Finding a Limit Using Polar Coordinates In Exercises 57-60, use polar coordinates and L'H6pitals Rule to find the limit.
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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_forwardMULTIVARIABLE CALCULUS, VECTOR CALCULUS, VECTORS COMPUTE THE LIMIT OR SHOW IT DOES NOT EXISTarrow_forward(a): Present the correct definition for Rotational, Divergent and Laplacian. (b): Introduce the Jacobian and the Jacobian of some function. (c): Present the correct definition of Double Integral.arrow_forward
- Limit and Continuity In Exercises , find the limit (if it exists) and discuss the continuity of the function. 14. lim (x, y)→(1, 1) (xy) /(x^2 − y^2 ) 16. lim (x, y)→(0, 0) (x^2 y) /(x^4 + y^2)arrow_forwardChannel flow The flow in a long shallow channel is modeled by the velocity field F = ⟨0, 1 - x2⟩, where R = {(x, y): | x | ≤ 1 and | y | < 5}.a. Sketch R and several streamlines of F.b. Evaluate the curl of F on the lines x = 0, x = 1/4, x = 1/2, and x = 1.c. Compute the circulation on the boundary of the region R.d. How do you explain the fact that the curl of F is nonzero atpoints of R, but the circulation is zero?arrow_forwardAnalysis problem Prove that f(x) = x ⋅ |x| is continuous at all points c in ℝ.arrow_forward
- Topological spacesarrow_forwardFinding the Derivative by the LimitProcess In Exercises 15–28, find the derivativeof the function by the limit process g(x) = −3arrow_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.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning