Consider the differential equation $\frac { d y } { d x } = x ^ { 2 } + y$ .If you use Euler's method to approximate the value of y when x = 2 on the solution curve that passes through (1,4), is your approximation an underestimate or overestimate?
A)An underestimate
B)An overestimate

Question

Quizplus · Accepted Answer

Euler's method uses tangent line approximations. Since the differential equation $\frac{dy}{dx} = x^2 + y$ is increasing in both $x$ and $y$, the solution curve is concave up. Therefore, the tangent line lies below the curve, making Euler's method an underestimate.

Consider the Differential Equation dydx=x2+y\frac { d y } { d x } = x ^ { 2 } + ydxdy​=x2+y

Consider the Differential Equation $\frac { d y } { d x } = x ^ { 2 } + y$