3 function. the green function is x^2 and the blue line indicates the two points which are 0 and 2. the blue line shows the secant line that connects the two. and the green line is the point c in this case it is 1, where the slope is equal to the secant line as u can see. green line slope(f'(c)) is equal to the blue secant line([f(b) - f(a)]/(b - a)). note the secant(blue) line is parallel with the green line, which represents f'(c).2) the mean value theorem only works on continues and differentiable functions because other wise the theorem can not give a guaranteed point C that exist in the function.
(C) if the function is discontinuous such as |1/x| the function does not have point c where the slope is equal to 0 and is parallel to the the secant line. this is what makes discontinuous function
unable to use the mean value theorem.(D) the function |x| is a continues function but not differentiable. in this graph the secant line crosses X=-2 , and x=3 and with the mean value theorem f'(c) should be equal to 1/5 but there are only two possible slopes which are -1 and 1, therefore c does not exist. which proves wrong the mean value theorem. that is why functions that are discontinuous or not differentiable are not guaranteed a f'(c) that is equal to the secant line.
=) Your clean layout makes your posts look very organized, but you should go back and re-read it sometimes too. First of all, your curve is red and the blue does not indicate two points, but a line.
ReplyDeleteSecond, and much more important, we're not just looking for ANY line with the same SLOPE as the secant line, otherwise, I can draw that line ANYWHERE, like you did. Do you see how your green line is floating??
As for your non working examples, can you set intervals on that first one and then use an example? I like the way you did that second one. That one looks great! Especially because you're the first to not have the slope be 0 and explain the only two slopes available.
Hmm, this is a little confusing ?
ReplyDeleteWhat are the intervals for the graphs??
The last part is a tad repetitive.
hey for the first graph, the green line doesnt touch the graph... that's supposed to be your tangent line right?
ReplyDeleteyou should check your 1 paragraph and explain the functions better because by saying "secant line of two points x = a and x = b." makes me think that there are two secant lines...you should say a secant line that goes through those points. but yeah it's your info it's very organized!
ReplyDelete