) Prove that the Point P(!,-) is a minimum Point of the curve y =lnx

Q: Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated…

A: To find the minimum distance from the given curve to the point using Lagrange multipliers method.…

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: Show that the normal line at any point of the circle x2 + y2 = a2 passes through the origin.

A:

Q: Determine the maximum curvature for the graph of f(x) = 6 In (3x). The maximum curvature is at x =

A:

Q: 4. Does there exist a simple closed unit-speed plane curve with curvature less than or equal to 2…

A: In a simple closed unit-speed plane curve, curvature =1r According to the question, Area of the…

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

A:

Q: Find the orthogonal trajectories of the family of curves. Sketch several members of each family y =…

A: We have given the curve; y = mx

Q: determine the maximum curvature for thegraph of each function.ƒ(x) = x/x + 1 for x >-1

A: Consider the function f(x)=xx+1 for x>-1. f'(x)=x+1-xx+12=1x+12f''(x)=-2x+13

Q: 3. For curve y ==x +4x2 - 12x , locate the possible maximum and minimum %3D points.

A:

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: Determine the maximum curvature for the graph of ​f(x)=5ln(8x).

A:

Q: What does it mean for two families of curves to be mutually orthogonal?

A: Two families of curve are said to be mutually orthogonal if they intersect at right angles. Angles…

Q: Find the orthogonal trajectories of the given family of curves: Circles through the origin with…

A:

Q: 8how that the square of the leng point on the curve by = (x + a)° , length of the subnormal at that

A: Subnormal and Sub tangent: Allow y=f(x) to be the differentiable curve at point P. Let the tangent…

Q: Show that the minimum distance from the origin to a point on theplane ax + by + cz = d is

A: Given: ax+by+cz=d we know that minimum distance p of a point x1,y1,z1 from given equation of plane…

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: Sketch a smooth connected curve y = f(x) with f(-2) = 8, f(0) = 4, f(2) = 0, f'(2) = f'(-2) = 0, %3D…

A:

Q: Can one draw two non-intersecting closed curves that each do not bound disks on RP2? How about on a…

A:

Q: Find an equation for the tangent plane to z-x-y = 3 at the point (2, 1, 8) Z - ) %3D

A: given z-x2-y2=3 at point 2,1,8

Q: 1. A such that the difference in distances between (x, y) and two fixed points (called is the set of…

A: A___ is the set of points (x, y) in a plane such that the difference in distances between (x, y) and…

Q: 2-Show that the regular curve a = a(s) is a straight line if and only if its curvature k is…

A: Here we have to show that a regular curve is a straight line if and only if, its curvature is…

Q: Straight lines with slope twice the x-intercept.

A: Let us assume x-intercept is a. That is, x = a when y  =0. So,  slope  =2a. Let us assume that the…

Q: Find the length of the curve y = In (1 – 2*) over the interval [-,].

A:

Q: Prove that there no line through the point (0,1) is tangent to the graph of y = x*.

A:

Q: Prove that the following curves are intersecting with a right angle: 5y-2x+ y -x²y =0 2y+5x+ x* –…

A:

Q: 3 Find the Minimum distance from the origin to the plane X+2y-z=5

A:

Q: dz -, n = 2,3, 4, ... where z = a is inside the simple closed curve C. (z – a)" Evaluate

A:

Q: Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated poi…

A:

Q: Find the points that are minimum to the origin and lie on the lines of intersection of planes: y +…

A:

Q: Find the minimum distance from the curve y = 2 to the point P(6,0)

A:

Q: Describe the traces of the curve x2 + 9z2 = 9 in the vertical planes (those parallel to the…

A: The traces of a surface are the cross-sections created when the surface intersects a plane parallel…

Q: Find the are bounded by the curve y=x²-4, the lines y=0 and x=4.

A:

Q: ii) Prove that the straight line y = x is the curve giving the shortest distance between the points…

A:

Q: 2. Use the guidelines to sketch the curve. Y In x x²

A: The given function is: y=ln xx2 Note that the domain of the function is (0,∞). First of all, let's…

Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: Show that the curve In (2X-1) , n> i has no stationary point

A:

Q: Let C be the curve of intersection of length of C from the origin to the poi

A:

Q: 5 Find the valume of the Sotids abtaind by-revelving the Yegion in the firstquadrand boundby the…

A: Sketch the graph and find the region. Using shell method, volume is evaluated.

Q: The equation of the curve in the tx-plane which passes through the point (2,3), and has, at every…

A: In this question, slope of curve and point is given and we have to find out equation of cuve.

Q: Find the minimum distance from the point (2, 1, 9) to the surface x + y + z = 9.

A:

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

A:

Q: Determine the maximum curvature for the graph of f(x) = 9 In (8x). The maximum curvature is at x=

A:

Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y²-x²y = 0 2y +…

A:

Q: Every plane curve has a unique parameterization. O False. OTrue.

A:

Q: 1 y == (x² + 4)°%2 ,0 sxS 2 (x2 + 4)°/2 ,0<x< 2 6. deur die berekening van die volgende bej…

A: The given function y is - y = 16x2+432     ........(1)Formula for calcualting the lenght of a curve…

Q: Suppose Chefs and cops have incone distribtion Lorenz curves qivn 3 f6)= {x*+ and which profession…

A: Assume, x be a random variable, whose distribution can be given by a Lorenz curve y=f(x). The…

Q: Find the length of the curve. (Hint: u-s

A:

Q: Determine the maximum curvature for the graph of f(x) = 7 In (2x). The maximum curvature is at x=

A: We are given function f(x)=7ln(2x) of the graph Formula for curvature is κ(x)=|f′′(x)(1+(f′(x))2)32|…

Q: 5. Show that the following orthogonal families of curves: are x² + y² : = ax, x² + y? = by

A:

Q: A Show that the curve F(t) = lies in a plane.

A:

Q: 1 Find the points have minimum distance From the origih to the Surface xyz =1.

A: Using the lagrangian ultiplies Let For critical points,

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.