Consider the curve, y with parameter (, and function f in 3 dimensional Euclidean space with coordinates I,y, z, and a point p: y= (sin 6, cos C,C), (1) f =z² - y² +z², (2) (3) p= (0,1,0). ompute df /dC along y. Evaluate it numerically at p.

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: Find the tangent plane to the graph of f(x, y) = x² + y² + 4x - 2y +1 at the point (0,0,1).

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Q: 3. Let f(x, y) = xy+ y². Find the equation of the tangent line in parametric form in the x-direction…

A: Solve the following

Q: Find the unit tangent vector of the given curve. r(t) = (7t cost-7 sin t)j + (7t sin t + 7 cost) k…

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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: B) r(t) = (4 cos t, 4 sin t, 16)

A: We consider x=rcos(theta) and  y=rsin(theta)  Therefore r^2=x^2+y^2

Q: For the curve defined by F(t) = (e¯', 4t, e') find the unit tangent vector, unit normal vector,…

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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: How would you evaluate |F-dr, where C is the curve with param etrizati on r(t) for astsb?

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Q: Find the equation of the tangent plane to z = e² + y + y² +8 at the point (0, 4, 269). Z= 0

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

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Q: Find an equation of the tangent plane to: 3x +y = z at P(-1,1,-2) %3D

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Q: Find the slope of the tangent line to the parametric curve 2te x(t) y(t) at the point P(0, 1). 2t +e…

A: xt=2t etyt=2t+etAt  P 0,1,  x=0,   y=1  ∴x=0⇒2t et=0             ⇒t=0   dxdt=ddt2t et= 2 t…

Q: Use the theorem to find the slope of the line x = 4t, y = 2t, for - ∞ < t < ∞.

A: According to the theorem, the slope is given by - dydx = dydtdxdt…

Q: 4. Consider the parametric curve x = cost, y = sin cost, te [-T, π]. (a) Show that (0,0) is a point…

A: Consider the parametric curve given by  and , where  belongs to .a) To show that  is a point of…

Q: Find the slope of the tangent line to the parametric curve x= t³ + 4t at the | y = t² + 3t point t =…

A: m= -253/52Explanation:Step 1: Step 2: Step 3: Step 4:

Q: Let C be the curve defined by r(t) = (2 cos t,t,3 sin t), where 0<t< 2t, (a) Use a table of values…

A: