Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus (MindTap Course List)
75E76EProof Prove that the power series n0(n+p)!n!(n+q)!xn has a radius of convergence of R= when p and q are positive integers.Using a Power Series Let g(x)=1+2x+x2+2x3+x4+ where the coefficients are c2n=1 and c2n+1=2 for n0 (a) find the interval of convergence of the series, (b) Find an explicit formula for g(x).79EProof Prove that if the power series n=0cnxn has a radius of convergence of R, then n=0cnx2n has a radius of convergence of R.81E1E2EFinding a Geometric Power Series In Exercises 3-6, find a geometric power series for the function, centered at 0, (a) by the technique shown in Examples 1 and 2 and (b) by long division. f(x)=14xFinding a Geometric Power Series In Exercises 3-6, find a geometric power series for the function, centered at 0, (a) by the technique shown in Examples 1 and 2 and (b) by long division. f(x)=12+xFinding a Geometric Power Series In Exercises 3-6, find a geometric power series for the function, centered at 0, (a) by the technique shown in Examples 1 and 2 and (b) by long division. f(x)=43+xFinding a Geometric Power Series In Exercises 3-6, find a geometric power series for the function, centered at 0, (a) by the technique shown in Examples 1 and 2 and (b) by long division. f(x)=25xFinding a Power Series In Exercises 7-18, find a power series for the function, centered at c . and determine the interval of convergence. f(x)=16x,c=1Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c, and determine the interval of convergence. f(x)=26x,c=2Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c, and determine the interval of convergence. f(x)=113x,c=0Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c. and determine the interval of convergence. h(x)=114x,c=0Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c, and determine the interval of convergence. g(x)=52x3,c=3Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c, and determine the interval of convergence. f(x)=32x1,c=213EFinding a Power Series In Exercises 7-18, find a power series for the function, centered at c, and determine the interval of convergence. f(x)=43x+2,c=3Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c , and determine the interval of convergence. g(x)4xx2+2x3,c=0Finding a Power Series In Exercises 7-18, find a power series for the function, centered at c , and determine the interval of convergence. g(x)3x83x2+5x2,c=017EFinding a Power Series In Exercises 7-18, find a power series for the function, centered at c, and determine the interval of convergence. f(x)=54x2,c=019E20E21E22EUsing a Power Series In Exercises 19-28, use the power series 11+x=n=0(1)nxn,| x |1 to find a power series for the function, centered at 0, and determine the interval or convergence. f(x)=ln(x+1)=1x+1dxUsing a Power Series In Exercises 19-28, use the power series 11+x=n=0(1)nxn,| x |1 to find a power series for the function, centered at 0, and determine the interval or convergence. f(x)=ln(1x2)=11+xdx11xdx25E26E27E28E29E30E31E32E33E34E35E36E37EUsing a Power Series In Exercises 37-40, use the power series 11x=n=08xn,| x |1 to find a power series for the function, centered at 0, and determine the interval of convergence. f(x)=x(1x)239E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E1E2E3EFinding a Taylor Series Explain how to use the series g(x)=ex=n=0xnn! to find the series for f(x)x2e3x Do not find die series.Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=e2x,c=0Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=e4x,c=0Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=cosx,c=4Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=sinx,c=4Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=1x,c=1Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=11x,c=2Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=lnx,c=1Finding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=ex,c=113EFinding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=ln(x2+1),c=015EFinding a Taylor Series In Exercises 5-16, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=tanx,c=0(firstthreenonzeroterms)17E18E19E20E21E22EUsing a Binomial Series In Exercises 21-26, use the binomial series to find the Maclaurin series for the function. f(x)=11x224E25E26EFinding a Maclaurin Series In Exercises 27-40, find the Maclaurin series for the function. Use the table of power series for elementary functions on page 674. f(x)=ex2/228E29E30E31E32E33E34E35E36EFinding a Maclaurin Series In Exercises 27-40, find the Maclaurin series for the function. Use the table of power series for elementary functions on page 674. f(x)==12(exex)=sinhxFinding a Maclaurin Series In Exercises 27-40, find the Maclaurin series for the function. Use the table of power series for elementary functions on page 674. f(x)=ex+ex=2coshx39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58EFinding a Limit In Exercises 59-62, use the series representation of the function f to find x0limf(x), if it exists. f(x)=1cosxx60E61E62E63E64EApproximating an Integral In Exercises 63-70, use a power series to approximate the value the definite integral with an error of less than 0.0001. (In Exercises 65 and 67, assume that the integrand is defined as 1 when x=0.) 01/4xln(x+1)dx66E67E68E69E70E71E72E73E74E75E76E77E78EProjectile Motion A projectile fired from the ground follows the trajectory given by y=(tan+gkv0cos)x+gx2ln(1kxv0cos) where v0 is the initial speed, is the angle of projection, g is the acceleration due to gravity, and k is the drag factor caused by air resistance. Using the power series representation ln(1+x)=xx22+x33x44+,1x1 verify that the trajectory can be rewritten as y=(tan)xgx22v02cos2kgx33v03cos3k2gx44v04cos4.80E81E82E83E84E85E86E87E88E89E90E91EUsing Fibonacci Numbers Show that the Maclaurin series for the function g(x)=x1xx2 is n=1Fnxn where Fn is the nth Fibonacci number with F1=F2=1 and Fn=Fn2+Fn1, for n3. (Hint: Write 11xx2=a0+a1x+a2x2+ and multiply each side of this equation by 1xx2.)93EMatching In Exercises 1-6, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]Matching In Exercises 16, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] 4x2y2=43RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50REHorizontal and Vertical Tangency In Exercises 49-52, find all points (if any) of horizontal and vertical tangency to the curve. Lse a graphing utility to confirm your results. x=2+2sin,y=1+cos52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RERectangular-to-Polar Conversion In Exercises 63-66, the rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for 02. (0,7)65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RE82RE83RE84RE85RE86RE87RE88RE89RE90RE91RE92RE93RE94RE95RE96REFinding the Area of a Polar Region In Exercises 97-100, find the area of the region. One petal of r=3cos598RE99RE100RE101RE102RE103RE104RE105RE106RE107RE108RE109RE110RE111RE112RE113RE114RE115RE116RE117RE118RE119RE120RE121RE122RE123RE124RE125RE126RE1PS2PS3PSFlight Paths An air traffic controller spots two planes at the same altitude flying toward each other (see figure). Their flight paths are 20 and 315. One plane is 150 miles from point P with a speed of 375 miles per hour. The other is 190 miles from point P with a speed of 450 miles per hour. (a) Find parametric equations for the path of each plane where t is the time in hours, with t=0 corresponding to the time at which the air traffic controller spots the planes. (b) Use the result of part (a) to write the distance between the planes as a function of t. (c) Use a graphing utility to graph the function in part (b). When will the distance between the planes be minimum? If the planes must keep a separation of at least 3 miles, is the requirement met?5PS6PSCornu Spiral Consider the cornu spiral given by x(t)=0tcosu22du and y(t)=0tsinu22du. (a) Use a graphing utility to graph the spiral over the interval t. (b) Show that the cornu spiral is symmetric with respect to the origin. (c) Find the length of the comu spiral from t=0 to t=a. What is the length of the spiral from t= to t=?8PS9PS10PS11PS12PS13PS14PS15PS16PS17PS