A solution curve of a differential equation is concave up on and concave down on
MATHEMATICS · Single correct · AP · Calculus BC · Differential Equations · Differential Equations
Problem
A solution curve of a differential equation is concave up on 0≤x≤1 and concave down on 1≤x≤2. A student makes two equal-length tangent-line steps—one from x=0 to x=1 and another from x=1 to x=2—always beginning each step at the most recently estimated point. Which of the following best explains why the first estimate is an underestimate while the second is an overestimate?
(a) Because the slope increases on [0,1] and decreases on [1,2], the tangent-line approximation changes from underestimating to overestimating.
(b) Error from the first Euler step carries into the second, so an initial underestimate forces the next estimate to overshoot in the opposite direction.
(c) On a concave-up interval a tangent line lies below the curve, whereas on a concave-down interval it lies above the curve.
(d) The switch occurs precisely at x=1 because the average rate of change equals the instantaneous rate of change only at points of inflection.