UD CALC (241 ONLY) W/1 TERM ACCESS >IB
8th Edition
ISBN: 9781337051545
Author: Stewart
Publisher: CENGAGE C
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Chapter 16.R, Problem 8TFQ
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Chapter 16 Solutions
UD CALC (241 ONLY) W/1 TERM ACCESS >IB
Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Prob. 7ECh. 16.1 - Prob. 8ECh. 16.1 - Prob. 9ECh. 16.1 - Prob. 10E
Ch. 16.1 - Prob. 11ECh. 16.1 - Match the vector fields F with the plots labelled...Ch. 16.1 - Match the vector fields F with the plots labelled...Ch. 16.1 - Match the vector fields F with the plots labeled...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Match the vector fields F on 3 with the plots...Ch. 16.1 - Prob. 19ECh. 16.1 - Prob. 20ECh. 16.1 - Prob. 21ECh. 16.1 - Find the gradient vector field of f. f(s,t)=2s+3tCh. 16.1 - Find the gradient vector field of f....Ch. 16.1 - Find the gradient vector field of f....Ch. 16.1 - Find the gradient vector field f of f and sketch...Ch. 16.1 - Find the gradient vector field f of f and sketch...Ch. 16.1 - Prob. 27ECh. 16.1 - Plot the gradient vector field of f together with...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - Match the functions f with the plots of their...Ch. 16.1 - A particle moves in a velocity field...Ch. 16.1 - Prob. 34ECh. 16.1 - The flow lines or streamlines of a vector field...Ch. 16.1 - a Sketch the vector field F(x,y)=i+xj and then...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Prob. 4ECh. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Prob. 14ECh. 16.2 - Evaluate the line integral, where C is the given...Ch. 16.2 - Prob. 16ECh. 16.2 - Let F be the vector fields shown in the figure. a...Ch. 16.2 - The figure shows a vector field F and two curves...Ch. 16.2 - Prob. 19ECh. 16.2 - Prob. 20ECh. 16.2 - Prob. 21ECh. 16.2 - Evaluate the line integral CFdr, where C is given...Ch. 16.2 - Prob. 23ECh. 16.2 - Prob. 24ECh. 16.2 - Prob. 25ECh. 16.2 - Prob. 26ECh. 16.2 - Use a graph of the vector field F and the curve C...Ch. 16.2 - Use a graph of the vector field F and the curve C...Ch. 16.2 - a Evaluate the line integral CFdr, where...Ch. 16.2 - a Evaluate the line integral CFdr, where...Ch. 16.2 - Find the exact value of Cx3y3zds, where C is the...Ch. 16.2 - a Find the work done by the force field...Ch. 16.2 - A thin wire is bent into the shape of a semicircle...Ch. 16.2 - A thin wire has the shape of the first-quadrant...Ch. 16.2 - a Write the formulas similar to Equations 4 for...Ch. 16.2 - Find the mass and center of mass of a wire in the...Ch. 16.2 - Prob. 37ECh. 16.2 - Prob. 38ECh. 16.2 - Find the work done by the force field...Ch. 16.2 - Find the work done by the force field...Ch. 16.2 - Prob. 41ECh. 16.2 - Prob. 42ECh. 16.2 - Prob. 43ECh. 16.2 - An object with mass m moves with position function...Ch. 16.2 - A 160-lb man carries a 25-lb can of paint up a...Ch. 16.2 - Prob. 46ECh. 16.2 - a Show that a constant force field does zero work...Ch. 16.2 - Prob. 48ECh. 16.2 - If C is a smooth curve given by a vector function...Ch. 16.2 - Prob. 50ECh. 16.2 - An object moves along the curve C shown in the...Ch. 16.2 - Experiments show that a steady current I in a long...Ch. 16.3 - The figure shows a curve C and a contour map of a...Ch. 16.3 - Prob. 2ECh. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.3 - Determine whether or not F is a conservative...Ch. 16.3 - The figure shows the vector field F(x,y)=2xy,x2...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - a Find a function f such that F=f and b use part a...Ch. 16.3 - Prob. 16ECh. 16.3 - Prob. 17ECh. 16.3 - Prob. 18ECh. 16.3 - Show that the line integral is independent of path...Ch. 16.3 - Show that the line integral is independent of path...Ch. 16.3 - Suppose youre asked to determine the curve that...Ch. 16.3 - Prob. 22ECh. 16.3 - Find the work done by the force field F in moving...Ch. 16.3 - Find the work done by the force field F in moving...Ch. 16.3 - Is the vector field shown in the figure...Ch. 16.3 - Is the vector field shown in the figure...Ch. 16.3 - If F(x,y)=sinyi+(1+xcosy)j, use a plot to guess...Ch. 16.3 - Let F=f, where f(x,y)=sin(x2y). Find curves C1 and...Ch. 16.3 - Show that if the vector field F=Pi+Qj+Rk is...Ch. 16.3 - Use Exercise 29 to show that the line integral...Ch. 16.3 - Determine whether or not the given set is a open,...Ch. 16.3 - Determine whether or not the given set is a open,...Ch. 16.3 - Determine whether or not the given set is a open,...Ch. 16.3 - Prob. 34ECh. 16.3 - Prob. 35ECh. 16.3 - a Suppose that F is an inverse square force field,...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Evaluate the line integral by two methods: a...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate the line integral...Ch. 16.4 - Use Greens Theorem to evaluate cFdr. Check the...Ch. 16.4 - Prob. 12ECh. 16.4 - Prob. 13ECh. 16.4 - Prob. 14ECh. 16.4 - Verify Greens Theorem by using a computer algebra...Ch. 16.4 - Verify Greens Theorem by using a computer algebra...Ch. 16.4 - Prob. 17ECh. 16.4 - A particle starts at the origin, moves along the...Ch. 16.4 - Prob. 19ECh. 16.4 - If a circle C with radius 1 rolls along the...Ch. 16.4 - Prob. 21ECh. 16.4 - Prob. 22ECh. 16.4 - Use Exercise 22 to find the centroid of a...Ch. 16.4 - Prob. 24ECh. 16.4 - A plane lamina with constant density (x,y)=...Ch. 16.4 - Prob. 26ECh. 16.4 - Use the method of Example 5 to calculate CFdr,...Ch. 16.4 - Prob. 28ECh. 16.4 - If F is the vector field of Example 5, show that...Ch. 16.4 - Complete the proof of the special case of Greens...Ch. 16.4 - Prob. 31ECh. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - Find a the curl and b the divergence of the vector...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - The vector field F is shown in the xy-plane and...Ch. 16.5 - Let f be a scalar field and F a vector field....Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Prob. 15ECh. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Prob. 17ECh. 16.5 - Determine whether or not the vector field is...Ch. 16.5 - Is there a vector field G on 3 such that curl...Ch. 16.5 - Prob. 20ECh. 16.5 - Show that any vector field of the form...Ch. 16.5 - Prob. 22ECh. 16.5 - Prob. 23ECh. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prob. 25ECh. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prob. 27ECh. