Show F(z,y) = (8ry³ + 8)i + (12z³y² +2e²)j is conservative by finding a potential function f for F, and use f to compute S F-dr, where C is the curve given by r(t) = (2 sin³ t)i + (sin" (5t)) j for 0 St≤m/2. 7-dr

Q: Find the length of the curve r(t) =〈2/5t^5/2−2/3t^3/2, t^2, 1/2t^2+t+ 7〉for 0≤t≤3.

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Q: 2. a). Find the velocity and acceleration of a particle whose position function is x(t)=…

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Q: Use Green's Thoerem to evaluate F dr. where F(r, y) = (-3Vx² + 2, 7 tan '(x)) and C is the triangle…

A: We have to evaluate by Green's theorem.

Q: Verify that fy = fx ху ух f (x, y) = 2x'y' -x'y b. f(x,y)=(x' + y²)° а.

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Q: 2. Evaluate f (x^²y³ − 2y)dx + (3x + x³y²)dy where is the curve given below oriented in clockwise…

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Q: Find r'(t), r"(t), r'(t) · r"(t), and r'(t) x r"(t). - r(t) = 3t²i – 4tj + ²t³k 2 (a) r' (t) (b)…

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Q: ind the fangent line for... 1x1) = +²+et ( (1) = sh (k) + t at t=0 B) r(e) = sinG + case at 6: T

A: Explaination of the answer is as follows

Q: 7. At which direction the directional derivative of f(x, y, z) = 2x² +4y² +9z² at P(1, 2, 1) has a…

A: To determine the direction in which the directional derivative of fx, y, z=2x2+4y2+9z2 at P1, 2, 1…

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: Suppose r'(t) has units of m/s. Explain why T(t) = r'(t)/ | r'(t) | is dimensionless (has no units)…

A: Suppose r'(t) has units of m/s. To explain why Let r(t) be a differential vector valued functionNow…

Q: Please find the requested derivative of the following functions. s²-4st+t² = 4, find 25 ds dt - ds…

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Q: Find f. f "(t) = 2et + 5 sin t, f(0) = 0, f(x) = 0 f(t) =

A: We can solve the given problem using the concept of application of integral.

Q: 12. Evaluate c Vx² + y² ds_along the curve r(f) = (4 cos t)i + (4 sin t)j + 3tk, –27 < t < 27.

A: According to the given information, it is required to evaluate the integral over the given curve.

Q: Parametrize the tangent line to the curve (t3 - 2t +3,t² +t-1, t² + 1) at the point t = -1.

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Q: Compute ∫C(2x+√y)ds, where Cis the curve from (0,0) to (2,4) along y=x^2.

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Q: Determine each of the following derivative values that involve functions that are defined in terms…

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Q: Use the method of Characteristic to solve U t tuux = 2t, u(x, 0) = 3x.

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Q: Write down the equation of motion for the point particle of mass m moving in the Kepler potential…

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Q: Find f. f"(t) = 5et + 4 sin t, f(0) = 0, f(1T) = 0 f(t)

A: topic- differential equation

Q: Compute r''(t) when r(t) = (+14,4t, sin t). r''(t) =

A: The first derivative of a vector function r(t) = (f(t), g(t), h(t)) is given by r'(t) = (f'(t),…

Q: Find the arclingth of A) r(6) = 4cos (0) OS OS T ... B) / X (²) = +²+1 2 y(t) = 2³² +2 1≤t≤ 3

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Q: 5. (а) Find the directional derivative of f at P in the direction of ā if f(x,y) = y In(x) ; P(1,4);…

A: To find The directional derivative of f at p in the direction of a.