Aim: Related Rates Course: Calculus Do Now: Aim: How do we find related rates when we have more than...

Aim: Related Rates Course: Calculus

Do Now:

Aim: How do we find related rates when we have more than two variables?

Find the points on the curve x2 + y2 = 2x +2y where the tangent is parallel to the x-axis.

(1, 1 + 21/2), (1, 1 – 21/2)


Related Rates involving TIME

Most quantities encountered in science or everyday life vary with time.

If two such quantities are related to each other by an equation, and if we know the rate at which one of them changes, then, by differentiating the equation with respect to time, we can find the rate at which the other quantity changes.

Implicit differentiation and the Chain Rule

dy dy dt

dx dt dx



Suppose a particle starts from the origin and moves along the curve y = x2.

8

6

4

2

g x = x2

•as it moves both x and y values change

•the rate of change for x is ½ unit/sec.

Suppose:

•How can we find the rate of change of y?

1unit/sec

2

dx

dt

dy dy dx

dt dx dt 2

dyx

dx

12 2

2

dy dxx x x

dt dt

dy

dt

rate of change of y equals x at that point.

at (2, 4)

2dy

dt

at (3, 9)

3dy

dt



Suppose that the diameter D and the area of a circle are differentiable functions of t. Write an equation that relates dA/dt and dD/dt.

2A r 2D r

?dA

dt ?

dD

dt2

drr

dt 2

dr

dt

1

2

dr dD

dt dt

12

2

dA dDr

dt dt

dDr

dt

2A r 2D r1

2D r

21

2A D

2

2

D

2

D dD

dt

2

4 2

dA dD D dDD

dt dt dt


Model Problem

A particle moves along the curve y = 3x2 – 6x so that the rate of change of the x coord. is

Find the rate of change of the y-coord. when the particle is at the origin.

2 unit/secdx

dt

6 6dy

xdx

dy dy dx

dt dx dt

2 units/secdx

dt

@ x = 0

?dy

dt

6 6 2 12 12dy

x xdt

@ x = 0 12 0 12 12 units/secdy

dt


Model Problem

A particle moves along a circle 25 = x2 + y2. If the particle is at (-4, 3) and the y-coord. is increasing so that

Find the rate of change of the x-coord.

2units/sec.dy

dt

2 2 25 0dx dy d

x ydt dt dx

2 units/secdy

dt?

dx

dt

Take derivative of x2 + y2 = 25 with respect to t

2 4 2 3 2 0dx

dt Substitute and solve

for dx/dt.

3 units/sec.

2

dx

dt


Model Verbal Problem Guidelines

1. Identify all given quantities and quantities to be determined. Make a sketch and label quantities.

2. Write an equation involving the variables whose rates of change either are given or are to be determined.

3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t.

4. After completing Step 3, substitute into the resulting equation all known values for the variable and their rates of change and then solve.



2

3V r h

volume in dependent on measurements of radius and height

If water were draining out of this cone, the volume V, the height h, and the radius r, of the water level would all be functions of time t.


The Draining Cone

implicitly differentiate the Volume formula in terms of time t.

dV

dt

2

3V r h

2 23

dh drr h r

dt dt

2 23

dh drr rh

dt dt

this reinforces the fact that the rate of Volume change is related to the changes in h and r.


The Draining Cone

If the water height is changing at a rate of -0.2 foot per minute and the radius is changing at the rate of -0.1 foot per minute, what is the rate of change in the volume when the radius r = 1 foot and the height h = 2 feet?

2 23

dV dh drr h r

dt dt dt

21 0.2 1 0 13

.2 2dV

dt

= -0.628 . . . cubic feet per min.

Chain Rule

2

3V r h

?

dV

dt 0.2

dh

dt 0.1

dr

dt


Model Problem

x and y are both differentiable functions of t and related by the equation y = x2 + 3. Find dy/dt when x = 1, given that dx/dt = 2 when x = 1.

y = x2 + 3 original equation

2 3d d

y xdt dt

differentiate with respect to t

2dy dx

xdt dt

Chain Rule

When x = 1 and dx/dt = 2, you have

2 1 42dy

dt


Do Now:


Suppose that the diameter D and the area of a circle are differentiable functions of t. Write an equation that relates dA/dt and dD/dt.


Do Now:


A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?


Model Verbal Problem


2A r

1dr

dt

Area formula for circle

rate of change

?, when 4dA

rdt

what to find?




2dA dr

dt dt

2A r


Chain Rule

Substitute

2 2dA d dr

r rdt dt dt

2 1 84dA

dt

When r = 4, the area is changing at a rate of 8 sq.ft./sec.



Air is being pumped into a spherical balloon at a rate of 4.5 cubic inches per minute. Find the rate of change of the radius when the radius is 2 inches.

