How do you find the Taylor polynomial of degree 4 for #f(x) = cosh(x)# about #x = 0 # by using the given Taylor polynomial for #e^x#?
and we can see that the terms of even order cancel each other, so that:
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To find the Taylor polynomial of degree 4 for ( f(x) = \cosh(x) ) about ( x = 0 ) using the given Taylor polynomial for ( e^x ), you can use the fact that ( \cosh(x) ) is related to ( e^x ).
Since ( \cosh(x) = \frac{e^x + e^{x}}{2} ), you can use the Taylor polynomial for ( e^x ) to find the Taylor polynomial for ( \cosh(x) ).
The Taylor polynomial for ( e^x ) about ( x = 0 ) up to degree 4 is:
[ P_4(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} ]
Now, substitute this polynomial into the expression for ( \cosh(x) ) and simplify to obtain the Taylor polynomial for ( \cosh(x) ) about ( x = 0 ) up to degree 4.
[ \cosh(x) = \frac{e^x + e^{x}}{2} = \frac{P_4(x) + P_4(x)}{2} ]
[ = \frac{1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + 1  x + \frac{x^2}{2}  \frac{x^3}{6} + \frac{x^4}{24}}{2} ]
[ = 1 + \frac{x^2}{2} + \frac{x^4}{24} ]
Thus, the Taylor polynomial of degree 4 for ( \cosh(x) ) about ( x = 0 ) is:
[ P_4(x) = 1 + \frac{x^2}{2} + \frac{x^4}{24} ]
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To find the Taylor polynomial of degree 4 for ( f(x) = \cosh(x) ) about ( x = 0 ) using the Taylor polynomial for ( e^x ), we can use the fact that ( \cosh(x) ) can be expressed in terms of exponential functions.

Recall that ( \cosh(x) = \frac{e^x + e^{x}}{2} ).

Expand ( e^x ) using its Taylor polynomial about ( x = 0 ). Let's denote this Taylor polynomial as ( P_4(x) ): [ P_4(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} ]

Replace ( e^x ) in the expression for ( \cosh(x) ) with its Taylor polynomial: [ \cosh(x) = \frac{P_4(x) + P_4(x)}{2} ]

Simplify the expression to obtain the Taylor polynomial for ( \cosh(x) ) about ( x = 0 ). This will be the Taylor polynomial of degree 4.

The resulting polynomial will represent the Taylor polynomial of degree 4 for ( \cosh(x) ) about ( x = 0 ) using the given Taylor polynomial for ( e^x ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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