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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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