x-0.5exp(0.5x) =0xo = 1 x = 0.5exp(0.5x) = 50d = 0.001

di xi F(xi) error remarks i xi

0 1 0.82436063535 0.213061319425 continue 0 501 0.82436063535 0.755053355356 0.091791235021 continue 1 9.2103403719762 0.755053355356 0.729336178633 0.035261073666 continue 2 5.8269479738563 0.729336178633 0.720017982749 0.012941615497 continue 3 4.9112810804394 0.720017982749 0.71667115111 0.004669968414 continue 4 4.5693640016955 0.71667115111 0.715472865163 0.001674816761 continue 5 4.4250424153246 0.715472865163 0.715044323015 0.000599322495 stop 6 4.360854088716

7 4.3316302212888 4.3181822924849 4.311963457281

xo = 0 f(xo) = -0.5 OKerror = 0.001

Δx = 1 |f(xi)|i Δx xi f(xi) error remarks0 1 0 -0.5 0.5 (-) continue1 1 1 0.17563936465 0.1756393646499 revert1 0.1 0.1 -0.42563554819 0.425635548188 (-) continue2 0.1 0.2 -0.35258545904 0.3525854590378 (-) continue3 0.1 0.3 -0.28091712136 0.2809171213641 (-) continue4 0.1 0.4 -0.21070137908 0.2107013790801 (-) continue5 0.1 0.5 -0.14201270834 0.1420127083439 (-) continue6 0.1 0.6 -0.07492940379 0.074929403788 (-) continue7 0.1 0.7 -0.0095337743 0.0095337742966 (-) continue8 0.1 0.8 0.054087651179 0.0540876511794 revert8 0.01 0.71 -0.00309032714 0.0030903271407 (-) continue9 0.01 0.72 0.00333529272 0.0033352927198 revert9 0.001 0.711 -0.00244696146 0.0024469614555 (-) continue10 0.001 0.712 -0.00180377413 0.0018037741319 (-) continue11 0.001 0.713 -0.00116076526 0.0011607652593 (-) continue12 0.001 0.714 -0.00051793493 0.0005179349269 STOP

Half Interval Method

xo- = 0 f(x-) = -0.5 OKProceed

x0+ = 1 f(x+) = 0.17563936465 OK Proceedi x- x+ xi F(xi) error remarks0 0 1 0.5 -0.1420127083439 Change x-1 0.5 1 0.75 0.0225042926909 0.333333333333 Change x+2 0.5 0.75 0.625 -0.0584189705869 0.2 Change x-3 0.625 0.75 0.6875 -0.0176130174629 0.090909090909 Change x-4 0.6875 0.75 0.71875 0.0025330682175 0.04347826087 Change x+5 0.6875 0.71875 0.703125 -0.0075182874003 0.022222222222 Change x-6 0.703125 0.71875 0.7109375 -0.0024871665863 0.010989010989 Change x-7 0.7109375 0.71875 0.71484375 2.4314225901E-05 0.005464480874 Change x+8 0.7109375 0.71484375 0.712890625 -0.0012310856604 0.002739726027 Change x-9 0.712890625 0.71484375 0.7138671875 -0.0006033005457 0.001367989056 Change x-10 0.7138671875 0.71484375 0.71435546875 -0.0002894718618 0.000683526999 STOP

x1 = 0 f(x-) = -0.5 OKProceedx2 = 1 f(x+) = 0.17563936465 OK

i xi xi+1 F(xi) F(xi+1) xi+2 F(xi+2)1 0 1 -0.5 0.1756393646499 0.740039769973 0.0161580684062 0 0.740039769973 -0.5 0.0161580684064 0.71687319764 0.0013280494213 0 0.71687319764 -0.5 0.001328049421 0.714974155613 0.0001081100864 0 0.714974155613 -0.5 0.0001081100863 0.714819597197 8.79381712E-06

x1 = 4 f(x-) = -0.30547195053 OKProceedx2 = 5 f(x+) = 1.091246980352 OK

i xi xi+1 F(xi) F(xi+1) xi+2 F(xi+2)1 4 5 -0.30547195053 1.0912469803517 4.218706816225 -0.097251934942 4.218706816225 5 -0.09725193494 1.0912469803517 4.282638110481 -0.027309983483 4.282638110481 5 -0.02730998348 1.0912469803517 4.300152767299 -0.007395684784 4.300152767299 5 -0.00739568478 1.0912469803517 4.304863898308 -0.001983026345 4.304863898308 5 -0.00198302634 1.0912469803517 4.306124816132 -0.00053029924

