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How to prove that $\dfrac{1}{(1-x)^3}$ is the generating function for the triangular numbers? The $n^{\text{th}}$ triangular number is defined as $T_n = \displaystyle{n+1 \choose 2}$.

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The two examples $y=f(x)=2x+1$ and $y=f(x)=x^2$ are both functions which can be evaluated at any value of $x$ from negative infinity to positive infinity. For many functions, however, it only …

Using the general form of a quadratic function, y = a × x 2 + b × x + c and the three coordinates labeled above, write a system of three equations to solve for the three unknowns, a,b and c.

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The function models the shape of the arch, where y is the height in feet and x is the horizontal distance from the base of the left side of the arch in feet. Convert the coefficient of x^2 to a …

Jan 22, 2025 · The function cosh ⁡ x \cosh x cosh x is the hyperbolic cosine, defined by cosh ⁡ x = e x + e − x 2 \cosh x=\frac{e^x+e^{-x}}{2} cosh x = 2 e x + e − x . Estimate the length of the …

Its shape can be modeled by the function y = 1260 − 315 (e 0.00418 x + e − 0.00418 x) y=1260-315\left(e^{0.00418 x}+e^{-0.00418 x}\right) y = 1260 − 315 (e 0.00418 x + e − 0.00418 x), …

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Sep 29, 2024 · Trigonometric substitution is an integration technique that allows us to convert algebraic expressions that we may not be able to integrate into expressions involving …

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