Ch 1 Intro to
Calculus
Simplify Radical Expressions
Rationalizing the Denominator
Use Limits to Determine the
slope and the equation of the
tangent to a graph
The process of changing a denominator
from a radical (square root) to a rational
number (integer)
Slope of Tangent
To rewrite a radical expression with a oneterm radical in the denominator, multiply
the numerator and denominator by the
one-term denominator.
The slope of the tangent to a curve at a
point P is the limiting slope of the secant
PQ as the point Q slides along the curve
toward P.
When the denominator of a radical
fraction is a two-term expression, you can
rationalize the denominator by
multiplying by the conjugate.
In other words, the slope of the tangent is
said to be the limit of the slope of the
secant as Q approaches P along the curve.
Slope of a Tangent as a Limit
Avg and Instantaneous Rate of
Change Related to Slope of
Secants and Tangents
The slope of the tangent to the graph y = f(
x) at point P(a, f(a)) is as the equation
shows if the limit exists.
Instantaneous Velocity
Understand and Evaluate Limits
Using Appropriate Properties
The velocity of an object with position
function s(t), at time t = a, is as the
equation shows.
Limits and Their Existence
Average Rate of Change
We say that the number L is the limit of a
function y f1x2 as x approaches the value
a, written as the equation shown above.
Otherwise, the limit does not exists.
The difference quotient, shown above, is
called the average rate of change in y with
respect to x over the interval from x = a to
x = a + h.
Instantaneous Rates of Change
There are some cases that we need to
remember.
Therefore, we can conclude that the
instantaneous rate of change in y = f(x)
with respect to x when x = a is the
equation shown above, provided that the
limits exists.
Properties of the Limit
Examine continuous Functions
Using Limits
If f is a polynomial function, then as it
shown above.
Substituting x = a into the limit can yield
the indeterminate from 0 / 0. If this
happens, you may be able to find an
equivalent function that is the same as the
function f got all values except at x = a.
Then, substitution can be used to find the
limit.
Continuity at a Point
To evaluate a limit algebraically, we can
used:
The function f(x) is continuous at x = a if f(a)
is defined and if the equation shows above.
Otherwise, f(x) is discontinuous at x = a.
Types of discontinuity
direct substitution
factoring
Point - a hole
rationalizing
Jump - piecewise functions
one-sided limits
Infinite - vertical asymptote
change of variable
Defined Continuity
Things need to know for
continuity
All polynomial functions are continuous
for all real numbers.
A rational function as shown above is
continuous at x = a if g(a) no equal to 0.
A rational function in simplified form has a
discontinuity at the zeros of the
denominator.
When the one-sided limits are not equal to
each other, then the limit at this point
does not exist and the function is not
continuous at this point.
f(a) is defined