1. Find the absolute minimum and maximum values taken by the func- tion f(x, y) = x² + y² – x – y +1 on the circle x? + y? = 1.

Q: 6. Find the shortest distance from B6,0,0) to the pane Ax + By tCz +D= 0 where A, B,C and D SR .

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Q: Suppose f(x, y, z) = In(7x + 9y+7z). Find an equation for the level surface of f that passes through…

A: Given Data: The level of the curve for fx,y,z=In 7x2+9y+7z2 is of the form,…

Q: 5. Prove that whole length of the curve x3 + y2/3 = a/3 is 6a.

A: Given that x23+y23=a23 To find the length of the curve:…

Q: 3) Find the exact length of the curve: x = y = In(1+t); 0<t<2.

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Q: 7-The tangent of the curve x^2+y^2- 2xy3D0 at the point (1,1), is * O a. y=2-x. O b. y=x-2. C.…

A: We find equation of tangent to the curve

Q: 1. Find the length of the curve y = 3 xi - () x , from x = 1 to x = 4.

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Q: 8. Show that the one parameter family of curves y² = 4k(k + x) (k E R) are self orthog- onal.

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Q: Find absolute maximum and minimum values for f (x, y) = x² + 14xy + y, defined on the disc D = {(x,…

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Q: 2. Prove that the following curves are intersecting with a right angle: 5y-2x+ y –x*y =0 2y+5x+x*…

A: Given curves are  5y-2x+y2-x2y=0and2y+5x+x4-x3y2=0

Q: Find constants a and b so that the minimum for the parabola f(x) = x- + ax + bis at the point (5,…

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Q: 1. Find the minimum distance from the point (0, 0, 0) to the plane x- y+z = 3.

A: perpendicular distance

Q: 1. Find the are length of the curve x (+ 2)/2 from y=0 to y-1

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Q: c) Find the minimum distance from the point (4,0,6) to the plane x- y +2 = 3

A: The solution is given below. .

Q: 1- Find the minimum and maximum values of the function f(x,y) = 2x2 + 3xy + 2y2 + x - y inside and…

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Q: Find the exact length of the curve. y= In{1 – x²), o s 1. 0 sxs! 7. I Ju =

A: We will find the exact length of the curve as shown below

Q: 6. Find the positively oriented simple closed curve C for which th value of the line integral I 4 –…

A: Consider the given integral. I=∫Cy3−ydx−2x3dy Apply the green theorem. ∫CPdx−Qdy=∬D∂Q∂x−∂P∂ydA…

Q: Find the two (x, y, z) points on the surface z² = x² + 3y² that are closest to the point (2, 2, 0)…

A: The surface is z2=x2+3y2. Obtain the distance of the point x, y, z to the point 2, 2, 0: Dx, y,…

Q: 1 in the first quadrant x>0, y>0. ху Find the minimum value of f (x, y) =x+8y+- -How do you know…

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Q: 9. Find the point (x, y) on the line y = 2x + 3 such that the distance from (x, y) to (1, 1) is as…

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Q: Consider the inmplicitly defined curve xy +3x2 4. %3D O Verify that the point (1, 1) is on the…

A: Consider, the implicitely defined curve                          xy2+3x2=4 We have to verify the…

Q: 3. Find the length of the curve y =x* +÷x¯² from x = 1 to x = 4

A: 3 Given function isfx=y=18x4+14x-2  from x=1 to x=4now we can writen as fx=x48+14x2 on 1,4The lenght…

Q: 16. Let h(x, y) = xy – 2x². Find the minimum and maximum values of h on the rectangle where 0 < x <…

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Q: Find the maximum and minimum values of f(x, y) = x² + 2y² on the quarter circle x² + y² ≤ 4 with r,…

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Q: Determine the x-coordinate of the local maximum of the curve y = (2x - 1)2 (3x+4) in the interval…

A: Given: y=2x-123x+4

Q: 18. Find the absolute maximum and absolute minimum values for the function f (x,y) = e-**-y* (x² +…

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Q: 4. Assume the function f(x, y) = æ/y² Is given with a condition (restriction) y – x + 1 = 0. Amongst…

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Q: б. (a) Sketch the level curves z = k for k = 0, 1, 2 if z = x2 – y?.

