increasing T at T =0

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Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: dy of the curve yln x + 5x² + y' = 13 at the point (1,2) dx b) Find

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

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Q: Compute xe" dx + x?y dy along the curve x 3t, y = t2, 0 < t < 1.

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Q: Find the length of the curve y = ln (sec x), 0 <I<÷.

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Q: u(x, y) = arctan a = (-1,0)

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Q: At the point (a, b) the curve T =t-4t, y=t2 +2t, t eR has a vertical tangent line. What is the sum a…

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Q: Find the equation of the tangent line to the parametric curve. x = t1/2 y = t2/3 at t = 1. Also…

A: The curve is defined by the parametric equations as x(t) = t1/2 and y(t) = t 2/3   Apply the chain…

Q: dy in terms of t: Differentiate parametrically to find dx 1 y = +2 t+1

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Q: Scy dx –x dy; C is the cardioid r=1- cos 0.

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Q: Q4] Find the equation of the tangent plane and normal line to the surface z + 8 = x ey cos z at…

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Q: y(z) along the curve given by r" = e*.

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Q: dy Sketch the direction field for = x – 2y. dx х — |

A: The direction field is as follows

Q: Find the lenght of the curve y = x- tnx, 1+/dy 2. dx %3D %3D

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Q: 3. Find the value of c such that the line y = x + 3 is tangent to the curve y = cvx. C =

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Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: Find dy/dx and d²y/dx2. x = et, y = te-t dy dx d²y dx2 For which values of t is the curve concave…

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Q: Tangent to the curve z? = 2xy + x2 , z = 3 ,at the point ( 1, 4,3)

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Q: dy Differentiate parametrically to find dx in terms of t: %3t2, y3 6t

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Q: Find the exact length of the curve (x=t?cost ly=t?sint Osts2n (n? + 4)3/2 - 4] 3 a. 4 + 4)3/2 – 1]…

A: The given implicit function is,  x=t2cost, y=t2sint0≤t≤2π

Q: Integrate to find the length of the curve. 8 S = 2 - + 4t dt 4 + 8 2t ++) dt = (? +t ln(t) 1 II

A: