Concept explainers
Finding a Limit Using Polar Coordinates In Exercises 57-60, use polar coordinates and L’H ô�pital’s Rule to find the limit.
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Chapter 13 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Complex Analysis - Limits Use the definition of limits to prove (6z-4)=2+6i as the limit of z approaches 1+iarrow_forwardCalculus I In the exercise f(x)= cos x + sin x; [0,2pi], find the following 1.) Search for critical points2.) Search if it grows or decreases3.) Search for local maximum and minimumarrow_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
- Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 4)arrow_forwardLimits of Rational FunctionsIn Exercises 13–22, find the limit of each rational function (a) asarrow_forwardMass of a box A solid box D is bounded by the planes x = 0, x = 3,y = 0, y = 2, z = 0, and z = 1. The density of the box decreases linearly in the positive z-direction and is given by ƒ(x, y, z) = 2 - z. Find the mass of the box.arrow_forward
- Symmetry 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_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_forwardComplex Analysis - Limits Evaluate [(z^3 + 1)/(z - 1)] as the limit of z approaches +infinityarrow_forward
- Channel 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_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_forwardPseudometric spacesarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage