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If \(f\) is a differentiable function such that
\(\int_0^x f(t) \,\mathrm{d}t = \big(f(x)\big)^2\) for all \(x\),
find a formula for \(f\).
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What’s the area of the region bound between
the graph of the function \(f(x) = 2\mathrm{e}^{-x}\sin(x)\)
and the positive \(x\)-axis?
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Prove that every nonzero coefficient of the Taylor series of
\( \left(1-x+x^2\right)\mathrm{e}^x \)
about \(x = 0\) is a rational number whose numerator (in
lowest terms) is either \(1\) or a prime number.
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Recalling that \(\ln(x) \lt \sqrt{x}\) for \(x \gt 1,\)
show that this series diverges.
\[\sum_{n=2}^\infty \frac{1}{\big(\ln(n)\big)^{\ln\left(\ln(n)\right)}}\]
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Consider the function \(\displaystyle f(x) = \frac{x}{1-x-x^2}\,.\)
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Show that the Maclaurin series of the function is \(\sum_{n=1}^\infty F_nx^n\)
where \(F_n\) is the \(n\)th Fibonacci number.
I.e., \(F_1 = F_2 = 1\) and \(F_n = F_{n-1} + F_{n-2}\).
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By writing \(f(x)\) as a sum of partial fractions and thereby obtaining the Maclaurin series in a
different way, find an explicit formula for the \(n\)th Fibonacci number.
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For a positive integer \(n,\) let
\[f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx).\]
Find the smallest \(n\) such that \(|f_n''(0)| \gt 2023.\)
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Let \(I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,\mathrm{d}x\).
For which integers \(m,\) \(1 \leq m \leq 10\) is \(I_m \neq 0?\)
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Find all differentiable functions \(f\colon\mathbf{R} \to \mathbf{R}\)
such that \[ f'(x) = \frac{f(x+n)-f(x)}{n} \]
for all real numbers \(x\) and all positive integers \(n\).
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Let \(f \colon \mathbf{R}_{\geq 0} \to \mathbf{R}\)
be a strictly decreasing continuous function
such that \(\lim_{x \to \infty} f(x) = 0\).
Prove that this integral diverges:
\[ \int\limits_0^\infty \frac{f(x) - f(x+1)}{f(x)}\,\mathrm{d}x \]
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Let \(f\) be an infinitely differentiable real-valued function
defined on the real numbers. If
\[ f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots, \]
compute the values of the derivatives \(f^{(k)}(0), k = 1, 2, 3, \dots.\)
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Let \(f\) be a three times differentiable function
(defined on \(\mathbf{R}\) and real-valued)
such that \(f\) has at least five distinct real zeros.
Prove that \(f + 6f'+ 12f''+8f'''\)
has at least two distinct real zeros.
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Suppose that \(f(x) = \sum_{i=0}^\infty c_i x^i\)
is a power series for which
each coefficient \(c_i\) is \(0\) or \(1\).
Show that if \(f\left(\frac 2 3\right) = \frac 3 2\),
then \(f(\frac 1 2)\) must be irrational.