(a) Find the slopes of the surface in the x- and y-directions at the given point. (b) What is the direction of maximum increase of z from this point? 25 x2-y2 %D (3,0, 4)

Q: Where are the relative extrema (local maximum and minimum) on the curve? y = 3x² – x A. (0, 0) only…

A: given curve is y=3x2-x3 find the local maxima and local minima

Q: Q:3 Find the minimum distance from the surface x + y°-z = 1 to the origin.

A: The minimum distance from the surface

Q: Using Lagrange multiplier, find the minimum value of f(x,y) = 3x +ly on the circle x2 + y2 = 1. Give…

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Q: Find two values of d such that the line 2x + y = d is tangent to the ellipse 4x2 + y2 = 25. Find the…

A: Consider the equation of the ellipse  We know:- slope=dydx We evaluate implicit differentiation of…

Q: 3. (8.3) find the area of the surface obtained by rotating the curve y = v4- x2, -1<x < I about the…

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Q: Find the area of the surface generated when the curve is revolved about the y-axis. y=4x-1, for 1 <…

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Q: 4. Determine the slope of the common tangent to the parabolas y=x² +1 and y=-x² - 1. y=x*2+1 (a, b)…

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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: 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 slopes of the surface in the x- and y-directions at the given point. h(x, y) = x2 − y2 (-2,…

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Q: Find the point(s) on the surface xyz = 1 closest to the origin.

A: Given, g(x,y,z): xyz=1 distance from origin is  d=x2+y2+z2 f(x,y,z)=x2+y2+z2  square of distance…

Q: 6. Find an equation of the tangent plane to the surface z = y² – x² at the point (-4, 5, 9).

A: Recall: Equation of the tangent plane to the surface z=f(x,y)  at (x0,y0,z0) is        fxx-x0 +…

Q: Bonus: 1. Find the area of the surface generated when the curve x2 - 12y + y2 = 0 is revolved about…

A: Given:          x2−12y+y2=0  is revolved about the y−axis for 2≤x≤10.

Q: 2. The slope of the surface = ry² in the r-direction at the point (2,3) is. slope of this surface in…

A: The slope of the surface z=fx, y in the x-direction at the point is denoted by the value of partial…

Q: 11. What is the equation of the surface revolved about the x-axis for 2² - xz = ln x + 3x? A. y²…

A: Introduction: When a surface is revolved around the coordinate axis, another surface is obtained. In…

Q: Find the absolute minimum and maximum points of the function g(x, y) = x² – 2y² + 8y over the region…

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Q: x2 + 4 Find the equation of the tangent plane to the surface z= 2 at (4,3,-3) | 9

A: The equation of the tangent plane to the surface at (4,3,-3) is, 6x + 2y + 3z = 7  The detailed…

Q: sketch a typical level surface for the function. ƒ(x, y, z) = ln (x2 + y2 + z2)

A: The given function is  ƒ(x, y, z) = ln (x2 + y2 + z2). In order to sketch the typical level surface…

Q: Q3: Find the points on the surface x^2+z^2-25=0, closest to the origin.

A: Distance formula is d2=x2+y2+z2 .......(1) From the given equation z2=25-x2 ....... (2)

Q: Q3/Find the equation for the tangent plane and normal line to the surface Z = 9-x² + y² at the point…

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Q: Given a surface f(x,y) = -2x² + (5)y². Find the derivative in the direction of v = at the point (7,…

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Q: The area of an ellipse with axes of length 2a and 2b is nab. The percent change in the area when a…

A: Area of ellipse initially  = Ai  = πab    Now, lengths of axis are changed,   They are  [ 2(1.0099)a…

Q: A bug crawls on the surface z = x2 - y2 directly above a path in the xy-plane given by x = ƒ(t) and…

A: We are given that the bug crawls on the surface z = x2 - y2 and x = ƒ(t) and y = g(t).  So,…

Q: Find the slopes of the surface in the x- and y-directions at the given point.

