APPLICATION: (I)TANGENT LINE (II)RELATED RATES (III)MINIMUM AND MAXIMUM VALUES CHAPTER 4.
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Transcript of APPLICATION: (I)TANGENT LINE (II)RELATED RATES (III)MINIMUM AND MAXIMUM VALUES CHAPTER 4.
APPLICATION:(I) TANGENT LINE
(II) RELATED RATES(III) MINIMUM AND MAXIMUM VALUES
CHAPTER 4
Tangent line
Consider a function , with point lying on the graph:
Tangent line to the function at is the straight line that touches
at that point. Normal line is the
line that is perpendicular to the tangent line.
xfy
11,yx
11,yx
xfy
Tangent Line
Normal Line
0dx
dy
y
x42 xy
mdx
dy
• Tangent Line Equation:
or
• Normal Line Equation:
or
11)( yxxmy
cmxy
11)(1 yxx
my
cxm
y 1
Example 1:1. Find the slope of the curve at the given
points
2. Find the lines that are tangent and normal to the curve at the given point.
)1,2;(2)(
3;12422
2
xyxybxxxya
)3,2;(1)(
2;322
2
yxyxbxxya
• A process of finding a rate at which a quantity changes by relating that quantity to the other quantities.
• The rate is usually with respect to time, t.
RELATED RATES
Example 2Suppose that the radius, r and area, of a circle are differentiable functions of t. Write an equation that relates to .
Answer:
2rA
dtdA
dtdr
dtdrr
dtdA
rdtdA
dtd
rA
2
: wrt t(1) ateDifferenti2
2 1
Example 3How fast is the area of a rectangle changing from one side 10cm long and the side increase at a rate of 2cm/s and the other side is 8cm long and decrease at a rate of 3cm/s?
Solution:
Differentiate (1) wrt t:
x
y
1 :rectangle of Area
/3,8At ./2,10At
xyA
scmdt
dxyscm
dt
dxx
scm
dt
dxy
dt
dyx
dt
dA
xydt
dA
dt
d
/4
283102
Example 4A stone is dropped into a pond, the ripples forming concentric circles which expand. At what rate is the area of one of these circles increasing when the radius is 3m and increasing at the rate of 0.6ms-1?
Solution:
Differentiate wrt t :
r
12 :circle of Area
/6.0,3At
rA
scmdt
drr
sm
dt
drr
dt
Ad
rdt
dA
dt
d
/6.3
6.032
2
2
2
Exercise 1A 13ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5ft/s.
(a) How fast is the top of the ladder sliding down the wall?
(b)At what rate is the area of the triangle formed by the ladder, wall and ground changing
(c) At what rate is the angle between the ladder and the ground changing?
Exercise 2The length l of a rectangle is decreasing at the rate of 2cm/s, while the width w is increasing at the rate 2cm/s. When l=12cm and w=5cm find the rates of change
(a) The area(b)The perimeter
Exercise 3:
When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01cm/min. At what rate is the plate’s area increasing when the radius is 50cm?
• Use 1st derivative to locate and identify extreme values(stationary values) of a continuous function from its derivative
Definition: Absolute Maximum and Absolute Minimum• Let f be a function with domain D. Then f has an
ABSOLUTE MAXIMUM value on D at a point c if:
ABSOLUTE MINIMUM
MAXIMUM & MINIMUM
Dxcfxf ),(
Dxcfxf ),(
• A point on the graph of a function y = f(x) where the rate of change is zero.
Example 6Find stationary points:
0dxdy
STATIONARY POINT
33)2(
3413
2
xxy
xxy
Let f be a function defined on an interval I and let x1 and x2 be any two points in I
1) If f (x1)< f (x2) whenever x1 < x2, then f is said to be increasing on I
2) If f (x1)> f (x2) whenever x1 < x2, then f is said to be decreasing on I
INCREASING & DECREASING
Suppose that f is continuous on [a,b] and differentiable on (a,b).
1) If f’(x)>0 at each point , then f is said to be increasing on [a,b]
2) If f’(x)<0 at each point , then f is said to be decreasing on [a,b]
],[ bax
],[ bax
1st DERIVATIVE TEST
The graph of a differentiable function y=f(x)
1) Concave up on an open interval if f’ is increasing on I
2) Concave down on an open interval if f’ is decreasing on I
CONCAVITY
Let y=f(x) be twice-differentiable on an interval I
1) If f”(x)>0 on I, the graph of f over I is concave up2) If f”(x)<0 on I, the graph of f over I is concave down
2ND DERIVATIVE TEST:TEST FOR CONCAVITY
• If y is minimum
Therefore (x,y) is a minimum point.
• If y is maximum
Therefore (x,y) is a maximum point.
02
2
dxyd
02
2
dxyd
MAXIMUM POINT & MINIMUM POINT
A point where the graph of a function has a tangent line and where the concavity changes is a POINT OF INFLEXION.
02
2
dx
yd
CONCAVITY
Example 5:Find y’ and y” and then sketch the graph of y=f(x)
5823 xxxy
Solution:Step 1: Find the stationary point
Therefore, the stationary points are:
2,3
4
0243
0823
823
0 : valueStationary
2
2
x
xx
xx
xxdx
dydx
dy
17,2&27
41,
3
4
Step 2 : Find inflexion point
Therefore, the inflexion points is:3
1
026
26
0 :valueInflexion
2
2
2
2
x
x
xdx
yddx
yd
27
209,
3
1
Step 3: 1st and 2nd Derivative Test
IntervalTest
Value , x -3 -1 0 2
+ - - +
Increasing Decreasing Decreasing Increasing
- - + +
Concave Concave down
Concave down
Concave up Concave up
2,
dxdy
2
2
dxyd
3
1,2
3
4,
3
1
,
3
4
Step 4: Test for maximum and minimum
At
Therefore, (-2, 17) is a maximum point.
At
Therefore, (4/3, -41/27) is a minimum point.
010
;2
2
2
dx
yd
x
010
;3
4
2
2
dx
yd
x
Example 6:Find y’ and y” and then sketch the graph of y=f(x)
104 34 xxy