Express the curvature of a twice-differentiable curve r = ƒ(u) in the polar coordinate plane in terms of ƒ and its derivatives.

Q: Find the curvature of y = sin(3T) at a = =.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: find the tangent plane to the surface z = Vy-x at the point (1,2,1)

A: Equation of tangent plane is z-c = df/dx(x-a)+df/dy(y-b) at point (a,b,c)

Q: Find the curvature and radius of curvature of the plane curve at the given value of x . y = x −4/ x,…

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Q: Find an equation of the tangent plane to the surface z = x^3- y^2 + 12x + 2y at the point (1, 2, 5).

A: Given: z = x^3- y^2 + 12x + 2y (1, 2, 5)

Q: Find the ordinate of the center of curvature of the curve y² = 4x at point (2, 4) from the x-axis.

A: Given option are wrong, option c should be '-16'.

Q: Find the curvature and radius of curvature of the plane curve at the given value of x.

A: We have,   It is known that radius of curvature is given as, Here K is curvature of the curve.…

Q: What is the curvature of the path of the motion at t=1

A: Since velocity is given we need to calculate  r''(t) only at t=1

Q: (b) Find the circle of curvature at (0, 1)of the curve y = x3 + 2x² + x +1

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Q: Find the slope of the tangent line at the point (1,−2,−3) on the curve of intersection of the…

A: Given surface is z=5-4x2-y2 The slope of tangent line to the curve that is the intersection of the…

Q: Find the curvature of y = sin(3x) at a = 프 4

A: We Know that The curvature measures how fast a curve is changing direction at a given point.

Q: Let P be a point on a regular surface S and let f and g be two curves on S passing through P and…

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Q: At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? y = ln…

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Q: Determine the equation of the tangent plane to the surface z = cos(x + 2y) at the point (2,-1,0).

A: Tangent plane to f(x, y) at (x0, y0) is given by below:

Q: Find the radii of curvature and torsion at a point of the curve    x^2+y^2=a^2 , x^2-y^2=az

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Q: A normal vector to the tangent plane for the at the point (1, 1, 1) is: y surface x =

A: A(1, 1, 1)z= xyf(x,y,z) =xy-zfx=yfy=xfz=-1Normal vector to the tangent plane for the surface;n^…

Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y = 9x 4…

A: Given curve is, y=9x-4x Differentiating this with respect to x we get,  dydx=9-4ddx(1x)⇒dydx=9+4x2…

Q: Find an equation of the tangent plane to the surface x = u² + 1, y = v³ + 1, z = u + v at the point…

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Q: Write a formula for the curvature of a plane curve with equation y = f(x). %3D

A: If the equation y=fx. Then the formula of radius curvature at the point is:   Rx=1+dydx232d2ydx2…

Q: Find the curvature and the radius of curvature of the curve y = x + 5x2 – 2x3 at the origin (0,0).

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Q: Determine the curvature of the graph of y = In x at the point where x = 1.

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Q: Find parametric equations for the tangent line to the curve a-tcost, y-t, z- tsint at the

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Q: Find an equation of the plane tangent to the surface sin(xyz) = at the point (T, 1,-)

A: sinxyz=12 at the point π, 1, 16. Consider, sinxyz=12⇒xyz=sin-112⇒xyz=π6⇒z=π6xy Differentiating…

Q: Find the equation for the tangent plane to the surface x2 - 5xyz + y² = 7z2 at the point (-1, -1,…

A: Simplify the given equation of the surface in the form F(x,y,z)=0. x2-5xyz+y2=7z2x2-5xyz+y2-7z2=0…

Q: Compute the mean curvature of the surface z = xy at the origin

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Q: Show that curvature at an inflection point of a plane curve y = f (x) is zero.

A: We have to show that curvature at an inflection point of a plane curve y=fx is zero. We know that if…

Q: Find the radius of curvature of 6y = x³ at the point (2,).

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Q: Find an equation of the tangent plane to the surface z = In(x - 2y) at the point (3, 1,0) %3D

A: Given that, z = ln (x - 2y) we have to find the equation of tangant at point (3, 1, 0).

Q: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

A: For tangent plane of the surface f(x,y,z)=k at a given point normal vector to tangent plane  will be…

Q: Find the curvature of the curve defined by the function y=3x^2 - 2x + 4 at the point x=2

A: Curvature is defined as : K(x) = |y"(x)| / {1 + [y'(x)]^2}^(3/2).  Note that :- |y"(x)| means the…

Q: Find the curvature x for the curve y = x - K = x² at the point x = 2.

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Q: Find the center of curvature of the curve y-Inx at (1, 0).

A: We Know that Centre of curvature for y=f(x) is κ=|f′′(x)|/(1+f′(x)3/2  -----(1)

Q: At what point does the curve y = e have maximum curvature? What happens to the curvature as x → ∞?

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Q: Write a formula for the curvature of a plane curve with equation y= f(x)

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Q: Find an equation for the tangent plane to the surface x = u² – v², y = u + v, 2 = u² + 4v at (-},2).

A: We have r(u,v) = (u2-v2)i + (u+v)j + (u2+4v)k

Q: 15. y= 2x +3 at x = 27

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Q: find the curvature of y=x^4 at (1,1).

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Q: Find the curvature of y = sin(-lx) at x = .

A: Solution:-

Q: Find the curvature of the plane curve y = -t at the point t = 2. к(2) -

A: Given:y = -t4differentiating with respect to t, we get:            y' =     d(-t4)dt…

Q: 8. Find the curvature of the vector-valued function r(t) = at the point t = -1.

A: given function is rt=3t2,-4t,1t find the curvature at t=-1 k=r→'t×r→''tr→'t3

Q: Find the equation of the tangent plane to the surface 2x^2 -xz + y^2 -zy = -5 at the point (1, 3, 4)

A: First we find out partial derivative at given point and then find the equation of the tangent plane…

Q: Find a plane curve a(s) endowed with a unit-speed prametrization by SE (-0, 0) such that its signed…

A: The curvature of the curve αs is given by: αs=∫s0scos(θ(u))du, ∫s0ssin(θ(u))du-----(1) in which…

Q: Find the point on the curve y = ex where the curvature is greatest.

A: Given: Find the point on the curve y = ex where the curvature is greatest.

Q: Let 3 : (3t, r = be a curve in 3-dimensional space having unit tangent and normal vectors u and .…

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Q: Find the slope of the tangent line to the curve that is the intersection of the surface z=x² - y…

A: The details solution is given below

Q: Determine the radius of the curvature of the plane curve given by y In sec x at the || point (0,0).

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Q: 2) Find and sketch the circle of curvature at the point on the curve y x where the tangent line has…

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