1. the proceeds of a $7,500.00, 10 percent simple discounted note for 85 days is
$7,325.34.
$7,677.08.
$7,674.66.
$7,322.92

  it depends (see below)

Step-by-step explanation:

Exact Interest

Exact interest is calculated on the basis of a 365-day year, so the exact interest on the loan would be ...

  I = Prt = $7500·0.10·(85/365) = $174.66

Then the proceeds would be ...

  $7500 -174.66 = $7,325.34 . . . . . matches first option

__

Ordinary Interest

Exact interest is calculated on the basis of a 360-day year, so the ordinary interest on the loan would be ...

  I = Prt = $7500·0.10·(85/360) = $177.08

Then the proceeds would be ...

  $7500 - 177.08 = $7,322.92 . . . . . matches last option

I suggest you check your reference materials (text or course examples) to see what year length is expected to be used in this calculation. Some on-line references show it one way; others show it the other way.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.