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CSU
MECH 102 -
Taylor Series Example
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Colorado State University- Fort Collins
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Taylor Series ExampleFunction:≔f((x))+sin((⋅2x))x2Derivative (slope):≔f'((x))→――ddxf((x))+⋅2x ⋅2 cos((⋅2x))Function approximation relative to a known point:≔i1≔xi2≔h0.5≔yi=f⎛⎝xi⎞⎠3.243≔x+i1=+xih2.5 ≔y+i1=+yi⋅f'⎛⎝xi⎞⎠h4.59Error in approximation:≔Et=−f⎛⎝x+i1⎞⎠y+i10.702=f⎛⎝x+i1⎞⎠5.291Illustration of results:tangent line:≔t⎛⎝xt⎞⎠+⋅f'⎛⎝xi⎞⎠⎛⎝−xtxi⎞⎠yi≔δ1 ≔xt,‥⎛⎝−xiδ⎞⎠xi⎛⎝+xiδ⎞⎠≔xf,‥00.01 31.72.553.44.255.15.956.87.658.500.859.351.41.61.8 2 2.22.42.62.811.2 3f⎛⎝xf⎞⎠t⎛⎝xt⎞⎠xfxtNon-Commercial Use OnlyRemainder calculation:≔f''((x))+⋅−4 sin((⋅2x))2second derivativeGuess ValuesConstraintsSolver≔ξxi＝Et⋅――f''((ξ))2h2≔ξ=((ξ))2.134≔R1=⋅――f''((ξ))2h20.702 =Et0.702Non-Commercial Use
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