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...

39
4.6 Related rates

TAGS:

Transcript of 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...

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)

4.6 Related rates

Page 2: 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)

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

Page 3: 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)

triples

3,4,5 5,12,13 6,8,10 7,24,25 8,15,17 9,12,15

Page 4: 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)

Implicit differentiation

Change wrt time

Each changing quantity is differentiated wrt time.

Page 5: 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)

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?

Page 6: 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)

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?

Page 7: 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)

Example

Circle

Area decreasing 4.8 pi ft^2/sec Radius decreasing 0.3 ft/sec

Find radius

Page 8: 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)

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?

Page 9: 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)

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?

Page 10: 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)

Same rectangle

Find rate of change of perimeter

Find rate of change of diagonal

Page 11: 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)

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

Page 12: 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)

Example

V= l w h

dV/dt=

Page 13: 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)

Example

A 25 ft ladder is leaning against a house. The bottom is being pulled out from the house at 2 ft/sec.

Page 14: 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)

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?

Page 15: 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)

Part b

Find the rate at which the area of the triangle formed is changing when the bottom is 7 ft. from the house.

Page 16: 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)

Part c

Find the rate at which the angle between the top of the ladder and the house changes.

Page 17: 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)

Spherical soap bubble

r= 10 cm air added at 10 cm^2/sec.

Find rate at which radius is changing.

Page 18: 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)

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

Page 19: 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)

Cylindrical tank with circular base Drained at 3 l/sec Radius=5

How fast is the water level dropping?

Page 20: 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)

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.

Page 21: 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)

Cone, r=7,h=12

Draining at 15 m^3/sec When r=3

How fast is the radius changing?

Page 22: 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)

Cone, r=10, h=7

Filled at 2 m^3/sec H=5m

How fast is the radius changing?

Page 23: 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)

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

Page 24: 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)

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?

Page 25: 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)

P 329

19

20

Page 26: 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)

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?

Page 27: 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)

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?

Page 28: 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)

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.

Page 29: 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)

P 326

30

32

31

Page 30: 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)

4.7 Mean Value Theorem

Sure you remember!!!

f’ ( c ) = f(b)-f(a)

b-a

Page 31: 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)

4.7 Mean Value Theorem

Sure you remember!!! And

Corollary 1 is the first derivative test for increasing and decreasing.

Page 32: 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)

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).

Page 33: 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)

Corollary 2

This is the converse of :

the derivative of a constant is zero.

Page 34: 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)

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).

Page 35: 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)

Definitions

Antiderivative

General antiderivative

Arbitrary constant

Page 36: 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)

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.

Page 37: 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)

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.

Page 38: 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)

Arbitrary constant

The constant C is called the

arbitrary constant.

Page 39: 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)

4.7 Initial value problems

Uses general antiderivatives

With “initial values”

To find the specific function of the family


P L A N YEA R( S ) 2 0 2 1 - 2 0 2 2 L O CA TIO N Pi k ev ...

P L A N YEA R( S ) 2 0 2 1 - 2 0 2 2 L O CA TIO N Pi k ev ...

> plot(cos(x) + sin(x), x=0..Pi); plot(tan(x), x=-Pi..Pi ... · > plot3d({sin(x*y), x + 2*y},x=-Pi..Pi,y=-Pi..Pi); ↵ c1:= [cos(x)-2*cos(0.4*y),sin(x)-2*sin(0.4*y),y]: ↵ c2:= [cos(x)+2*cos(0.4*y),sin(x)+2*sin(0.4*y),y]:

> plot(cos(x) + sin(x), x=0..Pi); plot(tan(x), x=-Pi..Pi ... · > plot3d({sin(x*y), x + 2*y},x=-Pi..Pi,y=-Pi..Pi); ↵ c1:= [cos(x)-2*cos(0.4*y),sin(x)-2*sin(0.4*y),y]: ↵ c2:= [cos(x)+2*cos(0.4*y),sin(x)+2*sin(0.4*y),y]:

2. Pi Industrial

2. Pi Industrial

A Set-membership Approach for Visible Light Positioning ...ceur-ws.org/Vol-2498/short36.pdf · tics Pi r by Pi r = 1 K r XK r j=1 RSS (i) j (2) where i= 1;2; ;K b. Then the distance

A Set-membership Approach for Visible Light Positioning ...ceur-ws.org/Vol-2498/short36.pdf · tics Pi r by Pi r = 1 K r XK r j=1 RSS (i) j (2) where i= 1;2; ;K b. Then the distance

Seismic Response and Structural Health Monitoring of the ... fileR 1 4 S W PI E R 1 5 S W HI N G E 1 5 H I N G E 15S W PI E R 1 8 S W PI E R 1 7 S E R 1 9 W PI 2 0 S W 2 1 S W HI N

