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Differential Calculus
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TableForm[errorvalues, TableSpacing {1,5}, TableHeadings {None, {" x"," error[x]"}}]
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x 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. error[x] 0. 1.53529 10 7 9.10987 10 6 0.0000967355 0.000509097 0.00182656 0.00514837 0.0122941 0.026016 0.0502191 0.0901862
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As x gets further from 1, the error gets larger.
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8.28 Let f(x) = ln x. Construct the Taylor polynomials of degrees 1, 2, 3, . . . , 10 about a = 1 and compute their value at x = 1.5. Determine the error in the approximations and express in a tabular form.
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SOLUTION
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f[x_] = Log[x]; exactvalue = f[1.5]; value[n_] Normal[Series[f[x], {x, 1, n}]]/. x 1.5 data = Table[{n, N[value[n]], exactvalue, N[Abs[value[n] exactvalue]]}, {n, 1, 10}]; TableForm[data, TableSpacing {1,5}, TableHeadings {None, {"n","
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n 1 2 3 4 5 6 7 8 9 10 Pn(1.5) 0.5 0.375 0.416667 0.401042 0.407292 0.404688 0.405804 0.405315 0.405532 0.405435 f(1.5) 0.405465 0.405465 0.405465 0.405465 0.405465 0.405465 0.405465 0.405465 0.405465 0.405465 Error 0.0945349 0.0304651 0.0112016 0.00442344 0.00182656 0.000777608 0.000338463 0.000149818 0.000067196 0.0000304603
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f(1.5) ","
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8.29 What is the coefficient of the x20 term of the Maclaurin series for sin( x 2 + 1)
SOLUTION
s = Series[Sin[x2 + 1], {x, 0, 20}]; SeriesCoefficient[s, 20] Sin[1] 3 628 800
C HA PTE R 9
Integral Calculus
9.1 Antiderivatives
An antiderivative of a function f is another function F such that F'(x) = f ( x). In Mathematica, the Integrate command computes antiderivatives. You will notice, however, that the constant of integration, C, is omitted from the answer.
Integrate[f[x], x] computes the antiderivative (indefinite integral) from the Basic Math Input palette may also be used.
f ( x ) dx. The symbol
Mathematica can compute antiderivatives of elementary integrals found in standard tables, but if unable to evaluate an antiderivative in terms of elementary functions, the software will try to express the antiderivative in terms of special functions. If this is not possible, Mathematica returns the antiderivative unevaluated.
EXAMPLE 1
Exp[x]Sin[x] x
Integrate[x^2 Exp[x] Sin[x], x]
1 x( ( 1+ x) Cos[x]+( 1 + x2)Sin[x]) 2 2
EXAMPLE 2
Sin[x ] x
Integrate[Sin[x^2], x]
This integral has no simple antiderivative, so Mathematica expresses 2 x it as a Fresnel sine integral: FresnelS( x ) = sin t dt 0 2
Fresnel 2 x 2
EXAMPLE 3
Sin[Sin[x]] x Sin[Sin[x]] x
or Integrate[Sin[Sin[x]], x]
Mathematica cannot evaluate this antiderivative.
Care must be taken when general antiderivatives involving parameters are requested.
EXAMPLE 4
x1+ n 1+ n
Integral Calculus
Of course, this result is valid only if n 1, but if the value of n is specified, Mathematica knows what to do. n = 1;
x
Log [x]
SOLVED PROBLEMS
9.1 Compute
SOLUTION
x dx .
Integrate x , x
x x
2x 3
9.2 Compute
SOLUTION
a 2 + x 2 dx
or Integrate a2 + x2 , x
a 2 + x 2 x
1 x a2 + x2 + 1 a2 Log x + a2 + x2 2 2
9.3 Compute
SOLUTION
1 du u a2
1 u u2 a2
Integrate[1/Sqrt[u2 a2], u]
Log u + a2 + u2
9.4 Compute
SOLUTION
tanh x dx .
or Integrate[Tanh[x], x]
Tanh[x] x
9.5 Evaluate (a)
SOLUTION
Log[Cosh[x]]
f '( x )dx and (b) g '(f ( x )) f '( x )dx .
f' [x] x
f[x]
g'[f[x]]f'[x] x
g[f[x]]
9.6 Construct a table of integrals for
SOLUTION
n anti[n] = sin[x] x :
sin
x dx n = 1, 2, 3, . . . , 10.
tablevalues = Table[{n, Together[anti[n]]}, {n, 1, 10}]; TableForm tablevalues, TableSpacing {1, 5},
TableHeadings None, "n" " Sin n x dx" ,
Integral Calculus
Sin[x] x
Cos[x] 1 (2 x Sin[2 x]) 4
1 ( 9 Cos[x]+ Cos[3 x]) 12 1 (12 x 8 Sin[2 x]+ Sin[4 x]) 32 1 ( 150 Cos[x]+ 25 Cos[3 x] 3 Cos[5 x]) 240 1 (60 x 45 Sin[2 x]+ 9 Sin[4 x] Sin[6 x]) 192
1 2 3 4 5 6 7 8 9 10
1225 Cos[x]+ 245 Cos[3 x] 49 Cos[5 x]+ 5 Cos[7 x] 4 2240 840 x 672 Sin[2 x]+ 168 Sin[4 x] 32 Sin[6 x]+ 3 Sin[8 x] 3072 39 690 Cos[x]+ 8820 Cos[3 x] 2268 Cos[5 x]+ 405 Cos[7 x] 35 Cos[9 x] [ 80 640
2520 x 2100 Sin[2 x]+ 600 Sin[4 x] 150 Sin[6 x]+ 25 Sin[8 x] 2 Sin[10 x] i 10 240
9.7 Use Manipulate to evaluate
SOLUTION
sin
x dx for 1 n 10.
Manipulate Sin[x]n x //Together, {n, 1, 10, 1}, ControlType RadioButton
9.2 Definite Integrals
A definite integral can be computed one of two ways: exactly, using the Fundamental Theorem of Calculus, or approximately, using numerical methods. You can instruct Mathematica which method you wish to use by choosing from two commands.
Integrate[f[x], {x, a, b}] computes, whenever possible, the exact value of f ( x ) dx . The a symbol on the Basic Math Input palette may be used as well. b NIntegrate[f[x], {x, a, b}] computes an approximation to the value of f ( x ) dx using a strictly numerical methods.
NIntegrate evaluates the integral using an adaptive algorithm, subdividing the interval of integration until a desired degree of accuracy is achieved. The interval is divided recursively until the value of AccuracyGoal or PrecisionGoal is achieved.
AccuracyGoal is an option that specifies how many digits to the right of the decimal point should be sought in the final result. AccuracyGoal effectively specifies the absolute error. The default for NIntegrate is AccuracyGoal Infinity, which specifies that accuracy should not be used as the criterion for terminating the numerical procedure. WorkingPrecision is an option that specifies how many digits of precision should be maintained in internal computations. The default value is approximately 16. PrecisionGoal is an option that effectively specifies the relative error. The default setting, PrecisionGoal Automatic, sets PrecisionGoal to half the value of WorkingPrecision. If defaults are not used, you should set PrecisionGoal to be less than the value of WorkingPrecision.
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