Calculus Of One Real Variable – By Pheng Kim Ving


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1. Approximations Of Values Of Functions 
In Section 8.3 we approximate the value of f at a + h by the relation:
Approximation [1.3] approximates the value of f at x by using
the values of f itself and of its
derivatives of all orders themselves
at c and the signed distance from x to c. It's of
course an extension of and improvement on approximation [1.2].



Fig. 1.2 
If a function that's more complicated than polynomials has a
Taylor or Maclaurinseries representation and x_{1} is a point in
the
interval of representation, then naturally an approximate sum, which is usually
a partial sum, of the series obtained at x_{1} is an
approximate value of the function at x_{1}. This
polynomial approximation of values of functions is used by calculators and
computers to approximately evaluate the transcendental functions.
We're asked to find an approximate value of the function e^{x} at x = 1/2 by using the Maclaurin series of e^{x} obtained at x = 1/2.
See Fig. 1.3.

Fig. 1.3 
Solution
We have:
1. As (d ^{n}/dx^{n}) e^{x} = e^{x} and thus (d ^{n}/dx^{n}) e^{x}_{x}_{=0} = e^{0} = 1 for all n = 0, 1, 2, ..., the series:
is the Maclaurin series of e^{x} obtained at x = 1/2.
2. Recall that s_{n}
is the sum of the first n terms of
the series, so, if the subscripts of the terms start from 0 up, then s_{n} is the
sum of the terms from subscript 0 up
thru to subscript n – 1. To make it clear that
the subscripts start from 0 up, we
express the series in the sigma
notation, where the subscript is at the sigma symbol.
3. The error is the size of a tail of the series;
here it's just the tail because all the terms are positive and thus any tail is
positive. In this case we find a
geometric series to be a bound for the error. The absolute value of the common
ratio
1/(2(n
+1)) of the geometric series is less than 1 for any one n
= 0, 1, 2, ..., so we can apply the formula for the sum of
such a series to it.
For any particular value n_{1} of n we have:
which is a
geometric series with first term 1/2^{n}(n!) and common ratio
1/2(n + 1), and, as 0 < 1/2(n + 1) < 1 for all n
= 0, 1,
2, ..., has a sum of:
etc. We see that g^{(}^{n}^{)}(0) doesn't exist for any n. The square root function has no Maclaurinseries representation.
6. For a function represented by a Taylor or
Maclaurin series on an interval, we can use a partial sum s_{n}
of the series obtained
at a point x
in that interval to approximate the value of the function at x. Recall that for the function e^{x}, that interval is the
entire real line.
The Magnitude Of The Next Term As An Error Bound
Find an approximation of sin 42^{o} accurate to 4
decimal places by using a Maclaurin series.
Notes
1. We have to convert degrees to radians. We're asked
to find an approximate value of the function sin x at x = x_{1}, where
x_{1} rad = 42^{o}.
2. For an accuracy of 4 decimal places, the error bound will be 0.00005; see Section 14.7 Part 1.
3. When an alternating series satisfies the
hypotheses of the AST (AlternatingSeries Test) and when we approximate its sum
by one of its partial sums, the
error is less than or equal to the magnitude of its next term; see Section
14.7 Part 4.
Solution
Clearly the series satisfies the conditions of the AST
(AlternatingSeries Test). The accuracy of 4 decimal places requires that
the error bound is 0.00005. So:
EOS
Note

Fig. 1.4 
Solution
EOS
Remarks 1.3
1. As (d ^{n}/dx^{n}) e^{x} = e^{x} and thus (d ^{n}/dx^{n}) e^{x}_{x}_{=1} = e^{1} = e for all n = 0, 1, 2, ..., the series:
3. Generally, when employing a Taylor series centered
at a point c other than 0 to approximate the
value f(x_{1}) of a function
f(x) at a point x_{1}, we choose as c a non0 point that's closest to x_{1} and where the value f(c) of f is known or
readily
calculated. See Fig. 1.5. This is
because the series will involve f(c). Usually the values of the derivatives of all
orders of f at
c are obtained
from the algebraic manipulation of f(x_{1}) to get the difference x_{1} – c.

