AboutScotto Expertise Any kind of mathematics (algebra, geometry, trigonometry, matrices, calculus, linear approximation, linear regression, linear programming, numerical analysis, probability, statistics, etc.). I also have answered some questions in Physics, Chemistry, and Biology. I would like to volunteer in all areas of Mathematics, not just calculus, and the other three courses that were mentioned.
Experience Experience in the area: I have tutored students in all areas of mathematics for over 20 years.
Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors.
Awards and Honors: I have passed Actuarial tests 100, 110, and 135.
Question could you give me the reason for choosing f'(x)=0;if f(X)is a real function?and also could you give me the proof of having
f'(C)=f(b)-f(a)/b-a in mean value theorem
Answer The reason for choosing f'(x)=0 if for finding the extreme points of the function since when an extreme point is reached, continuous functions go from a slope up to a slope down, making it 0 at the extreme point.
The mean value theorem assumes the function and its derivatives are both continuous over the interval [a,b] and the C is some point in between a and b. f'(C) is the slope of a straight line from (a,f(a)) to (b,f(b)).
Let's suppose that at the start the graph slopes up faster at a than the average slope. To come into b at f(b), the function needs to have a lesser slope than the average somewhere. Since the derivative is assumed to be continuous, then at some point in the interval the slope needs to be the average slope since it was greater at one end and lesser at the other.
If the graph slopes up slower than the average change at the start, at some point in the middle it must slope up faster to reach (b,f(b)). It must therefore (from continuity) be equal the average slope somewhere in the middle.
If the graph slope is equal to the average slope at the left side, we're done before we started.