A: The level curves of a function f of two variables are the curves with equations f(x, y) = c, where c…

Q: 1. Find the absolute minimum value of 0 < x < 4 f (x, y) = 4x² + 2xy – 2y? – 18x on the rectangular…

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Q: 1. Find the smallest distance from the point (0, c, 0) to the surface y = x² + z².

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Q: 10 نقاط Check whether the following points: {(0,0), G5.E=)} indicate a maximum, minimum or saddle…

A: Solution:-

Q: 1 Find the points of discontinuity of y = where и +и - 2 1 u = * - 1

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Q: 2. Find the absolute maximum and minimum values of f(x, y) = 4x+6y-(r²+y?) on the rectangular region…

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Q: Find the exact length of the curve y = In(1 – x²), 0 Sx S

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Q: Find the maximum and minimum values attained by f(x, y, z) = 2xyz on the unit ball x2 + y2 + z2 < 1.…

A: fx,y,z=2xyz on the unit ball x2+y2+z2≤1 gx,y,z=x2+y2+z2-1 ∇f=∂f∂x,∂f∂y,∂f∂z ∇g=2yz,2xz,2yx…

Q: (ii) Find the extremum of Z= x2 +y2 subject to x + 4y 2 and examine whether the %3D %3D stationary…

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Q: 2 Find the points on the curve ax+ 2bxy + ay = c,ay = c, 0<a<b<c,whose distance from the origin is…

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Q: 7.Find the maximum and minimum distances from (0,0) to the curve 5x2+6xy+5y2=8

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Q: st, enter DNE.) f(x, y) = x² + y² + 4x 10y6 minimum maximum point (x, y, z) = (x, y, z) = (x, y, z)…

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Q: A rectangular plate with an area of 36 cm? is to be produced on a lathe in such a way that the…

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Q: By minimizing the function ƒ(x, y, u, y) = (x - u)2 + (y - y)2 subject to the constraints y = x + 1…

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Q: 2) a) Find the absolute maximum and minimum values of the function f(x, y) = e*+y* in the disk x? +…

A: Note: We'll answer the first question since the exact one wasn't specified. Please submit a new…

Q: Show that the g(x,y) has minimum of 0 in the region 0 ≤ x ≤ 5 and 0 ≤ y ≤ 5, occurring at x =3 and y…

A: It is given that ;                   gx, y = x2+y-112 + x+y2-72   ..............(I) We have to show…

Q: 27. Find the length of the curve Vcos 2t dt y = from x = 0 to x = "/4.

A: The given curve is We know that by Second Fundamental theorem of calculus ,if F(x) is continuous on…

Q: Find the maximum and minimum attained by f(x, y) = 3xy + 2x + 3y in x2 + y < 1. You must use…

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Q: 3. Discuss how to evaluate f () dz , where C is a simple closed curve C=10 enclosing the origin.

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Q: Find the point at which the function f(r,y) = 2 in z+2 In y attains a maximum subject to the con-…

A: Maxima or minima: For any function f(x), 1) Evaluate f'(x)=0 for critical point. 2) Evaluate f''(x)…

Q: Suppose that the Celsius temperature at the point (x, y, z) on the sphere x2 + y2 + z2 = 1 is T =…

A: We have given; The temperature T at any point (x, y, z) in space isT = 400xyz2and the unit spherex2…

Q: Suppose that the Celsius temperature at the point (x,y,z) on the sphere x +y +z =1 is T= 1600xyz“.…

A: This is a problem related to maxima and minima. Based on the general formula we will answer the…

Q: Find the length of the curve defined by y = 3 In 3 from x = 8 to x = 9.

A: Use derivative of function and length of curve formula .

Q: 4 Determine the surface area of a funnel that is generated by revolving the graph of 1 y = f(x) =…

A: Given  y=fx=x3+112xand interval 1,2