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A direction field for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes $ h = 1 $ and $ h = 0.5. $ Will the Euler estimates be under-estimates or overestimates? Explain.

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The curve of the solution is downward So Euler's approximations are overestimates.In other words, tangents at any points on the curve will be above the curve, indicating an overestimation.

Calculus 2 / BC

Chapter 9

Differential Equations

Section 2

Direction Fields and Euler's Method

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A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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A direction field for a di…

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Consider the initial-value…

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Consider the differential …

in this problem, we are beginning to understand more about directional fields in their relation to differential equations and, more specifically, how to take approximations using Oilers method. Now let's first review what we're given. Essentially, what we need to do in this problem is we're given some boundaries and were given a directional field and we need to decide. Is this an overestimate or underestimate using our knowledge of oiler approximation? So we're given that we have h equal toe one and H equal 2.5. So what? This graph shows us these lines here are essentially the directional field that we were given. And these lines correspond to our boundaries of H equal to one and h equal 2.5. So what we can see here is that if these lines our our field or directional field is clearly an overestimate of our actual curves that we have here because what we notice here is most of that field that direction is lying above are curved boundaries above our functions. So what does this tell us? That means that in this case, are oilers. Method of approximation is an overestimate compared to the functions compared to those differential equations that have created this field. Another word to say this another way to say this pardon me is that if we take the tangent to any point of our curves, it's going to lie above our function. So again, this is an overestimate because of that fact that we see this tangent line above our curve. So I hope that this made sense to you. And I hope you understand a little bit more about the methodology behind Oiler approximation of differential equations and creating a directional field.

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