34

3V r

4.5dV

dt

Volume formula for sphere

rate of change

?, when 2dr

rdt

what to find?


Model Problem (con’t)

Air is being pumped into a spherical balloon at a rate of 4.5 cubic inches per minute. Find the rate of change of the radius when the radius is 2 inches.

34

3V r Volume formula for sphere

243

3

dV drr

dt dt


24dr

rdt

2

1

4

dr dV

dt r dt solve for dr/dt

2

1

4

0.09 inche

4.5

s per min.

2

dr

dt

substitute


Model Problem

A airplane is flying at an altitude of 6 mi. on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s = 10 miles, what is the speed of the plane?

x

6 mi.s

a2 + b2 = c2

when s = 10, x = 8 miles400ds

dt

rate of change

?,

when 10 and = ?

dx

dts x

what to find?

8

10

8


Model Problem

A airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s = 10 miles, what is the speed of the plane?

x

6 mi.s

x2 + 62 = s2 Pythagorean Theorem

2 2dx ds

x sdt dt


2

2

dx s ds

dt x dt solve for dr/dt

substitute

10

400

500 mi/hr

8

dx

dt


what to find?

Model Problem

Find the rate of change in the angle of elevation of the camera at 10 seconds after liftoff.

is the angle of elevation

when t = 10, s = 5000 feet

?, when 10

and 5000

dt

dts

s = 50t2 Position Equation

Substitution & Evaluation

100ds

tdt

Velocity of rocket


Model ProblemFind the rate of change in the angle of elevation of the camera

at 10 seconds after liftoff. tan2000

s


substitute s, and t and simplify

substitute ds/st = 100t and solve for d/dt

2 1sec

2000

d ds

dt dt

2 1cos 100

2000

dt

dt

2 2

cos 2000 / 2000s

2

2 2

2000 100

20002000

d t

dt s

2 radians per second

29

d

dt

2

2 2

100 102000

20005000 2000

d

dt


Do Now:


At a sand and grave plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?


?

Model Problem

In an engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when = /3.

connecting rodcran

k

xthe velocity of a piston is related to the angle of the crankshaft.

?, when 3

dx

dt

what to find?

rev 2 radians200 400

min 1 revolution min

rad

How do we relate x to ?

= /3

3 7

d

dt

1 revolution

2 radians


Model ProblemIn an engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when = /3.

How do we relate x to ?

Law of Cosines: b2 = a2 + c2 – 2ac cos

72 = 32 + x2 – 2(3)(x) cos


isolate dx/dt

0 2 6 sin cosdx d dx

x xdt dt dt

0 2 6cos 6 sindx dx d

x xdt dt dt

2 6cos 6 sindx d

x xdt dt

6 sin

6cos 2

dx x d

dt x dt

x

a b


Model ProblemIn an engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when = /3.

solve for x by substituting = /3

36 8

2400

16 16

2

9600 34018 inches per minute

13

dx

dt

72 = 32 + x2 – 2(3)(x) cos /3

49 = 9 + x2 – 6(x)(1/2)

0 = x2 – 3x - 40

0 = (x – 8)(x + 5) x = 8

solve for dx/dt by substituting x = 8 and

= /3

6 sin

6cos 2

dx x d

dt x dt


Model Problem

A circular pool of water is expanding at the rate of 16π in2/sec. At what rate is the radius expanding when the radius is 4 inches?


Model Problem

A 25-foot ladder is leaning against a wall and sliding toward the floor. If the foot of the ladder is sliding away from the base of the wall at a rate of 15 ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 7 feet from the ground?


Model Problem

A spherical balloon is expanding at a rate of 60π in3/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 in.


Model Problem

An underground conical tank, standing on its vertex, is being filled with water at the rate of 18π ft3/min. If the tank has a height of 30 feet and a radius of 15 feet, how fast is the water level rising when the water is 12 feet deep?

Model Problem

An underground conical tank, standing on its vertex, is being filled with water at the rate of 18π ft3/min. If the tank has a height of 30 feet and a radius of 15 feet, how fast is the water level rising when the water is 12 feet deep?


Model Problem

A rocket is rising vertically at a rate of 5400 miles per hour. An observer on the ground is standing 20 miles from the rocket’s launch point. How fast (in radians per second) is the angle of elevation between the ground and the observer’s line of sight of the rocket increasing when the rocket is at an elevation of 40 miles?


Model Problem

A man 2 meters tall walks at the rate of 2 meters per second toward a streetlight that’s 5 meters above the ground. At what rate is the tip of his shadow moving?

Aim: Related Rates Course: Calculus Do Now: Aim: How do we find related rates when we have more than...

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Transcript of Aim: Related Rates Course: Calculus Do Now: Aim: How do we find related rates when we have more than...