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Prove the identity, assuming that the appropriate...Ch. 16.5 - Let r=xi+yj+zk and r=|r|. Verify each identity. a...Ch. 16.5 - Let r=xi+yj+zk and r=|r|. Verify each identity. a...Ch. 16.5 - Let r=xi+yj+zk and r=|r|. If F=r/rp, find div F....Ch. 16.5 - Use Greens Theorem in the form of Equation 13 to...Ch. 16.5 - Prob. 34ECh. 16.5 - Recall from Section 14.3 that a function g is...Ch. 16.5 - Prob. 36ECh. 16.5 - This exercise demonstrates a connection between...Ch. 16.5 - Maxwells equations relating the electric field E...Ch. 16.5 - We have seen that all vector fields of the form...Ch. 16.6 - Determine whether the points P and Q lie on the...Ch. 16.6 - Determine whether the points P and Q lie on the...Ch. 16.6 - Prob. 3ECh. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Prob. 5ECh. 16.6 - Identify the surface with the given vector...Ch. 16.6 - Use a computer to graph the parametric surface....Ch. 16.6 - Prob. 8ECh. 16.6 - Prob. 9ECh. 16.6 - Prob. 10ECh. 16.6 - Prob. 11ECh. 16.6 - Prob. 12ECh. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Match the equations with the graphs labeled IVI...Ch. 16.6 - Prob. 17ECh. 16.6 - Prob. 18ECh. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Find da parametric representation for the surface....Ch. 16.6 - Prob. 27ECh. 16.6 - Prob. 28ECh. 16.6 - Prob. 29ECh. 16.6 - Prob. 30ECh. 16.6 - a What happens to the spiral tube in Example 2 see...Ch. 16.6 - Prob. 32ECh. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Find an equation of the tangent plane to the given...Ch. 16.6 - Prob. 35ECh. 16.6 - Prob. 36ECh. 16.6 - Prob. 37ECh. 16.6 - Prob. 38ECh. 16.6 - Prob. 39ECh. 16.6 - Prob. 40ECh. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the cone...Ch. 16.6 - Find the area of the surface. The surface...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Prob. 48ECh. 16.6 - Find the area of the surface. The surface with...Ch. 16.6 - Find the area of the surface. The part of the...Ch. 16.6 - Prob. 51ECh. 16.6 - Prob. 52ECh. 16.6 - Prob. 53ECh. 16.6 - Prob. 54ECh. 16.6 - Prob. 55ECh. 16.6 - Prob. 56ECh. 16.6 - Prob. 57ECh. 16.6 - Prob. 58ECh. 16.6 - a Show that the parametric equations...Ch. 16.6 - a Show that the parametric equations...Ch. 16.6 - Find the area of the part of the sphere...Ch. 16.6 - The figure shows the surface created when the...Ch. 16.6 - Prob. 63ECh. 16.6 - a Find a parametric representation for the torus...Ch. 16.7 - Let S be the surface of the box enclosed by the...Ch. 16.7 - Prob. 2ECh. 16.7 - Prob. 3ECh. 16.7 - Suppose that f(x,y,z)=g(x2+y2+z2), where g is a...Ch. 16.7 - Evaluate the surface integral. S(x+y+z)dS, S is...Ch. 16.7 - Evaluate the surface integral. SxyzdS, S is the...Ch. 16.7 - Prob. 7ECh. 16.7 - Evaluate the surface integral. S(x2+y2)dS, S is...Ch. 16.7 - Evaluate the surface integral. Sx2yzdS, S is the...Ch. 16.7 - Prob. 10ECh. 16.7 - Evaluate the surface integral. SxdS, S is the...Ch. 16.7 - Evaluate the surface integral. SydS, S is the...Ch. 16.7 - Evaluate the surface integral. Sz2dS, S is the...Ch. 16.7 - Evaluate the surface integral. Sy2z2dS, S is the...Ch. 16.7 - Prob. 15ECh. 16.7 - Evaluate the surface integral. Sy2dS, S is the...Ch. 16.7 - Prob. 17ECh. 16.7 - Evaluate the surface integral. S(x+y+z)dS, S is...Ch. 16.7 - Evaluate the surface integral. SxzdS, S is the...Ch. 16.7 - Prob. 20ECh. