functions , Numerical Methods & Approximations Flashcards
(22 cards)
What is an even and odd function
create a linear approximation of a function f(x) around a specific point (x0) with 1 variable
what is the linear approximation of X^2 at (1,1)
Explain why exact integration methods are not typically used within industry
eg.
substitution
partial fractions
by parts
They are difficult to replicated within software
Not a general method, code is required for every different method.
Function may be too complicated to be integrated analytically
Function may not be integrated analytically
What is the formula for the rectangular rule approximation
Using the rectangular rule find an approximate solution to the following problems:
What is the formula for the Trapezoid rule approximation
Using the trapezoid rule find an approximate solution to the following problems:
Compare your answer to the exact result
what is the formula for the Simpson rule (composite) approximation for a composite with 4 sub-intervals
what is the general formula for the Simpson rule (composite) approximation
Using simpsons rule find an approximate solution to the following problems:
Compare your answer to the exact result
why does Simpsons rule result in an exact answer when the equation is quadratic?
Simpsons rule assumes the function can be approximated by a second degree polynomial.
Weights of 1,4,1 perfectly account for the contributions of even and odd intervals.
Explain which situations the rectangular rule is best for, comment on its accuracy.
Simple approximations
Constant/linear functions
Quick and rough estimations
Minimal computational effort
Low accuracy
Explain which situations the trapezoid rule is best for, comment on its accuracy.
simple approximations
Linear/constant functions
more but still minimal computational effort then rectangular rule
More accurate then rectangular rule
Explain which situations the Simpsons rule is best for, comment on its accuracy.
approximation for smooth curves
Functions that can be approximated well by parabolas
High accuracy for smooth curves and quadratic like behaviour
More computational effort needed.
Explain which numerical method would you use to approximate the integral of a stress strain curve
Rectangular rule.
Curve is smooth but wont be approximated well by parabolas.
Non-linear regions which can cause the trapezoid to over or under estimate
create a linear approximation of a function f(x) around a specific point (x0) with 2 variables
find a linear approximation of F(x,y) = X^2 + y^2 at (1,1,2)
Derive the newton raphson method of root aproximation
What important things must you remmeber while using it?
The initial guess must be sufficiently close to the actual root to converge-> might diverge, converge to a different root or get stuck in a cycle
Find the relationship between vector F in the x’-y’ and x-y co
ordinate systems. What is the 2D Coordinate transformation matrix?
