Please help! I’m really stuck. Marvin the fly starts at (0,0). Each step, Marvin moves one unit right or one unit up. He is trying to get to the point (5,7). However, at (4,3) there is a frog that will eat him if he goes through that point. In how many ways can Marvin reach (5,7)?

The question that you asked is kind of confusing but I think that I have it.

Step-by-step explanation:

So, as you said Marvin moves by a 1 by 1. 1 to the right and then 1 up. But there is an obstacle at (4,3). To evade that obstacle I would suggest going up four, and then over 1. To be honest, there are a ton of ways. All you have to do is just steer clear of (4,3).

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.