Fig. 1.5 c is non0
and is closest to x_{1} such that f(c) is known
or 
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2. Approximations Of Definite Integrals 
For the definition of integral functions see Section 9.4. They're functions defined by integrals.
Find the Maclaurin series of the integral function:
and write the series in the sigma notation.
We can try to integrate (sin t)/t first,
obtaining a function say F(x), which is an antiderivative of (sin
x)/x, then find
the series
of F(x). We can also try to find the series of (sin
t)/t first, then
integrate that series to obtain the series of the given integral
function. Because the series of (sin t)/t represents (sin t)/t, the
integral of that series represents the integral of (sin t)/t. Since
integrating a series is easier than integrating (sin t)/t, we'll
adopt the second approach.
Solution
EOS
Approximations Of Definite Integrals
Approximate the definite integral:
with error < 0.001, by using series.
Solution
EOS
1. In Example 2.2, the series:
2. For many functions that are expressible as simple
combinations of elementary functions, their antiderivatives aren't simple
combinations of elementary
functions. They can't be antidifferentiated by elementary techniques. However
we can often find
their Taylor series,
antidifferentiate these series to obtain the series of the antiderivatives, and
employ the latter series to
approximate definite integrals of
the original functions.
1. Determine the Maclaurin series of the integral function:
with error < 0.001.
Solution
EOS
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3. Indeterminate Forms 
^{{3.1}} Section 8.6 Theorem 2.1.
Evaluate the following limit:
Solution
EOS
In Example 3.1, we can use the series representation of cos x because it's valid for all x, and thus for all x near 0.
Calculate this limit by using series:
Solution
EOS
Problems & Solutions 
1. Approximate the value of e^{0.1} with error < 0.0001 by using a Maclaurin series.
2. Estimate the value of cos 10^{o} with error < 0.0001 by using a Maclaurin series.
3. Use a Maclaurin series to find an approximate value of ln 0.6 accurate to 2 decimal places.
The estimated value is accurate to 2 decimal places if error < 0.005, which is true if:
2. The subscripts of the terms of the series start
from 1 up. We express the series in the sigma notation to make this clear. So
s_{n}
is the sum of the terms from subscripts 1 up thru to n.
The corresponding tail of the series starts from term with
subscript n
+ 1 up.
4. Find the Maclaurin series of the integral function:
and write the series in the sigma notation.
Solution
5. Approximate the definite integral:
with error < 1/100,000, by
using series.
Solution
We can get by with n = 2. Thus:
6.
a. Determine the Maclaurin series of the integral function:
with error < 0.001.
7. Evaluate this limit:
8. Consider the system of equations:
b. Determine all the coefficients of the series in terms of a_{0} and a_{1}.
c. Solve the system of equations.
Solution
y' = a_{1} + 2a_{2}x + 3a_{3}x^{2} + 4a_{4}x^{3} + 5a_{5}x^{4} + ...,
y'' = 2a_{2} + 6a_{3}x + 12a_{4}x^{2} + 20a_{5}x^{3} + ...,
xy' = x(a_{1} + 2a_{2}x + 3a_{3}x^{2} + 4a_{4}x^{3} + 5a_{5}x^{4} + ...) = a_{1}x + 2a_{2}x^{2} + 3a_{3}x^{3} + 4a_{4}x^{4} + 5a_{5}x^{5} + ...,
0 = y'' + xy' + y,
= (2a_{2} + 6a_{3}x + 12a_{4}x^{2} + 20a_{5}x^{3} + ...) + (a_{1}x + 2a_{2}x^{2} + 3a_{3}x^{3} + ...) + (a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + ...)
= (a_{0} + 2a_{2}) + 2(a_{1} + 3a_{3})x + 3(a_{2} + 4a_{4})x^{2} + 4(a_{3} + 5a_{5})x^{3} + ...,
a_{n} + (n + 2)a_{n}_{+2} = 0, for all n = 0, 1, 2, ... .
and if n is odd then:
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