Chapter Two: Numbers and Functions Section One: Operations with
... Think about the graphs of the previous three problems. How can we tell by looking at a graph is something is a function? If one point is directly over another, this means that an x-value is paired with multiple y’s. This would not be a function. Therefore, if a vertical line can be drawn anywhere on ...
... Think about the graphs of the previous three problems. How can we tell by looking at a graph is something is a function? If one point is directly over another, this means that an x-value is paired with multiple y’s. This would not be a function. Therefore, if a vertical line can be drawn anywhere on ...
Math 2030 Lecture: Attracting Fixed Points
... the Mean Value Theorem from calculus holds. By the hypothesis, |F 0 (x0 )| < λ < 1 for some λ. By the continuity of the derivative at x0 , there is a δ > 0 such that F 0 [x] < λ on the interval I = (x0 − δ, x0 + δ). Applying the conclusion of the Mean Value Theorem, the [x0 ] = F 0 [c] for some c in ...
... the Mean Value Theorem from calculus holds. By the hypothesis, |F 0 (x0 )| < λ < 1 for some λ. By the continuity of the derivative at x0 , there is a δ > 0 such that F 0 [x] < λ on the interval I = (x0 − δ, x0 + δ). Applying the conclusion of the Mean Value Theorem, the [x0 ] = F 0 [c] for some c in ...
C1M4 Inverse Functions and Logarithms Each summer a new group
... Inverse Functions and Logarithms Each summer a new group of incoming students is inducted into the U.S. Naval Academy, they become Fouth Class Midshipmen or plebes, and identification numbers called alpha numbers are assigned. Since this year is 2000, and it is hoped that these students will graduate ...
... Inverse Functions and Logarithms Each summer a new group of incoming students is inducted into the U.S. Naval Academy, they become Fouth Class Midshipmen or plebes, and identification numbers called alpha numbers are assigned. Since this year is 2000, and it is hoped that these students will graduate ...
Math131A Set 2 June 30, 2013
... Do exercise 7.4 in Ross. 8. Proofs of limits of sequences 8.1. Let (sn ) be a sequence of nonnegative real numbers converging to s. (a) Prove that s ≥ 0. ...
... Do exercise 7.4 in Ross. 8. Proofs of limits of sequences 8.1. Let (sn ) be a sequence of nonnegative real numbers converging to s. (a) Prove that s ≥ 0. ...
1 Sets, functions and counting
... Most functions we meet will be quite well behaved. In particular, we will be able to construct an inverse function by restricting to parts where the given function is increasing or decreasing. Useful trick: To draw the graph of f −1 reflect the graph of f about the line y = x. ...
... Most functions we meet will be quite well behaved. In particular, we will be able to construct an inverse function by restricting to parts where the given function is increasing or decreasing. Useful trick: To draw the graph of f −1 reflect the graph of f about the line y = x. ...
Slide 1
... A sequence {an} converges to L (a real number) if, given any positive distance from L, we can go far enough out in the sequence so that every term from there out is within that given distance from L. We write limnan = L or {an} L . If {an} does not converge, we say it diverges. ...
... A sequence {an} converges to L (a real number) if, given any positive distance from L, we can go far enough out in the sequence so that every term from there out is within that given distance from L. We write limnan = L or {an} L . If {an} does not converge, we say it diverges. ...
Lecture17.pdf
... Interestingly, continuity at point a is a requirement for f ( a ) to be a local extremum, but continuity at a is not a requirement for f ( a ) to be a global extremum. Consequently, it is worth mentioning that in Figure 1 f ( b ) is not considered either a global nor a local minimum on the interval ...
... Interestingly, continuity at point a is a requirement for f ( a ) to be a local extremum, but continuity at a is not a requirement for f ( a ) to be a global extremum. Consequently, it is worth mentioning that in Figure 1 f ( b ) is not considered either a global nor a local minimum on the interval ...
f(x)
... A function value f(a) is called a relative minimum of f if there exist an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≤ f(x) A function value f(a) is called a relative maximum of f if there exist an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≥ f(x) ...
... A function value f(a) is called a relative minimum of f if there exist an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≤ f(x) A function value f(a) is called a relative maximum of f if there exist an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≥ f(x) ...
Full text
... integrated using standard formulas, but the results are not particularly simple. Finally, we note that the above ideas may be carried out to extend general second-order recurring sequences to continuous functions, as indicated in Section 2. However, because of increased complexity, we do not state t ...
... integrated using standard formulas, but the results are not particularly simple. Finally, we note that the above ideas may be carried out to extend general second-order recurring sequences to continuous functions, as indicated in Section 2. However, because of increased complexity, we do not state t ...