Lecture co4 math21-1

CO5
Analyze correctly and solve properly application
problems concerning the derivatives to include
writing equation of tangent/normal line, curve
tracing ( including all types of algebraic curves and
cusps), optimization problems, rate of change and
related-rates problems (time-rate problems).

OBJECTIVE:
At the end of the lesson, the students
should be able to illustrate properly and solve
application problems involving the derivatives
as an instantaneous rate of change or as slope
of the tangent line.

Applications : Equations of Tangent and Normal Lines
Applicationof the Concepts of the Derivative and Continuity on Curve Tracing( Include all types of the Algebraic
curves,cusps)
Optimization Problems: Applied Maxima/Minima Problems
Rate of Change Problems; Related-Rate Problems (Time-Rate Problems)
Long Quiz 4 Coverage

APPLICATIONS of the DERIVATIVES

Lesson 1 : SLOPE OF A CURVE, TANGENT Line, and
NORMAL LINE

DEFINITION:
The derivative of at point P on the curve is
equal to the slope of the tangent line at P, thus the derivative of
the function with respect to x at any x in its domain is
defined as:
)(xfy =
)(xfy =
0 0
( ) ( )
lim lim
x x
dy y f x x f x
dx x x∆ → ∆ →
∆ + ∆ −
= =
∆ ∆
provided the limit exists.

))(,( 11 xfxP ))(,( 22 xfxQ
)(xfy =
xxx
xxx
∆+=
−=∆
12
12
y∆
tangent line
secant line
x
y
The derivative of a function is equal to the slope of the
tangent line at any point along the curve.

Sample Problems
1. Find the slope of the curve y = x3
− 2x2
− x at −1, -2( ) .
2. Locate the points on the curve of y =
x−1
x+1
, x ≠ −1( ) at which the slope is 2?
3. What is the equation of the tangent line and of the normal line to y = −3x2
+ 2x+1 at -1, -4( ) ?
Locate the points at which the tangent is horizontal.
4. Find the point where the normal line to y = x+ x at 4,6( )
crosses the y-axis.
5. Find the tangent line to x2
+ 2y = 8 that is a) parallel , b) perpendicular to 2x-y = 4.

Lesson 2: Calculus-based Analysis of Properties of
Function; Graphs of Function

ANALYSIS OF FUNCTIONS :
INCREASING and DECREASING FUNCTIONS, ROLLE’S THEOREM,
MEAN VALUE THEOREM, CONCAVITY , POINT OF INFLECTION ,
RELATIVE/ ABSOLUTE EXTREMA

0 2 4
increasing decreasing increasing constant
The term increasing, decreasing, and constant are used to describe
the behavior of a function as one travels left to right along its graph.
An example is shown below.
INCREASING and DECREASING FUNCTIONS

y
x
•
•
Each tangent line
has positive slope;
function is increasing
y
x
•
•
Each tangent line
has negative slope;
function is decreasing
y
x
Each tangent line
has zero slope,
function is constant
• •

1. Find the intervals on which the function f(x) is
increasing/ decreasing. Given f(x) as follows:
a) f(x) = x2
− 4x + 3
b) f(x) = x4
−1
c) f(x) =
2x −1
x2
−1
Sample Problems

ROLLE’S THEOREM
AND
THE MEAN-VALUE THEOREM

ROLLE’S THEOREM
Let f be continuous on a closed interval [ a, b ] and
differentiable on the open interval ( a, b). If f(a) = 0 and f( b ) =
0 then there is at least one value c such that .f '(c) = 0

THE MEAN-VALUE THEOREM
Rolle’s Theorem is a special case of a more general result, called the
Mean-value Theorem.
Let f be a continuous function on the closed interval [ a, b ] and is
differentiable on the open interval ( a, b ). Then there is at least one
value c such that
Figure 4.8.5
f '(c) =
f (b)− f (a)
b− a

Sample Problems
1. Find the two x-intercepts of the function
and confirm that f’(c) = 0 at some point between those
intercepts.
2. Show that the function satisfies the hypotheses of
the mean-value theorem over the interval [0,2], and find all
values of c in the interval (0,2) at which the tangent line to the
graph of f is parallel to the secant line joining the points (0,f(0))
and (2,f(2))
f x( ) =
1
4
x3
+ 1
( ) 4x5xxf 2
+−=

CONCAVITY, POINT of INFLECTION,
RELATIVE/ABSOLUTE EXTREMA, GRAPHS

CONCAVITY
Although the sign of the derivative of f reveals where
the graph of f is increasing or decreasing , it does not
reveal the direction of the curvature.
There are two ways to characterize the concavity of a
differentiable f on an open interval:
• f is concave up on an open interval if its tangent lines
have increasing slopes on that interval and is concave
down if they have decreasing slopes.
• f is concave up on an open interval if its graph lies
above its tangent lines and concave down if it lies below
its tangent lines.

•
•
•
•
•
concave
up
y
x
••
•
• •concave
down
x
y
increasing slopes
decreasing slopes
Geometrically illustrated as follows:

INFLECTION POINTS
Points where the curve changes from concave up
to concave down or vice-versa are called points of
inflection.

The points x1, x2, x3, x4, and x5 are critical points. Of
these, x1, x2, and x5 are stationary points ( f’(x) = 0) and
X3 and x4 are points of non-differentiability ( f’(x) is
undefined or does not exist).

FIRST DERIVATIVE TEST FOR CRITICAL POINTS
A relative extremum may occur at critical points.
A function has a relative extremum at a critical points
where f’(x) changes sign from left to right of the said
point.
For a continuous function at a critical point x = c , a
relative maximum occurs at x = c whenever the
derivative changes from (+) to (–) going left to right of x =
c . A relative minimum occurs at x = c whenever the
derivative changes from (–) to (+).

SECOND DERIVATIVE TEST
• A function f has a relative maximum at stationary point if the
graph of f is concave down on an open interval containing that
point ( f”(x) <0).
• A function f has a relative minimum at stationary point if the
graph of f is concave up on an open interval containing that
point ( f”(x) >0).

Steps in Curve Tracing
1. If the equation is given in the form of f( x, y) = 0, solve for
y (or y2
).
2. Subject the equation to the test of symmetry.
3. Determine the x and y intercepts.
4. Locate the critical points and determine the maxima/
minima.
5. Determine the asymptotes if any. Also determine the
intersection of the curve with the horizontal asymptotes.
Note: The curve may intercept the horizontal asymptotes but
not the vertical asymptotes.

6. Divide the plane into regions by drawing light vertical lines
through the intersection on the x-axis.
Note: All vertical asymptotes must be considered as dividing
lines.
7. Find the sign of y on each region using the factored form of
the equation to determine whether the curve lies above
and/or below the x-axis.
8. Trace the curve. Plot a few points if necessary.

SAMPLE PROBLEMS
Work-out a Calculus-based analysis of properties then sketch
the graph of y = f(x):
1. f x( ) = x3
−3x +1
3. f x( ) = 3x
5
3 −15x
2
3
2. f x( ) = 3x5
− 5x3
4. x2
− xy −3y−1= 0
7. y = x2
−1( ) x − 2( )
2
8. x2
y − 4y = 8
9. xy − x +3 = 0
10. xy + x2
− 2 = 0
11. x2
y − x3
−1= 0
12. y2
= x3
+ 4x2
− 9x −36
13. y2
=
x2
− 4
x2
− 9
5. y = 3x − x3
6. y =
4x
4+ x2

Steps in solving maximum and minimum problems.
1.Draw the figure and label accordingly. indicate the given parts
and the unknown parts.
2.Make the necessary representation for the variables used.
3.Determine the quantity, the dependent variable, to be
maximized or minimized.
4.Find a formula relating the variables.
5.If possible, express the dependent variable in terms of a single
independent variable.
6.Differentiate the dependent variable in terms of the
independent variable.
7.Determine the critical values.
8.Answer the problem completely.

SAMPLE PROBLEMS:
1. Divide 120 into two parts such that the product of one part and
the square of the other is a maximum. Find the numbers.
2. A box is to be made from a piece of cardboard 16in.x10in. by
cutting equal squares out of the corners and turning up the sides.
Find the volume of the largest box that can be made this way.

RR
x
y
2y
2x
3. Find the area of the largest rectangle that can be inscribed in a
circle of a given radius, R = 10 inches.