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Evaluate the surface integral SFdS for the given...Ch. 16.7 - Prob. 33ECh. 16.7 - Prob. 34ECh. 16.7 - Prob. 35ECh. 16.7 - Find the flux of F(x,y,z)=sin(xyz)i+x2yj+z2ex/5k...Ch. 16.7 - Prob. 37ECh. 16.7 - Prob. 38ECh. 16.7 - Find the centre of mass of the hemisphere...Ch. 16.7 - Find the mass of a thin funnel in the shape of a...Ch. 16.7 - Prob. 41ECh. 16.7 - Let S be the part of the sphere x2+y2+z2=25 that...Ch. 16.7 - Prob. 43ECh. 16.7 - Prob. 44ECh. 16.7 - Use Gausss Law to find the charge contained in the...Ch. 16.7 - Prob. 46ECh. 16.7 - Prob. 47ECh. 16.7 - Prob. 48ECh. 16.7 - Prob. 49ECh. 16.8 - A hemisphere H and a portion P of a paraboloid are...Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS...Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS....Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS....Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS....Ch. 16.8 - Use Stokes Theorem to evaluate ScurlFdS...Ch. 16.8 - Use Stokes Theorem to evaluate cFdr. In each case...Ch. 16.8 - Prob. 8ECh. 16.8 - Use Stokes Theorem to evaluate cFdr. In each case...Ch. 16.8 - Use Stokes Theorem to evaluate cFdr. In each case...Ch. 16.8 - a Use Stokes Theorem to evaluate cFdr, where...Ch. 16.8 - a Use Stokes Theorem to evaluate cFdr, where...Ch. 16.8 - Verify the Stokes Theorem is true for the given...Ch. 16.8 - Verify that Stokes Theorem is true for given...Ch. 16.8 - Verify that Stokes Theorem is true for given...Ch. 16.8 - Let C be a simple closed smooth curve that lies in...Ch. 16.8 - A particle moves along line segments from the...Ch. 16.8 - Evaluate C(y+sinx)dx+(z2+cosy)dy+x3dz where C is...Ch. 16.8 - Prob. 19ECh. 16.8 - Suppose S and C satisfy the hypotheses of Stokes...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Verify that the Divergence Theorem is true for the...Ch. 16.9 - Prob. 3ECh. 16.9 - Prob. 4ECh. 16.9 - Prob. 5ECh. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Prob. 7ECh. 16.9 - Prob. 8ECh. 16.9 - Prob. 9ECh. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Prob. 14ECh. 16.9 - Use the Divergence Theorem to calculate the...Ch. 16.9 - Use a computer algebra system to plot the vector...Ch. 16.9 - Use a Divergence Theorem to evaluate SFdS, where...Ch. 16.9 - Let F(x,y,z)=ztan1(y2)i+z3ln(x2+1)j+zk. Find the...Ch. 16.9 - A vector field F is shown. Use the interpretation...Ch. 16.9 - a Are the points P1 and P2 sources or sinks for...Ch. 16.9 - Prob. 21ECh. 16.9 - Prob. 22ECh. 16.9 - Verify that div E=0 for the electric field...Ch. 16.9 - Prob. 24ECh. 16.9 - Prob. 25ECh. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prove each identity, assuming that S and E satisfy...Ch. 16.9 - Prob. 30ECh. 16.9 - Suppose S and E satisfy the conditions of the...Ch. 16.9 - Prob. 32ECh. 16.R - Prob. 1CCCh. 16.R - a What is a conservative vector field? b What is...Ch. 16.R - Prob. 3CCCh. 16.R - a Define the line integral of a vector field F...Ch. 16.R - Prob. 5CCCh. 16.R - Prob. 6CCCh. 16.R - Prob. 7CCCh. 16.R - Write expressions for the area enclosed by a curve...Ch. 16.R - Prob. 9CCCh. 16.R - Prob. 10CCCh. 16.R - Prob. 11CCCh. 16.R - Prob. 12CCCh. 16.R - Prob. 13CCCh. 16.R - Prob. 14CCCh. 16.R - Prob. 15CCCh. 16.R - Prob. 16CCCh. 16.R - Prob. 1TFQCh. 16.R - Prob. 2TFQCh. 16.R - Prob. 3TFQCh. 16.R - Prob. 4TFQCh. 