4 (a) Let F=xy²i+x²yj. (i) Calculate the value of the line integral [F.dr, where C is the path y=x¹², 0≤x≤1; y=1/x, 1≤x≤2.

Q: (1 point) Evaluate the line integral y dx + x dy where C is the parameterized path x 1 x = t², y =…

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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: 2. Find and classify all the extrema of the function f(s, t) = (4s² + 2t²)e¯

A: 2

Q: Let F(t) = (- t + 1, t – 2t", - 2t). Find the line (L) tangent to r (t) at the point (- 15, – 24, –…

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Q: Evaluate (2x-y)dx+(x+3y)dy along the path C: curve on y = 1-x² from (0,1) to (1,0).

A: Given the line integralI=∫C(2x−y)dx+(x+3y)dyCurve y=1−x2

Q: 2. Show that the function u p(ry) + Vry v(y/x) provides the equation Pu = 0. ду? dx²

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Q: 10. Find the derivative of y = V3x + 2. c) d) 3x a) 2V3x+2

A: Using derivatives

Q: "; Find y as a function of x if x?y" + 5æy' – 21y = x°, y(1) = 7, y(1) = -5. y =

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Q: The value of the line integral fydx + xydy over the straight-line path C from (0, 0) to (1, 2)…

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Q: Solve: 2 0 -1 0 2 -6 1 -2 -3 -2 0 1 -5

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Q: 4. a) Find an approximation for f (x, y) = at the point (1.97,1.02). V10-r-5y b) Check Euler's…

A: (a) Approximate f(x,y)=410-x2-5y2 at the point (1.97,1.02) (b) Check Euler Theorem for…

Q: Find the length of the curve y = In(e - 1) – In(e + 1), In 2 sxsIn 3.

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Q: If y In(z- y) then y at the point where y = 0 is equal to

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Q: The level curves of the function f (x, y) = V2x2 + y? – 1 are

A: Please refer to the image below

Q: 5. Find the length of the curve given by y= Vt* – 1 dt and 1<x<2.

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Q: Evaluate the line integral where F y = 2x² from (-1, 2) to (2,8).

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Q: Draw the level curves of f(x,y)= x² + 9y² for k=0, k=1, k=9.

A: Given curve is, fx,y=x2+9y2

Q: (47) Let C be the curve y = 5x where 1 ≤x≤9. Calculate the length of C.

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

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Q: 3. (ey + 1)²e-dx + (ex + 1)³e¯*dy = 0

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Q: Q.8 Find the gradient Vw ofthe function w = In /x² + y² + z². %3D

A: Given function W=ln(√(x2+y2+z2)) Now we will find the gradient as shown below

Q: 3. X is N(5,9). Obtain the pdf of Y=3X-5 and Z=5-X.

A: The X is N(5,9).Know that if X is normally distributed then the PDF of X isWant to calculate the PDF…

Q: Evaluate S (xy + x + 2) ds along the curve r(t) = 2ti + tj + (3 - 2t)k, Osts 1.

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Q: Sc (xy+x+z) ds along the curve FO 2i ++(8 – 20)k where 0 st 1. Calculate

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Q: 19. Differentiate the following functions. g) y = 2 h) y = xsin-1x+ V1- x2

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