4. Find the altitude of the largest circular cylinder that can be
inscribed in a circular cone of radius R=5” and height H=10”.
H=10”
10-y
y
x
R=5”

5. A rectangular field of fixed area is to be enclosed and
divided into three lots by parallels to one of the sides. What
would be the relative dimensions of the field to make the
amount of fencing a minimum?
L
W

6. Find the most economical proportions for a quart
can.
h
r
d

7. Find the proportion of the circular cylinder of largest
volume that can be inscribed in a given sphere.
r
a
2ah
h/2
C
a
r

1. What number exceeds its square by the maximum amount?
2. The sum of two numbers is “K”. find the minimum value of
the sum of their squares.
3. A rectangular field of given area is to be fenced off along the bank
of a river. If no fence is needed along the river, what are the
dimensions of the rectangle that will require the least amount of
fencing?
4. A Norman window consists of a rectangle surmounted by a
semicircle. What shape gives the most light for a given perimeter?
5. A cylindrical glass jar has a plastic top. If the plastic is half as
expensive as the glass per unit area, find the most economical
proportions for the glass.
6. Find the proportions of the circular cone of maximum volume
inscribed in a sphere.
7. A wall 8 feet high and 24.5 feet from a house. Find the shortest
ladder which will reach from the ground to the house when
leaning over the wall .
ADDITIONAL SAMPLE PROBLEMS:

If a particle is moving along a straight line according to the
equation of 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 particle at time “t” is the derivative
of “s” with respect to “t”.
s = f (t)
There are many problems in which we are concerned with the
rate of change of two or more related variables with respect to
time, in which it is not necessary to express each of these variables
directly as function of time. For example, we are given an equation
involving the variables x and y, and that both x and y are functions
of the third variable t, where t denotes time.

Since the rate of change of x and y with respect to t is
given by and , respectively, we differentiate both
sides of the given equation with respect to t by applying
the chain rule.
When two or more variables, all functions of t, are related
by an equation, the relation between their rates of
change may be obtained by differentiating the equation
with respect to t.
dx
dt
dy
dt

Example 1
A 17 ft ladder is leaning against a wall. If the bottom of the ladder
is pulled along the ground away from the wall at the constant rate of
5 ft/sec, how fast will the top of the ladder be moving down the wall
when it is 8 ft above the ground?
sec
ft
5
dt
dx
=
x
17 ft.
?
dt
dy
ft8y
=
=
y

Example 2
A balloon leaving the ground 60 feet from an observer, rises
vertically at the rate 10 ft/sec . How fast is the balloon receding from
the observer after 8 seconds?
Viewer
60 feet
h
L
sec
ft
10
dt
dh
=
?
dt
dL
sec8t
=
=

R
sec
ft
4
dt
dR
−=
x
20ft
Example3
A man on a wharf of 20 feet above the water pulls in a rope, to
which a boat is attached, at the rate of 4 ft/sec. At what rate is the
boat approaching the wharf when there is 25 feet of rope out?

Example 4
Water is flowing into a conical reservoir 20 feet deep and 10 feet
across the top, at the rate of 15 ft3
/min . Find how fast the surface is
rising when the water is 8 feet deep?
5 feet20feet
h
r
10 feet
min
ft
15
dt
dV 3
=

Example 5
Water is flowing into a vertical tank at the rate of 24 ft3
/min .
If the radius of the tank is 4 feet, how fast is the surface rising?
h
4 feet
min
ft
24
dt
dV 3
=

Example 6
A triangular trough is 10 feet long, 6 feet across the top, and 3
feet deep. If water flows in at the rate of 12 ft3
/min, find how fast
the surface is rising when the water is 6 inches deep?
min
ft
12
3
h
6 feet
10 feet
3feet
x

Example 7
A train, starting at noon, travels at 40 mph going north.
Another train, starting from the same point at 2:00 pm travels
east at 50 mph . Find how fast the two trains are separating at
3:00 pm.
80miles
x
2pm
B
C
D
A
L
y
3pm
3pm
hr
mi
40
dt
dy
=
hr
mi
50
dt
dx
=
12pm
2pm

Lecture co4 math21-1

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