3. Consider the circle as a vector-valued function r(t) = cos(t)i + sin(t)j. Calculate limt→+∞ r(t) (if it exists).

Q: Evaluate [F·dr if F(x, y)=(4e³)ī +(4xe³)j and the curve C' is line segments going from (0,0) to…

A: To evaluate ∫C F⇀·dr⇀ if F⇀x, y=4eyi⇀+4xeyj⇀ and the curve C is line segments from 0, 0 to 5, 4 to…

Q: 2. For this entire page, let r(t) = (4 sin(t), 3t, 4 cos(t)) (a) Circle the correct graph for the…

A: The objective of this question is to understand the properties of the vector function r(t) = (4…

Q: Sketch the graph of 7(t) = sin(t)i – 2cos(t)j–tk in 3-space; then find and draw the tangent vector…

A: Given, r→t=sinti^-2costj^-tk^

Q: Let 7(t) = (t + 4, – e*, 3 sin(3t)) - Find the unit tangent vector T (t) at the point t = :0 T(0) =

A: We have to find

Q: Find the vector equation of the tangent line to F(t) = (2 sin 2t, cos 2t, – 6t) at the point where…

A: Consider the given vector. rt=2sint,cos2t,-6t   Find the derivative of the vector with respect to t.…

Q: Let f (x, y) = e3x-49). Find the equation for the tangent plane to the graph of f at the point…

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Q: Let 7(t) = (– 5t4 – 4, – 2e“, 2 sin(3t)) Find the unit tangent vector T (t) at the point t = 0 T (0)…

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Q: Let C₁ be the straight line from the point (1,0) to the point (0,1) in Figure 1. Let C₂ be an…

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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: Answer the following questions involving the vector-valued function 7 (t) cos(at),sin(at)): (a)…

A: Please see the answer below

Q: (z)dz

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Q: Consider the tangent line to the curve r(t) = t2i −(1/(t + 1))j +(4 − t2)k at the point P(4, 1, 0).…

A: Given the tangent line to the curve r→t=t2i^−1t+1j^+4−t2k^  at the point P4,1,0. We have to…

Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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Q: 4. Find the perimeter of a rectangle whose area is 84 m². The length is 5 m more than its width.

A: Let the length is L and width is b. Given: L=b+5 Area of a rectangle=84m2 As we know that the area…

Q: Let C be the curve in the x-y plane defined by the equation (x – h) + v – k)* =? where h, k, and r…

A: Equation of the curve is x-h2+y-k2=r2 & a, b is a point on the given curve. a). Slope of the…

Q: 4t Let (t) = ( - t +2, -3e-4, -5 sin( - 2t)) t¹ Find the unit tangent vector T (t) at the point t =…

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Q: Write the given expression in an expanded form. Assume that variables represent positive numbers.…

A: Given, lnxx3-7x2+9   We have to write the above expression in expanded form.

Q: Consider the parametric curve C: x=t³ + 1, y = t² – t,0 <t < 3. Find the equation of the tangent(s)…

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Q: Find T, N, and K for the plane curve r(t) = (4cost + 4t sin t)i + (4sin t-4t cost)j, t> 0.

A: