Mr. C.S.Satheesh, M.E.,
Routh Array Hurwitz Criterion
determining whether all the roots of a polynomial have negative real part or not.
characteristic equation.
the coefficients of the polynomial be positive.
coefficients are zero or negative
purely imaginary roots so the system is limitedly or marginally stable.
remaining roots lies on left half of S plane.
Mr. C.S.Satheesh, M.E.,
Routh Array Hurwitz Criterion
determining whether all the roots of a polynomial have negative real part or not.
characteristic equation.
the coefficients of the polynomial be positive.
coefficients are zero or negative
purely imaginary roots so the system is limitedly or marginally stable.
remaining roots lies on left half of S plane.
En esta unidad 1 se evidenciará la solución de la actividad del paso dos para profundizar y contextualizar el conocimiento con la finalidad de desarrollar las habilidades de pensamiento matemático funcional, haciendo uso del lenguaje algebraico. Aportando la comprensión de conceptos y procesos matemáticos; por medio de ejercicios matemáticos y diapositivas sobre cada una de las temáticas propuestas en cada ejercicio.
Introduction to Calculus
The course sounds intimidating starting from its name to its content. The first step is the most difficult and challenging and it must be taken care so that we can get all the concepts right. Our plan is to present calculus in a graphical stand point to help students visualize the entire thing. We will also lay the ground work from logic and let us say basic commonsense so that developing the formula wont be so difficult. Lastly present the result with clear and understandable example. The sequence of topics will be as follows:
The graphical version of the limit
the tangent line
differentiation of some algebraic function
the fundamental theorem of calculus.
Differentiation of trigonometric functions
Integration of trigonometric functions
Differentiation of logarithmic function
Integration of logarithmic functions
Differentiation of the natural logarithmic function
Integration of the natural logarithmic function
Taylors polynomial
The chain rule.
Differentiation of multi variable function using the chain rule.
Integration using the chain rule.
the graphical version of limits
imagine a line containing infinitely many points. Since it is a line, we do not question ourselves whether it is totally continuous all through out. What if there are instances where some points of this line does not exist? Making the line reach its limit/end. It is as if that point vanished and gone for eternity.
Let us examine.
Consider the function,
f(x)=(x-1)/(x-1)
The table below shows the value of f(x).
x -5 -4 -3 -2 -1 0 1 2 3
f(x)=(x-1)/(x-1)
1 1 1 1 1 1 0/0 1 1
Results
We get an undefined value 0/0 when x is equal to 1 and get a value of f(x)=1 for other values of x. this means that at x = 1 the function is not defined or has no meaning. Also at that point the function is not continuous.
Interpretation
Limit of a function could mean the last value outputted by f(x) which is defined before it reaches a critical value x→a that will yield the function being discontinuous.
lim┬(x→a)〖f(x)=L〗
We can also see that our critical value can be positive or negative hence we can also write the formula for the limit of the function as;
lim┬(x→a^+ )〖f(x)=L〗
lim┬(x→a^- )〖f(x)=L〗
Worth to Ask
if such a number L, exist then, does that also mean that the function will be discontinuous from that value right away?
En esta unidad 1 se evidenciará la solución de la actividad del paso dos para profundizar y contextualizar el conocimiento con la finalidad de desarrollar las habilidades de pensamiento matemático funcional, haciendo uso del lenguaje algebraico. Aportando la comprensión de conceptos y procesos matemáticos; por medio de ejercicios matemáticos y diapositivas sobre cada una de las temáticas propuestas en cada ejercicio.
Introduction to Calculus
The course sounds intimidating starting from its name to its content. The first step is the most difficult and challenging and it must be taken care so that we can get all the concepts right. Our plan is to present calculus in a graphical stand point to help students visualize the entire thing. We will also lay the ground work from logic and let us say basic commonsense so that developing the formula wont be so difficult. Lastly present the result with clear and understandable example. The sequence of topics will be as follows:
The graphical version of the limit
the tangent line
differentiation of some algebraic function
the fundamental theorem of calculus.
Differentiation of trigonometric functions
Integration of trigonometric functions
Differentiation of logarithmic function
Integration of logarithmic functions
Differentiation of the natural logarithmic function
Integration of the natural logarithmic function
Taylors polynomial
The chain rule.
Differentiation of multi variable function using the chain rule.
Integration using the chain rule.
the graphical version of limits
imagine a line containing infinitely many points. Since it is a line, we do not question ourselves whether it is totally continuous all through out. What if there are instances where some points of this line does not exist? Making the line reach its limit/end. It is as if that point vanished and gone for eternity.
Let us examine.
Consider the function,
f(x)=(x-1)/(x-1)
The table below shows the value of f(x).
x -5 -4 -3 -2 -1 0 1 2 3
f(x)=(x-1)/(x-1)
1 1 1 1 1 1 0/0 1 1
Results
We get an undefined value 0/0 when x is equal to 1 and get a value of f(x)=1 for other values of x. this means that at x = 1 the function is not defined or has no meaning. Also at that point the function is not continuous.
Interpretation
Limit of a function could mean the last value outputted by f(x) which is defined before it reaches a critical value x→a that will yield the function being discontinuous.
lim┬(x→a)〖f(x)=L〗
We can also see that our critical value can be positive or negative hence we can also write the formula for the limit of the function as;
lim┬(x→a^+ )〖f(x)=L〗
lim┬(x→a^- )〖f(x)=L〗
Worth to Ask
if such a number L, exist then, does that also mean that the function will be discontinuous from that value right away?
8. • Vertical line(s) that graphs approach, but never touch • After you factor a rat’l function and reduce, VA’s come from factors in the denominator that DO NOT reduce/cancel. •Set those factors = 0 and solve for x. x=that value(s) will be your VA
10. • Point(s) in graphs that are undefined. • After you factor a rat’l function and reduce, holes come from factors in the denominator that DO reduce/cancel. •Set those factors = 0 and solve for x. There will be a hole where x = that number.
12. • HA’s are line(s) that graphs approach, but never touch • These lines show the end behavior of graphs on the left and right sides. •In a rational function, after reducing, if a denominator exists, there is a HA in 2 cases: 1. The denominator’s greatest exponent is > the numerator’s greatest exponent. In this case, the HA is always y=0 2. The denominator’s greatest exponent = the numerator’s greatest exponent. In this case, the HA is the ratio of the coefficients of your greatest terms.
14. • Slant line(s) that graphs approach, but never touch • These lines show the end behavior of graphs on the left and right sides. •In a rational function, after reducing, if a denominator exists, there is a SA if: -The numerator’s greatest exponent is exactly one more than the denominator’s greatest exponent. • To find the SA, do long division and ignore the remainder. Remember, it is a line so your answer should be in the form y=mx+b.