x1 = 9x2 = 12

i xi xi+1 F(xi) F(xi+1) xi+2 F(xi+2)1 9 12 36.00856565026 189.71439674637 8.297191940082 23.375307837792 12 8.297191940082 189.7143967464 23.375307837794 7.776843543125 16.640033883583 8.297191940082 7.776843543125 23.37530783779 16.640033883585 6.491281142026 6.3477955949044 7.776843543125 6.491281142026 16.64003388358 6.3477955949038 5.69840333277 2.938589632628

5 6.491281142026 5.69840333277 6.347795594904 2.9385896326282 5.014976594696 1.1220546634636 5.69840333277 5.014976594696 2.938589632628 1.122054663463 4.592831041977 0.3764160616477 5.014976594696 4.592831041977 1.122054663463 0.3764160616467 4.379721948632 0.0872635398338 4.592831041977 4.379721948632 0.376416061647 0.0872635398332 4.315407607746 0.0102173260689 4.379721948632 4.315407607746 0.087263539833 0.0102173260681 4.306878692961 0.00033907381210 4.315407607746 4.306878692961 0.010217326068 0.0003390738125 4.306585935541 1.39239371E-0611 4.306878692961 4.306585935541 0.000339073812 1.3923937061E-06 4.306584728386 1.91039184E-1012 4.306585935541 4.306584728386 1.39239371E-06 1.9103918447E-10 4.306584728221 013 4.306584728386 4.306584728221 1.91039184E-10 0 4.306584728221 0f(x) tanx - tanhx = 0 f'(x) sec^2x -sech^2x x^3-3.23x^2-5.54x+9.84=0

x1 = 5 Newton-Raphson Methodi xi xi+1 error, d remarks

0 5 5.352472635059 0.065852300253 continue 01 5.352472635059 6.188237830418 0.135057057964 continue 12 6.188237830418 7.273637518454 0.149223780438 continue 23 7.273637518454 7.115744041851 0.022189313679 continue 24 7.115744041851 7.070875186524 0.006345587236 continue 35 7.070875186524 7.068588008944 0.00032356923 STOP 46 7.068588008944 7.068582745656 7.44602936E-07 STOP 47 7.068582745656 7.068582745629 3.91907203E-12 STOP 58 7.068582745629 7.068582745629 0 STOP 69 7.068582745629 7.068582745629 0 STOP 7

10 7.068582745629 7.068582745629 0 STOP 7789

1011

ln(4x^2) = x

F(xi) error remarks9.210340371976 4.428681023791 continue5.826947973856 0.580645719389 continue4.911281080439 0.18644155739 continue4.569364001695 0.074828155213 continue4.425042415324 0.032614735143 continue4.360854088716 0.014719209885 continue4.331630221288 0.006746621003 continue4.318182292484 0.003114256855 continue4.311963457281 0.001442228178 continue4.309081078948 0.00066890789 stop

xo = 4 f(xo) = -0.305472 OKerror = 0.001

Δx = 1 |f(xi)|i Δx xi f(xi) error remarks0 1 4 -0.30547195053 0.30547195 (-) continue1 1 5 1.09124698035 1.09124698 revert1 0.1 4.1 -0.21604944685 0.21604945 (-) continue2 0.1 4.2 -0.11691504372 0.11691504 (-) continue3 0.1 4.3 -0.00757080141 0.0075708 (-) continue4 0.1 4.4 0.11250674972 0.11250675 revert4 0.01 4.31 0.00394508948 0.00394509 revert4 0.001 4.301 -0.00642405017 0.00642405 (-) continue5 0.001 4.302 -0.00527622528 0.00527623 (-) continue6 0.001 4.303 -0.00412732621 0.00412733 (-) continue7 0.001 4.304 -0.00297735243 0.00297735 (-) continue8 0.001 4.305 -0.00182630339 0.0018263 (-) continue9 0.001 4.306 -0.00067417855 0.00067418 STOP

