1. Evaluate f(xy+2²)ds where Cis an arc of the helix C x = cost, y=sint, z = t which joins the points (1,0,0) & (-1,0,7).

Q: Estimate the area under the line y= 2x+1 using six rectangles and right endponits over the interval…

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Q: 6. Consider the parametric curve x(t) = 1 + cost+cos 2t, y(t) = sint + sin 2t = sin t(1+2 cos t)…

A: Given that                Then            At  , we need to find the value of .

Q: Find DF of fcxiy)=x²e° at the point (-2,0)-

A: Given function is fx,y=x2ey. We have to find the ∇f at the point -2,0. We know that ∇f=∂f∂xi+∂f∂yj…

Q: How do the paths r1(t) = (cost,sin t) and r2(t) = (sin t, cost)around the unit circle differ?

A: Given :                   r1(t) = cos t , sin tr2(t) = sin t , cos t

Q: Which of the following is dy for the curve dx defined by x° – y³ = 2xy? 3x2-2y а. 2x+3y2 3x2-2y b.…

A: Find the derivative of the given curve and match with the correct option.

Q: 8. For the curve given by r(t)=√√² +8 i + In(t² + 8)j + tk 2 find: a. r'(t)

A: Introduction: For a vector curve rt=ati+btj+ctk, the tangent line at the point of contact t=t0 is…

Q: Suppose that the temperature T at point (x, y, z) is given by T(x, y, z) = sin(xyz) and that an…

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Q: At how many point(s) does the helix with vector equation r(t) : (cos t, sin t, t) intersect the…

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Q: c) A curve has a parametric equation x = 2t - 1 and y = t* -t'. dy Evaluate dx? 1 at t = 2

A: Given, The parametric equations x=2t3-1, y =t4-t3.

Q: Show that the function f(x, y) x4 + y? has no limit as (x, y) approaches (0,0).

A: Note: Since we can solve at most one problem at a time, we have solved the first one. Kindly…

Q: 3. Given F(x,y)=(y,2x), find F-T ds where curve C is the parabola y=x from point (0,0) to point…

A: We use line integration to evaluate this.

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 =

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Q: if y=sin (x), z=cos (x), v=exp(x), where could you plot y, z, v as shown below!

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Q: Sketch the curve r(t)=f°i+tj and indicate the orientation. Find r'(t) and sketch r(1) & r°(1).

A: First we will plot the given parameters.

Q: The helix F(t) = 4cos(t)i + 2 sin(t)j + 5tk is shown below. By setting the last component to zero…

A: Given that,Equation of a helix isr→(t)=4cos(t)i^+2sin(t)j^+5tk^The projection of this curve into the…

Q: F(t) = %3D a. elliptical helix |F(t) = b. conical helix %3D ]F(t) = c. line d. circular helix…

A: We will use standard parametric form of these curve to match   For circle x = a cos(t) y = a sin(t)…

Q: Find the length of the curve traced out by the path c(t) Answer= = (cos(3t), sin(3t), 6t) for 0 ≤ t…

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Q: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…

Q: 1. Let y be the part of the curve y =r² running from (-1,1) to (2,4). Calculate ry dr + (x+ y) dy.

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Q: 4- Find the direction and direction from the point (1, 3) for which the values off do not change,…

A: 4)Given isfx,y=e2y arc tan3y3x at point 1,3A function  fx,y has no change in any direction that is…

Q: Find the domain of the vector function 7(t) = Domain: {t| <t< }

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Q: 1) Let 7(t) = (t, 2 cos t, 2 sin t), -4 <t< 4. Find the length of the curve.

A: In

Q: 10. Find the value of 2 for the parametric curve xx= y = 2t2 -6 at the point (1.-6).

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Q: 2 + v - , 2+z+ 11. 0. di

A: