Problem
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Answer
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Comment
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| 4 |
True |
The error term involves f''''(c). For a cubic polynomial, the fourth derivative is the zero function. Thus, the error is zero and the Maclaurin polynomial is exact. |
| 5 |
True |
Every Maclaurin polynomial for an even function is even. This means it has only even powers of x. |
| 6 |
True |
If f is an even (differentiable) function, then f' is an odd function. Then, f'(-0) = -f'(0). This is true only when f'(0)=0. |
| 8 |
False |
While this is true in theory, in practice limitations on the ability of the calculator to represent numbers means that the difference of two close numbers might be zero when computed on the calculuator. |
| 9 |
False |
In fact, very few integrals can be evaluated in terms of elementary functions. |
| 10 |
False |
The error for the Trapezoid Rule is E[10] = -(5)^3/12/10^2 * f''(c) = -750/1200*c for some value of c in (0,5). Thus, E[10]<0 and the Trapezoid Rule gives a value larger than the exact value of the definite integral. |
| 11 |
True |
The error for the Parabolic Rule with n=10 involves f''''(c). Note that f'''(x)=6 and f''''(x) = 0. Thus, the error term is 0 and the Parabolic Rule gives the exact value of this integral. |
| 13 |
True |
The largest values of the absolute value of each term are exp(0)=1, 2^2=4, and sin(pi/2)=1; thus, the entire function cannot exceed 1+4+1=6. |