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The Taylor series is a polynomial of infinite degree used to represent functions like sine, cube roots, and the exponential function. They're how some calculators (and Physicists) make approximations.
Let \(f(x)\) be a function such that \(f(0)=3, f'(0)=1,\) and \(f(3)=1.\) Using a quadratic Taylor polynomial of \(f(x),\) we can approximate the value of \(f''(0)\) as \[f''(0) \approx -\frac{A}{B},\] where \(A\) and \(B\) are coprime integers. Find the value of \(AB.\)
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Using the quadratic Taylor polynomial of \( f(x) = \frac{1}{x} \) at \( x=5, \) find the approximate value of \( \frac{1}{7} \) multiplied by \(5^3.\)
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For the linear Taylor polynomial \(g(x)=ax+b\) of \(f(x)=-\frac{1}{x-1}\) at \(x=2,\) find the error \(e\) defined by \[e=\int_{2}^{3}{(g(x)-f(x))^2 dx}.\]
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Using the quadratic Taylor polynomial of \( f(x) = \ln{x} \) at \( x=4, \) we can approximate the value of \( \ln{9}\) in the form \[\ln{A}+B, \] where \(A\) and \(B\) are rational numbers. Find the value of \( 2A^2B.\)
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Let \(f(x)\) be a function such that \(f(0)=1, f'(0)=2,\) and \(f(3)=1.\) Using a quadratic Taylor polynomial of \(f(x),\) we can approximate the value of \(f''(0)\) as \[f''(0) \approx -\frac{A}{B},\] where \(A\) and \(B\) are coprime integers. Find the value of \(AB.\)
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