xo- = 4 f(x-) = -0.305472 OKProceed

x0+ = 5 f(x+) = 1.09124698 OK Proceedi x- x+ xi F(xi) error remarks0 4 5 4.5 0.24386792 Change x+1 4 4.5 4.25 -0.0635513 0.05882353 Change x-2 4.25 4.5 4.375 0.08145149 0.02857143 Change x+3 4.25 4.375 4.3125 0.00684089 0.01449275 Change x+4 4.25 4.3125 4.28125 -0.0288743 0.00729927 Change x-5 4.28125 4.3125 4.296875 -0.0111475 0.00363636 Change x-6 4.296875 4.3125 4.3046875 -0.0021861 0.00181488 Change x-7 4.3046875 4.3125 4.30859375 0.00231916 0.00090662 STOP

error remarksChange x+

0.032316136813 Change x+0.002656098842 Change x+0.000216220173 STOP

error remarksChange x-

0.014928016939 Change x-0.0040730313 Change x-

0.001094373973 Change x-0.000292819618 STOP

error remarkscontinue

0.066909973702 continue0.19804448043 continue

0.139140345629 continue

0.136277154074 continue0.09191401749 continue

0.048658133061 continue0.014903422048 continue0.001980300675 continue6.79790036E-05 STOP2.80304328E-07 STOP3.84636695E-11 STOP

0 STOPx^3-3.23x^2-5.54x+9.84=0 Use half-interval method to compute the cube root of 75

0.001 with four decimal places accuracy.Incremental Search Method

xi f(xi) error, d remarks xi-0 9.84 1 continue 0 41 2.07 1 1 continue 1 42 -6.16 1 0.5 revert 2 4

1.1 1.1687 0.1 0.81818181818 continue 3 4.1251.2 0.2688 0.1 0.08333333333 continue 4 4.18751.3 -0.6237 0.1 0.07692307692 revert 5 4.1875

1.21 0.179118 0.01 0.07438016529 continue 6 4.2031251.22 0.089516 0.01 0.00819672131 continue 7 4.21093751.23 0 0.01 0.0081300813 continue 8 4.214843751.24 -0.089424 0.01 0.00806451613 revert 9 4.21679688

1.231 -0.008946639 0.001 0.00731112916 revert 10 4.216796881.2201 0.088620404301 0.0001 0.00893369396 continue 11 4.216796881.2202 0.087724817208 0.0001 8.1953778E-05 stop 12 4.217041021.2203 0.086829238727 0.0001 8.1947062E-05 stop 13 4.217163091.2204 0.085933668864 0.0001 8.1940347E-05 stop 14 4.217163091.2205 0.085038107625 0.0001 8.1933634E-05 stop 15 4.21716309

Dx

Use half-interval method to compute the cube root of 75with four decimal places accuracy. f(x)=x^3-75 0.00009

xi+ f(xi-) f(xi+) xi+1 f(xi+1) error remarks5 -11 50 4.5 16.125 continue

4.5 -11 16.125 4.25 1.765625 0.25 continue4.25 -11 1.765625 4.125 -4.8105469 0.125 continue4.25 -4.810546875 1.765625 4.1875 -1.5715332 0.0625 continue4.25 -1.57153320313 1.765625 4.21875 0.08468628 0.03125 continue

4.21875 -1.57153320313 0.08468628 4.203125 -0.7465019 0.015625 continue4.21875 -0.74650192261 0.08468628 4.2109375 -0.3316789 0.0078125 continue4.21875 -0.33167886734 0.08468628 4.21484375 -0.1236892 0.00390625 continue4.21875 -0.12368923426 0.08468628 4.21679688 -0.0195497 0.001953125 continue4.21875 -0.01954973489 0.08468628 4.21777344 0.03255621 0.0009765625 continue

4.21777344 -0.01954973489 0.03255621 4.21728516 0.00650022 0.00048828125 continue4.21728516 -0.01954973489 0.00650022 4.21704102 -0.0065255 0.00024414063 continue4.21728516 -0.00652551219 0.00650022 4.21716309 -1.2835E-05 0.00012207031 continue 4.21728516 -1.2835293E-05 0.00650022 4.21722412 0.00324364 6.1035156E-05 stop4.21722412 -1.2835293E-05 0.00324364 4.2171936 0.00161539 3.0517578E-05 stop

4.2171936 -1.2835293E-05 0.00161539 4.21717834 0.00080128 1.5258789E-05 stop