Calculus help needed pleeease!!

Question

Given f(x) > 0 with f ′ (x) < 0, and f ″(x) < 0 for all x in the interval [0, 1] with f(0) = 1 and f(1) = 0.3, the left, right, trapezoidal, and midpoint rule approximations were used to estimate .  The estimates were 0.7915, 0.8405, 0.8410, 0.8421 and 0.8895, and the same number of subintervals were used in each case. Match the rule to its estimate.



Question options:

Trapezoidal:

Right Endpoint:

Left Endpoint:

Midpoint:

Actual Area:

1.	0.8895
2.	0.7915
3.	0.8421
4.	0.8405
5.	0.8410

Answer 1

Since f'(x) < 0, f is decreasing on the interval

Since f"(x) < 0, the function is also concave down.

Since f is decreasing and concave down, the left endpoint method is the only one that overestimates. So, the left endpoint rule gives 0.8895.

The right endpoint rule would give the "worst" underestimate, 0.7915.

The trapezoid rule value is the average of the left and right endpoint estimates, 0.8405

That leaves 0.8421 and 0.8410.

I'll let you decide which is which.