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.

Consider a function \(f(x)\) such that \( f(1) = 4\) and \( f'(x) = x^3 \). Using Euler's method with step size \(1,\) what is the resulting approximation of \(f(9)-f(5)?\)

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Consider a function \(f(x)\) satisfying \( f(1) = 4\) and \(f'(x) = x^2 \). Using Euler's method with step size \(1,\) what is the resulting approximation of \( f(6) \)?

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Consider a function \(f(x)\) satisfying \( f(1) = 3\) and \( f'(x) = \frac{1}{x(x+1)}. \) Using Euler's method with step size \(1,\) what is the resulting approximation of \( f(10) \)?

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Consider a function \(f(x)\) such that \( f(1) = 5, f'(x) = 4x^2 - x \). Using Euler's method with step size \(1,\) what is the value of \(a\) that gives the approximation \(f(a) \approx 348\)?

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Consider a function \(f(x)\) such that \( f(1) = 10\) and \( f'(x) = x^2 + x \). Using Euler's method with step size \(1,\) what is the resulting approximation of \(f(6) + f(12)\)?

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