16.R - Prob. 5TFQCh. 16.R - Prob. 6TFQCh. 16.R - Prob. 7TFQCh. 16.R - Prob. 8TFQCh. 16.R - Prob. 9TFQCh. 16.R - Prob. 10TFQCh. 16.R - Prob. 11TFQCh. 16.R - Prob. 12TFQCh. 16.R - Prob. 13TFQCh. 16.R - A vector field F, a curve C, and a point P are...Ch. 16.R - Evaluate the line integral. cxds, C is the arc of...Ch. 16.R - Evaluate the line integral. cyzcosxds,...Ch. 16.R - Evaluate the line integral. cydx+(x+y2)dy, C is...Ch. 16.R - Prob. 5ECh. 16.R - Evaluate the line integral. cxydx+eydy+xzdz, C is...Ch. 16.R - Prob. 7ECh. 16.R - Evaluate the line integral. cFdr, where...Ch. 16.R - Prob. 9ECh. 16.R - Find the work done by the force field...Ch. 16.R - Show that F is a conservative vector field. Then...Ch. 16.R - Prob. 12ECh. 16.R - Prob. 13ECh. 16.R - Show that F is a conservative and use this fact to...Ch. 16.R - Verify that Greens Theorem is true for the line...Ch. 16.R - Prob. 16ECh. 16.R - Use Greens theorem to evaluate cx2ydxxy2dy, where...Ch. 16.R - Prob. 18ECh. 16.R - Show that there is no vector field G such that...Ch. 16.R - Prob. 20ECh. 16.R - Prob. 21ECh. 16.R - If f and g are twice differentiable functions,...Ch. 16.R - If f is a harmonic function, that is, 2f=0, show...Ch. 16.R - a Sketch the curve C with parametric equations...Ch. 16.R - Prob. 25ECh. 16.R - Prob. 26ECh. 16.R - Prob. 27ECh. 16.R - Prob. 28ECh. 16.R - Evaluate the surface integral. sFdS, where...Ch. 16.R - Prob. 30ECh. 16.R - Verify that Stokes Theorem is true for the vector...Ch. 16.R - Prob. 32ECh. 16.R - Use Stokes Theorem to evaluate cFdr, where...Ch. 16.R - Use the Divergence Theorem to calculate the...Ch. 16.R - Prob. 35ECh. 16.R - Compute the outward flux of...Ch. 16.R - Prob. 37ECh. 16.R - Let F(x,y)=(2x3+2xy22y)i+(2y3+2x2y+2x)jx2+y2...Ch. 16.R - Find sFndS, where F(x,y,z)=xi+yj+zk and S is the...Ch. 16.R - Prob. 40ECh. 16.R - Prob. 41ECh. 16.P - Let S be a smooth parametric surface and P be a...Ch. 16.P - Find the positively oriented simple closed curve C...Ch. 16.P - Let C be a simple closed piecewise-smooth space...Ch. 16.P - Investigate the shape of the surface with...Ch. 16.P - Prove the following identity:...Ch. 16.P - The depicts the sequence of events in each...
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- Provethat a) prove that for any irrational numbers there exists? asequence of rational numbers Xn converg to S. b) let S: RR be a sunctions-t. f(x)=(x-1) arc tan (x), xe Q 3(x-1) 1+x² x&Q Show that lim f(x)= 0 14x C) For any set A define the set -A=yarrow_forwardQ2: Find the interval and radius of convergence for the following series: Σ n=1 (-1)η-1 xn narrow_forward8. Evaluate arctan x dx a) xartanx 2 2 In(1 + x²) + C b) xartanx + 1½-3ln(1 + x²) + C c) xartanx + In(1 + x²) + C d) (arctanx)² + C 2 9) Evaluate Inx³ dx 3 a) +C b) ln x² + C c)¾½ (lnx)² d) 3x(lnx − 1) + C - x 10) Determine which integral is obtained when the substitution x = So¹² √1 - x²dx sine is made in the integral πT π π a) √ sin cos e de b) √ cos² de c) c Ꮎ Ꮎ cos² 0 de c) cos e de d) for cos² e de πT 11. Evaluate tan³xdx 1 a) b) c) [1 - In 2] 2 2 c) [1 − In2] d)½½[1+ In 2]arrow_forward12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forwardQuestion 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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