Study Guide for Stewart's Multivariable Calculus, 8th
8th Edition
ISBN: 9781305271845
Author: Stewart, James
Publisher: Brooks Cole
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Chapter 13.1, Problem 3PT
To determine
To choose: The appropriate option for the value of
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4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard theorem for a suitable domain D. b. Then estimate the interval of existence of the solution.
(a) Show that f(x,y)=In(x² + y²) Satisfies the Laplace equation in two dimensional rectangular co-ordinates
(b) Compute all the first and second derivatives of f(x,y) = e3x+4cosxy
(c) Given Z= f(x,y). State the conditions for the minimum, maximum and saddle points of Z. Hence investigate the stationary values of Z=x³-6xy+y³
Let V be the vector space of functions which have B={sin ?, cos ?} as a basis, and let D be the differential operator on V. Find the characteristic polynomial Δ(t) of D.
Chapter 13 Solutions
Study Guide for Stewart's Multivariable Calculus, 8th
Ch. 13.1 - For r(t)=sinticost4j,r(3)= a) 32i+22j b) 22i+32j...Ch. 13.1 - The curve given by r(t) = 2i + tj + 2tk is a: a)...Ch. 13.1 - Prob. 3PTCh. 13.1 - Prob. 4PTCh. 13.1 - Prob. 5PTCh. 13.2 - For r(t)=t3i+sintj(t2+2t)k,r(0)= a) j 2k b) 3i ...Ch. 13.2 - True or False: [r(t)s(t)]=r(t)s(t)+s(t)r(t).Ch. 13.2 - Prob. 3PTCh. 13.2 - Prob. 4PTCh. 13.3 - Prob. 1PT
Ch. 13.3 - Prob. 2PTCh. 13.3 - Prob. 3PTCh. 13.3 - Prob. 4PTCh. 13.3 - Prob. 5PTCh. 13.3 - True or False: If f is twice differentiable and x0...Ch. 13.3 - Prob. 7PTCh. 13.4 - Prob. 1PTCh. 13.4 - Find the position function for which (at t = 0)...Ch. 13.4 - The force needed for a 10-kg object to attain...Ch. 13.4 - Prob. 4PT
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- 5. Prove that the equation has no solution in an ordered integral domain.arrow_forwardCalculus Define T:P2R by T(p)=01p(x)dx What is the kernel of T?arrow_forward(a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then (b) evaluate ux, u y, and uz at the given point (x, y, z). u = e^(qr) sin-1 p, p = sin x, q = z^2 ln y, r = 1/z; (x, y, z) = (pai/4, 1/2, -1/2)arrow_forward
- prove that Z[x]/(x2+1)≃Z[i], where Z[i]={a+b√-1|a,b∈Z}.show only injectivity and surjectivity in terms of φ:Z[x]/(x^2+1)→Z[i]. (For example 2+3x+x3(x2+1)→2+3i)arrow_forwardProve that arctan x + arctan y = arctan(x + y) / (1 − xy), xy ≠ 1. Use the formula derived to show that arctan (1/2) + arctan (1/3) = /4arrow_forwardDetermine Taylor's formula of order 1 and center a = (0, 0) of f(x)=ex1sin(x2).arrow_forward
- Show that linear differential operators with constant coefficients obey the commutative law. That is, show that (D−a)(D−b)f=(D−b)(D−a)fD−aD−bf=D−bD−af for any twice-differentiable function f and any constants a and b. The result extends at once to any finite number of factors.arrow_forwardH3. A, B and C are points in space relative to 0 such that AB 1 CO and AC 1 BO. Prove that BC 1 AO.arrow_forwardLet T: R2→R2 such that T(1, 0) = (1, 0) and T(0, 1) = (0, 0). (a) Determine T(x, y) for (x, y) in R2. (b) Give a geometric description of T.arrow_forward
- 1. (a) It is known that a complex function f(z) = u(x, y) + iv(x, y) is differentiable at a point z_0 ∈ C if and only if the Cauchy-Riemann equations hold at z_0 and the firstpartial derivatives of u and v are continuous at z_0. Use this to find all points of C at whichf(z) = (x^3) − 3(y^2)+2x+i(3(x^2)y−(y^3)+2y).is differentiable, and find the derivatives at these points.(b) Determine now all points of C at which the function f in part (a) is analytic.arrow_forwardFill in the blank: If f is differentialble on (0,3), f(0) =2 and f(3) = 14 then by the Mean Value Theorem there exists some 0<s<3 such that f'(s)=________arrow_forwardFind the isolated singularities for the holomorphic functions f, g, h. Determine the types and compute the residue. [Types include: essential, pole, removable] f1(z)=z5 cos(1/z) f2(z)=ez / (z (z-1)2)arrow_forward
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