Seismic Response and Structural Health Monitoring of the ... fileR 1 4 S W PI E R 1 5 S W HI N G E 1 5 H I N G E 15S W PI E R 1 8 S W PI E R 1 7 S E R 1 9 W PI 2 0 S W 2 1 S W HI N

PID Compensator Design Using Root Locus Methodsfaculty.citadel.edu/potisuk/elec312/ppt/piddsgn.pdf · 7 Control Action: Proportional+Integral (PI) 2 2 2 2 1 2 3 4 1 sRC sRC R R R

PID Compensator Design Using Root Locus Methodsfaculty.citadel.edu/potisuk/elec312/ppt/piddsgn.pdf · 7 Control Action: Proportional+Integral (PI) 2 2 2 2 1 2 3 4 1 sRC sRC R R R

PI 232 F - PI 232 FI - PI 232 BC - PI 232 R · TITAN PI 232 F - PI 232 FI - PI 232 BC - PI 232 R L’azienda si riserva di modificare le caratteristiche dei prodotti in qualsiasi

PI 232 F - PI 232 FI - PI 232 BC - PI 232 R · TITAN PI 232 F - PI 232 FI - PI 232 BC - PI 232 R L’azienda si riserva di modificare le caratteristiche dei prodotti in qualsiasi

Embedded programming on r pi and arduino

Embedded programming on r pi and arduino

2 Step PI PI Report Step 2 Summary - mlit.go.jp

2 Step PI PI Report Step 2 Summary - mlit.go.jp

August 1, 2019 - USDA ARS...PI 588306 Aminia United States PI 588320 Henryetta United States PI 588348 Ill 878‐2 United States PI 588318 Xenia United States PI 588317 Gaillard 2

August 1, 2019 - USDA ARS...PI 588306 Aminia United States PI 588320 Henryetta United States PI 588348 Ill 878‐2 United States PI 588318 Xenia United States PI 588317 Gaillard 2

SC-022-D/PI/R-2014 Resolución final

SC-022-D/PI/R-2014 Resolución final

Perinatal Nutrition and Environmental Exposures: Effects ... · Karen E. Peterson (contact PI, Project 2 Co-PI), Vasantha Padmanabhan (co-PI, Project 2 Co-PI), Dana Dolinoy (Project

Perinatal Nutrition and Environmental Exposures: Effects ... · Karen E. Peterson (contact PI, Project 2 Co-PI), Vasantha Padmanabhan (co-PI, Project 2 Co-PI), Dana Dolinoy (Project

Compulsory Critique NEO PI R

Compulsory Critique NEO PI R

Package ‘insol’ - The Comprehensive R Archive Network · PDF filePackage ‘insol ’ February 20, 2015 ... degrees(seq(0,2*pi,pi/2)) doshade 11 doshade Cast shadows Description

Package ‘insol’ - The Comprehensive R Archive Network · PDF filePackage ‘insol ’ February 20, 2015 ... degrees(seq(0,2*pi,pi/2)) doshade 11 doshade Cast shadows Description

PI;m PI;m  · 2019-06-26 · PI;m PI;m . Created Date: 7/15/2015 2:51:32 PM

PI;m PI;m  · 2019-06-26 · PI;m PI;m . Created Date: 7/15/2015 2:51:32 PM

Pi PPii Pi 2222 EmbeddedEmbeddedEmbedded · Pi PPii Pi 2222 EmbeddedEmbeddedEmbedded 502V2S ... Raspberry Pi A+, B+, Pi 2 or Pi 3, even Pi-Zero! DIGITAL PASS THROUGH – 24-bit, 48Khz

Pi PPii Pi 2222 EmbeddedEmbeddedEmbedded · Pi PPii Pi 2222 EmbeddedEmbeddedEmbedded 502V2S ... Raspberry Pi A+, B+, Pi 2 or Pi 3, even Pi-Zero! DIGITAL PASS THROUGH – 24-bit, 48Khz

© Copyright 2014 OSIsoft, LLC.cdn.osisoft.com/corp/en/media/presentations/2014/RegionalSeminars/R… · The PI Server Package 4 PI Analytics Asset Based, PI ACE, PE, Tot. PI Asset

© Copyright 2014 OSIsoft, LLC.cdn.osisoft.com/corp/en/media/presentations/2014/RegionalSeminars/R… · The PI Server Package 4 PI Analytics Asset Based, PI ACE, PE, Tot. PI Asset

Effectiveness of NEO–PI–R Research Validity Scales for ... Bagby Effectiveness of NEO-PI-R validity... · Thus, the available evidence suggests that the NEO–PI–R and NEO–FFI

Effectiveness of NEO–PI–R Research Validity Scales for ... Bagby Effectiveness of NEO-PI-R validity... · Thus, the available evidence suggests that the NEO–PI–R and NEO–FFI

pi P c r * U r - ERIC

pi P c r * U r - ERIC

PI-Phone Using Raspberry Pi-2

PI-Phone Using Raspberry Pi-2