1) Introduction Flashcards
Give the two definitons for a function to be continuous
- Given a function f : D → R, where D ⊂ R, f is said to be continuous at a point c ∈ D if limx→c f (x) = f (c).
- Given a function f : D → R, D ⊂ R, f is said to be continuous at a point c ∈ R if for any ϵ > 0 there exists δ > 0 such that |x −c| < δ ⇒ |f (x)− f (c)| < ϵ.
What is the Intermediate Value Theorem
If there is a function f that is continous on a closed set [a,b] and d is any number between f(a) and f(b), then there exists at least one number c in the interval [a,b] such that f(c) = d
When is a function differentiable
What is Taylor’s Theorem
What is the Mean Value Theorem
What is Rolle’s Theorem
Let f be continuous on [a,b] and differentiable on (a,b), and such that f (a) = f (b). Then there exists a number ξ ∈ (a,b) such that f’(ξ) = 0.
What are the two things to keep in mind when designing an algorithm
- Computational complexity: A measure of the time it takes for an algorithm to compute an approximation
- Computational Accuracy: A measure of the error between the approximation x- and the true solution x. It can be measured either through absolute error, or relative error
What are some sources of errors
- Truncation Errors
- Approximation Errors
What is Horner’s Algorithm
How many multiplications are required to evaluate a degree n polynomial naively and by Horner’s method
- Naively: 1/2n(n+1) multiplcations
- Horner’s Method: n multiplcations
How do we describe big-O notation
What is absolute error and relative error
Absolute Error=∣x−x̄∣
Relative Error = ∣x−x̄∣ / ∣x∣
Where x̄ is an approximation of x
What is Machine Precision
Machine precision is an upper bound on the relative approximation error due to rounding in floating point arithmetic
What is Catastrophic Cancelation
When subtracting two closely valued approximations, each having low relative errors, results in an answer with a significantly high relative error
