4.6 related rates. useful formulae a^2 +b^2 = c^2 cube v= s^3 sphere v= 4/3 pi r^3 sa=4 pi r^2 cone...
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![Page 1: 4.6 Related rates. Useful formulae a^2 +b^2 = c^2 Cube V= s^3 Sphere V= 4/3 pi r^3 SA=4 pi r^2 Cone V= 1/3 pi r^2 h Lateral SA= pi r (r^2 + h^2)^(1/2)](https://reader030.vdocument.in/reader030/viewer/2022032803/56649e265503460f94b1551c/html5/thumbnails/1.jpg)
4.6 Related rates
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Useful formulae
a^2 +b^2 = c^2 Cube V= s^3 Sphere V= 4/3 pi r^3 SA=4 pi r^2 Cone V= 1/3 pi r^2 h Lateral SA= pi r (r^2 + h^2)^(1/2) Right circular cylinder V=pi r^2 h Lateral SA= 2 pi r h Circle A= pi r^2 C= 2 pi r
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triples
3,4,5 5,12,13 6,8,10 7,24,25 8,15,17 9,12,15
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Implicit differentiation
Change wrt time
Each changing quantity is differentiated wrt time.
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Example
the radius of a circle is increasing at 0.03 cm/sec. What is the rate of change of the area at the second the radius is 20 cm?
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Example
A circle has area increasing at 1.5 pi cm^2/min. what is the rate of change of the radius when the radius is 5 cm?
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Example
Circle
Area decreasing 4.8 pi ft^2/sec Radius decreasing 0.3 ft/sec
Find radius
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Example
What is the radius of a circle at the moment when the rate of change of its area is numerically twice as large as the rate of change of its radius?
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Example
The length of a rectangle is decreasing at 5 cm/sec. And the width is increasing at 2 cm/sec. What is the rate of change of the area when l=6 and w=5?
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Same rectangle
Find rate of change of perimeter
Find rate of change of diagonal
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Example
The edges of a cube are expanding at 3 cm/sec. How fast is the volume changing when:
e= 1 cm
e=10 cm
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Example
V= l w h
dV/dt=
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Example
A 25 ft ladder is leaning against a house. The bottom is being pulled out from the house at 2 ft/sec.
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Part a
How fast is the top of the ladder moving down the wall when the base is 7 ft. from the end of the ladder?
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Part b
Find the rate at which the area of the triangle formed is changing when the bottom is 7 ft. from the house.
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Part c
Find the rate at which the angle between the top of the ladder and the house changes.
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Spherical soap bubble
r= 10 cm air added at 10 cm^2/sec.
Find rate at which radius is changing.
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Rectangular prism
Length increasing 4 cm/sec Height decreasing 3 cm/sec Width constant When l=4.w=5,h=6
Find rate of change of SA
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Cylindrical tank with circular base Drained at 3 l/sec Radius=5
How fast is the water level dropping?
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Cone-shaped cup
Being filled with water at 3 cm^3/sec
H=10, r=5
How fast is water level rising when level is 4 cm.
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Cone, r=7,h=12
Draining at 15 m^3/sec When r=3
How fast is the radius changing?
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Cone, r=10, h=7
Filled at 2 m^3/sec H=5m
How fast is the radius changing?
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Water drains from cone at the rate of 21 ft^3/min. how fast is the water level dropping when the height is 5 ft?
Cone, r=3, h=8
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A hot-air balloon rises straight up from a level field.
It is tracked by a range-finder 500 ft from lift-off. When the range-finder’s angle of elevation is pi/4, the angle increases at 0.14 rad/min. How fast is the balloon rising?
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P 329
19
20
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A 5 ft girl is walking toward a
20 ft lamppost at the rate of 6 ft/sec.
How fast is the tip of her shadow moving?
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A 6 ft man is moving away from the base of a streetlight that is 15 ft high.
If he moves at the rate of 18 ft/sec., how fast is the length of his shadow changing?
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A balloon rises at 3 m/sec. from a point on the ground 30 m from an observer.
Find rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 m above ground.
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P 326
30
32
31
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4.7 Mean Value Theorem
Sure you remember!!!
f’ ( c ) = f(b)-f(a)
b-a
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4.7 Mean Value Theorem
Sure you remember!!! And
Corollary 1 is the first derivative test for increasing and decreasing.
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Corollary 2
If f’(x)=0 for all x in (a,b) then there is a constant ,c, such that
f (x) = c,
for all x in (a,b).
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Corollary 2
This is the converse of :
the derivative of a constant is zero.
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Corollary 3
If F’(x)=G’(x) at each x in (a,b), then there is a constant,c, such that
F(x)=G(x)+c for all x in (a,b).
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Definitions
Antiderivative
General antiderivative
Arbitrary constant
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antiderivative
A function F is an anti-derivative of a function f over an interval I if
F’(x)=f(x)
At every point of the interval.
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General antiderivative
If F is an antiderivative of f, then the family of functions F(x)+C (C any real no.) is the general antiderivative of f over the interval I.
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Arbitrary constant
The constant C is called the
arbitrary constant.
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4.7 Initial value problems
Uses general antiderivatives
With “initial values”
To find the specific function of the family