It is given that f(x) = −2x2 + 10
Consider the following three curves:
(1) y = f(x)
(2) y = f(x+1)
(3) the curve y = f(x+1) reflected in the line y = 6
The trapezium rule is used to estimate the area under each of these three curves between x = 0 and x = 1.
State whether the trapezium rule gives an overestimate or underestimate for each of these areas.
- A(1) underestimate, (2) underestimate, (3) underestimate
- B(1) underestimate, (2) underestimate, (3) overestimate
- C(1) underestimate, (2) overestimate, (3) underestimate
- D(1) underestimate, (2) overestimate, (3) overestimate
- E(1) overestimate, (2) underestimate, (3) underestimate
- F(1) overestimate, (2) underestimate, (3) overestimate
- G(1) overestimate, (2) overestimate, (3) underestimate
- H(1) overestimate, (2) overestimate, (3) overestimate
Show the answer and worked solution
answer · B
- A(1) underestimate, (2) underestimate, (3) underestimate
- B(1) underestimate, (2) underestimate, (3) overestimate
- C(1) underestimate, (2) overestimate, (3) underestimate
- D(1) underestimate, (2) overestimate, (3) overestimate
- E(1) overestimate, (2) underestimate, (3) underestimate
- F(1) overestimate, (2) underestimate, (3) overestimate
- G(1) overestimate, (2) overestimate, (3) underestimate
- H(1) overestimate, (2) overestimate, (3) overestimate
The trapezium rule joins points on the curve by chords, so it overestimates where the curve is convex (bending upwards) and underestimates where it is concave. Curve (1) has f''(x) = −4 < 0, so it is concave and the estimate is too small. Curve (2) is only a horizontal translation of (1), so its shape, and therefore the answer, is unchanged. Reflecting in the horizontal line y = 6 sends y to 12 − y, giving y = 12 − f(x+1) = 2(x+1)2 + 2, which is convex, so (3) is an overestimate.