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First Order Differential Equations

These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

Euler's Method - Small Step Size

         

Consider a function \(f(x)\) satisfying \( f(1) = 7\) and \( f'(x) = x^2 \). Using Euler's method with step size \(\frac{1}{2},\) what is the resulting approximation of \( f(8) \)?

Consider a function satisfying \( f(1) = 12\) and \( f'(x) = x^2 + x \). If applying Euler's method with step size \(\frac{1}{11}\) gives the approximation \(f(8) \approx \frac{a}{b},\) where \(a\) and \(b\) are positive coprime integers, what is the value of \(a-b\)?

Consider a function \(f(x)\) such that \( f(1) = 4\) and \(f'(x) = x^3 \). Using Euler's method with step size \(\frac{1}{3},\) what is the resulting approximation of \(f(16)-f(7)?\)

Consider a function \(f(x)\) such that \( f(1) = 11\) and \( f'(x) = 2x^2 - x \). Using Euler's method with step size \(\frac{1}{2},\) what is the resulting approximation of \(f(8) + f(16)\)?

Consider a function \(f(x)\) such that \( f(1) = 12\) and \( f'(x) = \displaystyle{\frac{1}{x(x+2)}}. \) Using Euler's method with step size \(\frac{1}{5},\) if the approximation for \(f(8) \) is \(\displaystyle{\frac{a}{b}}\) for coprime positive integers \(a\) and \(b\), what is the value of \(a+b?\)

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