Copland competes in 27 swimming events this season. He wins either first place or second
place in each event. Copland has 3 more first-
place wins than second-place wins. In how
many events did he win first place, and in how
many did he win second place?

15 wins 12 2nd because subtract 3 from 27 that's 24 divide that by 2 its 12 so you add the extra 3 to 12 and that's 17

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.