A: z=25-x2-y2, 3, 0, 4 Partially differentiating with respect to x, we get ∂z∂x=1225-x2-y2·∂25-x2-y2∂x…

Q: Find the positively oriented simple closed curve C for which the value of the line integral     (y3…

A: The line integral is  ∫C(y3 − y) dx − 9x3dy The expression for Green's theorem is  ∫CP(x,y) dx…

Q: The roof of the Monterrey Campus gymnasium has the shape of the hyperbolic paraboloid given by the…

A: Here we have to use directional derivative  Directional derivative of a function f(x,y,z) in the…

Q: 1. 2. 3. 4. 5. 6. Match the equation to the graph of the surface. 1. z = x² + y

A: The given equation is z=x2+y2 This will form a 3D graph as there are 3 variables

Q: find the surface area of the portion of the curve x²+y²=4 from x=0 to x=2 when it is revolved about…

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Q: Q1: Find the maximum and minimum point on the surface f(x,y)= 5xy-7x -y² +3x -6y +2

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Q: Find the points on the surface y2 = 4 + xz that are closest to the origin. (Enter your answers from…

A: Consider the given surface equation y2=4+xz.  The distance d between the origin 0,0,0 and any point…

Q: Rotate the graph of g(x)=e^(−6x) on the interval [0,8] about the x-axis to generate a surface with…

A: Given that g(x)=e-6x To find the surface area by rotating the graph of g(x) on the interval [0,8]:

Q: Display the values of the functions in two ways: (a) by sketching the surface z = ƒ(x, y) and (b) by…

A: Consider the given function f(x,y)=y2

Q: 4. Parameterize the curve of intersection of the surfaces z = x² + y² and y² + (z – 1)² = 1.

A: parameterisation

Q: (a) Find the area of the surface generated by the given curve about x-axis y = V, (b) Find the area…

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Q: (a) Find an equation of the tangent plane to the surface at the given point. xyz = 12, (1, 3, 4) (b)…

A: Given: xyz=12 (1,3,4)

Q: (b)Let f(x, y) = / 3x + 2y. Find the slope of the surface z f(x, y) in the x-direction at the point…

A: Slope of z= f(x, y) in the y- direction at the point (4,2) Slope of in the y - direction means…

Q: Q3. Find the point(s) on the surface xyz 1 closest to the origin. %3D

A: Given: To find the points on the surface xyz=1 is given as,

Q: An equation of the tangent plane to the surface 2+y at the point p(1, 7, 3) is 2+x

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Q: Given a surface f(x,y) = -3x² + (3)y². Find the derivative in the direction of v = at the point (5,…

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Q: If ƒ(x, y) = 100 ( y + 1) represents the population density of a planar region on Earth, where x and…

A: Given: The reginal population is f(x,y)=100(y+1) And the curve is f(x,y)=100(y+1)

Q: Display the values of the functions in two ways: (a) by sketching the surface z = ƒ(x, y) and (b) by…

A: Entering x and y values in the function will give us the exact plot of how the function goes. From…

Q: 1. Sketch the graph of the level curves of the surface z = y + x^2 when k = −2, k = 0, and k = 2

A: Level curves problem.....

Q: Q3: Find the points on the surface x^2+z^2-16=0, closest to the origin.

A: The equation of the surface is x2+z2-16=0.

Q: Q/ Find the points have minimum distance from the origin to the surface. xyz²=1

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Q: Find the equation of the xy-trace, the equation of the xz-trace and the equation of the yz-trace of…

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Q: (b)Let f(x, y) = /3x +2y. Find the slope of the surface z = f(x,y) in the x-direction at the point…

A: It is given that, f(x,y)=3x+2y. It is known that the slope of any function is defined as,…

Q: 45. What is the maximum slope for the curve y= In(x +9)? %3D O A A 3 OB In 9 OC 3 OD There is no…

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Q: (a) Find an equation of the tangent plane to the surface at the given point. z = x² - y², (5, 3, 16)…

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Q: Find the slopes of the surface in the x- and y-directions at the given point.

A: Given surface, gx,y=4-x2-y2 and point 1,1,2

Q: (a) Find the slope of the surface, f(x,y)=1-(x-1)² -(y-2)² at the point (1, 2, 1) in the x-direction…

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