Let satisfy with

MATHEMATICS · Single correct · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Let y=y(x)y=y(x) satisfy dydx=y−2x\dfrac{dy}{dx}=y-2x with y(0)=1y(0)=1. A tangent-line approximation is used with step size h=0.5h=0.5. The first estimate y^1\hat y_1 approximates y(0.5)y(0.5), and the second estimate y^2\hat y_2 approximates y(1.0)y(1.0) by repeating the procedure starting from (0.5,y^1)(0.5, \hat y_1). Because y′′(x)=y′−2y''(x)=y'-2, which description of the two estimates is correct? (a) Both y^1\hat y_1 and y^2\hat y_2 are underestimates of the actual solution values. (b) Both y^1\hat y_1 and y^2\hat y_2 are overestimates of the actual solution values. (c) y^1\hat y_1 is an overestimate and y^2\hat y_2 is an underestimate of the actual solution values. (d) y^1\hat y_1 is an underestimate and y^2\hat y_2 is an overestimate of the actual solution values.

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