Consider the solution with y(0)= 0 to the differential equation $\frac{d y}{d x}=\frac{7}{1+x^{2}}$ .If you use Euler's method with 1 million steps to approximate the exact value of $y ( 1 )$ , will your approximation be an over- or underestimate?
A)An overestimate
B)An underestimate

Question

Quizplus · Accepted Answer

Euler's method approximates the solution to a differential equation by taking small steps along the curve, using the slope at the beginning of each step. For the given differential equation, $\frac{d y}{d x}=\frac{7}{1+x^{2}}$, the slope decreases as $x$ increases because the denominator $1+x^{2}$ becomes larger, making the fraction smaller. When using Euler's method with a positive step size, starting from $x=0$, the initial slope used for the first step is the steepest, and it overestimates the slope for subsequent steps. This leads to an overestimation of the final value of $y(1)$ because the method consistently uses slopes that are slightly larger than the actual slope of the curve at each step.

Consider the Solution with Y(0)= 0 to the Differential Equation dydx=71+x2\frac{d y}{d x}=\frac{7}{1+x^{2}}dxdy​=1+x27​

Consider the Solution with Y(0)= 0 to the Differential Equation $\frac{d y}{d x}=\frac{7}{1+x^{2}}$