Center of Curvature Use the result of Exercise 67 to find the center of curvature for die curve at die given point,
(a)
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Chapter 12 Solutions
Multivariable Calculus
- Normal Vector and Tangent Plane Find an equation of the tangent plane to the surface 4x2+ 9y2 - z2 = 16 at P = (2, 1, 3).arrow_forwardCentroids and moment of inertiaarrow_forwardNormal component Let C be the circle of radius 2 centered at theorigin with counterclockwise orientation. Give the unit outwardnormal vector at any point (x, y) on C.arrow_forward
- How do you define and calculate the velocity, speed, direction of motion, and acceleration of a body moving along a sufficiently differentiable space curve? Give an examplearrow_forwardWork by a constant force Evaluate a line integral to show thatthe work done in moving an object from point A to point B in thepresence of a constant force F = ⟨a, b, c⟩ is F ⋅ AB.arrow_forwardsketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. Surface: x2+y2=4; z=x2 Parameter: x=2sin(t)arrow_forward
- Integral representing arc length of vector function. Please give correct solution with integer values of A B C D!arrow_forward(a) Find the centroid of a right circular cone with height and base radius . (Place the cone so that its base is in the -plane with center the origin and its axis along the positive -axis.) (b) Find the moment of inertia of the cone about its axis (the -axis).arrow_forwardMass from density data The following table gives the density (in units of g/cm2) at selected points (in polar coordinates) of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method.arrow_forward
- Circular crescent Find the center of mass of the region in the first quadrant bounded by the circle x2 + y2 = a2 and the lines x = a and y = a, where a > 0.arrow_forwardA thin plate of constant density δ = 1occupies the region enclosed by the curve y = 36/(2x + 3) andthe line x = 3 in the first quadrant. Find the moment of the plateabout the y-axis.arrow_forwardHow do you find the tangent line at a point on a level curve of a differentiable function ƒ(x, y)? How do you find the tangent plane.and normal line at a point on a level surface of a differentiable function ƒ(x, y, z)? Give examples.arrow_forward
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