Mathematics
Mathematics
Problem
A differentiable function is decreasing and concave down on . Explain in one or two sentences whether the trapezoidal approximation of using equal subintervals is an overestimate or an underestimate, and justify your conclusion.Intuitive explanation
When a function is decreasing and concave down, its graph curves downward while falling, which means the straight-line segments connecting consecutive points (the trapezoids) bulge above the actual curve. Since the trapezoids cover more area than the region under , the trapezoidal rule produces an overestimate of the integral.
Intuitive explanation
When a function is decreasing and concave down, its graph curves downward while falling, which means the straight-line segments connecting consecutive points (the trapezoids) bulge above the actual curve. Since the trapezoids cover more area than the region under , the trapezoidal rule produces an overestimate of the integral.