Finding a Limit Using Polar Coordinates In Exercises 51-56, use polar coordinates to find the limit. [Hint: Let
and
implies
.]
Trending nowThis is a popular solution!
Chapter 13 Solutions
Multivariable Calculus
- Surface Area The roof over the stage of an open air theater at a theme park is modeled by f(x, y) = 25[1 + e−(x2+y2)1000 cos2(x2 + y2/ 1000 )] where the stage is a semicircle bounded by the graphs of y = √502 − x2 and y = 0. Use a computer algebra system to approximate the number of square feet of roofing required to cover the surface.arrow_forwardStokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. F = sin 3y i + cos 7x j; C is the rectangle with vertices at (0, 0),(pi/7,0),(pi/7,pi/3) and (0,pi/3) a) 0 b) 2/3π c) - 2/3π d) -4/3 πarrow_forward
- (a) Find the minimum and maximum xcoordinates of points on the cardioid r =1−cosθ.(b) Find the minimum and maximum ycoordinates of points on the cardioid in part (a).arrow_forwardSymmetry Principle Let R be the region under the graph of y = f (x) over the interval [−a, a], where f (x) ≥ 0. Assume that R is symmetric with respect to the y-axis. (a) Explain why y = f (x) is even—that is, why f (x) = f (−x). (b) Show that y = xf (x) is an odd function. (c) Use (b) to prove that My = 0. (d) Prove that the COM of R lies on the y-axis (a similar argument applies to symmetry with respect to the x-axis).arrow_forwardConsider the vector function: (see attached) a) Find the domain of r. b) Find the limit as t -> 0 of lim r(t). (see attached) c) Find the parametric equations for the line tangent to the space curve described by the vector function at t = 1.arrow_forward
- true or false? prove your answer a) If f and fg have limits at p, then g has a limit at p.arrow_forwardLine integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The flux line integral of F = ⟨ex - y, ey - x⟩, where C is theboundary of {(x, y): 0 ≤ y ≤ x, 0 ≤ x ≤ 1}arrow_forwardMULTIVARIABLE CALCULUS, VECTOR CALCULUS, VECTORS COMPUTE THE LIMIT OR SHOW IT DOES NOT EXISTarrow_forward
- lim(x,y)->(0,0) = (1+x2+y2-cos(x2+y2))/(x2+y2)arrow_forwardLine integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful.arrow_forwardHeat flux Suppose a solid object in ℝ3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -k∇T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is∇ ⋅ F = -k∇⋅ ∇T = -k∇2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning