PARABOLA
Math 14
Plane and Solid Analytic
Geometry
OBJECTIVES:
At the end of the lesson, the student is
expected to be able to:
define conic section
identify the different conic section
describe parabola
convert general form to standard form of
equatio
DIFFERENTIATION OF
HYPERBOLIC
FUNCTIONS
OBJECTIV
ES: and graph hyperbolic functions;
define
prove some exercises on identities and
differentiation formulas;
explain hyperbolic functions and their
derivatives; and
evaluate problems on derivatives of
hyper
THE
DIFFERENTIALS
(Applications)
EXAMPLE 1: Use differentials to approximate the
changeintheareaofasquareifthelengthofitsside
increasesfrom6cmto6.23cm.
Letx=lengthofthesideofthesquare.
Theareamaybeexpressedasafunctionofx,
whereA=x2.
dA
ThedifferentialdAis
THE
DIFFERENTIALS
Consider a function defined by y=f(x) where x
is the independent variable. In the four-step rule
we introduced the symbol x to the denote the
increment of x. Now we introduce the symbol dx
which we call the differential of x. Similarly,
SLOPE OF A CURVE,
TANGENT, and NORMAL
LINE
OBJECTIV
ES:
determine the slope of a curve and the
derivative of a function at a specified point;
solve problems involving slope of a curve;
determine the equations of tangent and
normal lines using different
INDETERMINATE
FORMS
OBJECTIVE
S:
define, determine, enumerate the
different indeterminate forms of
functions;
apply the theorems on
differentiation in evaluating limits of
indeterminate forms of functions
using LHopitals Rule.
x2 4x 3
Recall: Evaluateth
MAXIMA and MINIMA
PROBLEMS
OBJECTIV
identify the quantity to be maximized or
ES:
minimized,
apply the knowledge of derivatives and
critical points in solving maximum and
minimum problems and
solve maximum and minimum problems
with ease and accuracy.
A
RELATED RATES
PROBLEMS
If a particle is moving along a straight line
s of t )
according to the equation f (motion
,
since the velocity may be interpreted as a rate of
change of distance with respect to time, thus we
have shown that the velocity of the par
THE DERIVATIVE IN
GRAPHING AND
APPLICATIONS
ANALYSIS OF FUNCTIONS 1:
INCREASING and DECREASING
FUNCTIONS, ROLLES
THEOREM, MEAN VALUE
THEOREM, CONCAVITY and
POINT OF INFLECTION
OBJECTIVES:
define increasing and decreasing functions;
define concavity and d
DIFFERENTIATION OF
INVERSE
HYPERBOLIC
FUNCTIONS
OBJECTIVE
S:
define and graph inverse hyperbolic
functions;
prove some exercises on the logarithmic
equivalents of inverse hyperbolic functions
differentiate inverse hyperbolic functions.
TRANSCENDENTAL
F
DIFFERENTIATION OF
INVERSE
TRIGONOMETRIC
FUNCTIONS
OBJECTIV
derive the formula for the derivatives of the
ES:
inverse trigonometric functions;
apply the derivative formulas to solve for the
derivatives of inverse trigonometric
functions; and
solve prob
TOPIC
DIFFERENTIATION
NAME
COURSE /SEC
PROG / YEAR
ROOM
SCHEDULE
FACULTY
INSTRUCTIONS Answer the following items on the space provided. Differentiate the following
problems showing step by step solution each space
SOLUTION
SOLUTION
FUNCTIONS
OPERATIONS ON FUNCTIONS
OBJECTIVES:
perform operations on functions;
determine the domain of the given functions;
determine the domain of the resulting
functions
after performing the operations on functions;
and
define a composite function a
FUNCTIONS
GRAPHS OF FUNCTIONS;
PIECEWISE DEFINED FUNCTIONS;
ABSOLUTE VALUE FUNCTION;
GREATEST INTEGER FUNCTION
OBJECTIV
sketchthegraphofafunction;
ES:
determinethedomainandrangeofa
functionfromitsgraph;and
identifywhetherarelationisafunction
or
notfromits
LIMITS
OF
FUNCTIONS
CONTINUITY
DEFINITION: CONTINUITY OF A
FUNCTION
Definition 1.5.1 (p. 110)
If one or more of the above conditions fails to
hold
at C the function is said to be discontinuous.
Theorem 1.5.3 (p. 113)
EXAMPLE
x2 - x - 6
f(
1. Given the fun
FUNCTIONS
OBJECTIVES:
define functions;
distinguish between dependent and
independent variables;
represent functions in different ways;
and
evaluate functions
define domain and range of a function;
and
determine the domain and range of a
function
DEFINIT
LIMITS
OF
FUNCTIONS
INFINITE LIMITS; VERTICAL
AND HORIZONTAL
ASYMPTOTES; SQUEEZE
THEOREM
OBJECTIVES:
define infinite limits;
illustrate the infinite limits ; and
use the theorems to evaluate the limits of
functions.
determine vertical and horizontal asymp
THE DERIVATIVE AND
DIFFERENTIATION OF ALGEBRAIC
FUNCTIONS
OBJECTIVES:
to define the derivative of a function
to find the derivative of a function by
increment method (4-step rule)
to identify the different rules of differentiation
and distinguish one from
HIGHER ORDER
DERIVATIVES
AND
IMPLICIT
DIFFERENTIATION
OBJECTIV
ESdefine higher derivatives;
to :
to apply the knowledge of higher derivatives
and implicit differentiation in proving
relations;
to find the higher derivative of algebraic
functions; and
t
LIMITS
OF
FUNCTIONS
LIMITS OF FUNCTIONS
OBJECTIVES:
define limits;
illustrate limits and its theorems; and
evaluate limits applying the given
theorems.
define one-sided limits
illustrate one-sided limits
investigate the limit if it exist or not
using t
DIFFERENTIATION OF
LOGARITHMIC
FUNCTIONS
OBJECTIVE
S:
differentiate and simplify logarithmic
functions using the properties of logarithm,
and
apply logarithmic differentiation for
complicated functions and functions with
variable base and exponent.
TRANS
DIFFERENTIATION OF
EXPONENTIAL
FUNCTIONS
OBJECTIV
ES:
apply the properties
of exponential
functions to simplify differentiation;
differentiate
functions
involving
exponential functions; and
solve problems involving differentiation of
exponential functio
Definitions and algorithms for solving linear systems
A system of equations is in echelon form provided
1) Each equations leading coefficient is 1.
2 ) Each equations leading term is strictly to the right of the leading term in
the equation above.
X + 2y