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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.4: The Dot Product

Problem 12E

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Question 5
Find r'(t) r" (t) given the following vector function.
r(t) = (4t² +2t)i+(5t² +5t}j
O 33 +164t
O 66+164t
O 33+82t
O 66+1641²
O 66+82t
Question 6
Find the indefinite integral below.

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2.10 : (a) Express the vector fieldH = xy*2zax + x*2yzay+xyz*2a2in cylindrical and spherical coordinates. (b) In both cylindrical and spherical coordinates, determine H at (3, —4, 5).
A particle starts at the point (-2,0), moves along the x-axis to (2,0) and then along the semicircle y=radical(4-x^2) to the starting point. Use Green’s Theorem to find the work done on this particle by the force field F(x,y) =(2x,x^3+3xy^2).
True or False and explain1- For any two non parallel and non orthogonal vectors a and b with angle θ between them, it holds that cosθ(a.b) = sinθ(axb). 2- Ifr(t)=⟨−4cos(2t),3sin(3t),ln(2t)⟩,then the ∫r(t)dt is equal to⟨−2sin(2t),−cos(3t),tlnt−t⟩+C,where C is a vector constant of integration.
a) Calculate the line integral of the vector field F(x, y) = yi − 5xj from the point (0, 3) to the point (3, 0)(i) along the connecting line C1 between the points.(ii) along the arc C2 (shorter or quarter circle) of the circle centered at the origin.b) Does the vector field F have a potential?(The ratio of answers is π/2.)
Evaluate the integral of vector function F(x,y) = x - y where the boundary curve is the triangle formed by the points (1,3/2), (2,2) e (2,1).
A particle is under the influence of the force F= (-cosh (4x4) + xy)i + (e-y + x)j. The corner points move once in the counterclockwise direction on the rectangular curve in (1.1), (1.7), (3.1) and (3.7). Find the work done.
a) Plot a direction field and some integral curves for xy'-2y=-1 on the rectangular region {-1<or equal x <or equal 1,-0.5<or equal y <or equal 1.5}. what do the all integrals cuves have in common? b) show that the general solution of (A) on(-infinity,0) and (0,positive infinity) is y=1/2+cx2
or this problem, consider a particle traveling within the force field F = < -y,x,1/2 > along the parametrized curve r(t) = < t cos(t),t sin(t),1/2t > from the point (0,0,0) to the point (2pi,0,pi) Explain why the work done moving the particle along the path in this force field is positive. Compute the work done on a particle traveling along the given parametrized curve within the force field.
Δ denotes the Laplace operator defined by Δφ = ∂2φ/∂x2 + ∂2φ/∂y2 + ∂2φ/∂z2 . Show that if F is the gradient of a harmonic function, then curl(F) = 0 and div(F) = 0.
A particle starts at the point (-1, 0), moves along the x-axis to (1, 0), and then along the semicircle y = √(1 - x2 )to the starting point. Use Green's Theorem to find the work done on this particle by the force field F(x, y) = ‹3x, x3 + 3xy2›.
Suppose C is the curve from (0, 0) to (2, 0) to (2, 3) to (0, 3) to (0, 0). Find the work done by the vector fieldF(x, y) = <x^(3)−2y^(2), x + cos(√y)> on a particle moving along C.
Find the gradient vector field for the scalar function f(x,y)=sin(2x)cos(6y). Enter the exact answer in component form. ∇(x,y)=

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