Drill: find the derivative of the following

• 2xy + y2 = x + y• 2xy’ +2y + 2yy’ = 1 + y’• 2xy’ + 2yy’ – y’ = 1 – 2y• y’(2x + 2y – 1) = 1 – 2y• y’ = (1-2y)/(2x + 2y -1)

• xsin(y)= 1 – xy• xcos(y)y’ + sin (y) = -(xy’ +y)• xcos(y)y’ + sin (y) = -xy’ –y• xcos(y)y’+xy’= -y – sin (y)• y’ (xcos (y) + x) = -y – sin (y)• y’ = - (y + sin(y))/(xcos(y) + x)

Related Rates

Section 4.6

Objectives

• Students will be able to– solve related rate problems.

Strategy for Solving Related Rate Problems

• Understand your problem• Develop a mathematical model of the problem • Write an equation relating the variable whose rate of

change you seek with the variables whose rate of change you know

• Differentiate both sides of the equation implicitly with respect to time t.

• Substitute values for any quantities that depend on time

• Interpret the solution

Finding related rate equations

• Assume the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that related dV/dt and dr/dt

3

34 rV

dtdrr

dtdV 24

Finding related rate equations

• Assume that the radius and height h of a cone are differentiable functions of t and let V = volume of the cone. Find an equation that relates dV/dt, dr/dt and dh/dt.

hrV 2

31

)(31 2hrV

)2(31 2

dtdrrh

dtdhr

dtdV

Example 1 Finding Related Rate EquationsAssume that the width a and the length b of a rectangle are functions of t and let A be the area of the rectangle. Find an equation that relates . and ,,

dtdb

dtda

dtdA

Example 1 Finding Related Rate EquationsAssume that the width a and the length b of a rectangle are functions of t and let A be the area of the rectangle. Find an equation that relates . and ,,

dtdb

dtda

dtdA

abA

abdtdA

dtd

adtdbb

dtda

dtdA

Example 2 A Rising BalloonA hot-air balloon rising straight up from a level field is tracked by a range finder 700 feet from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.16 radians per minute. How fast is the balloon rising at that moment?

6

Example 2 A Rising BalloonA hot-air balloon rising straight up from a level field is tracked by a range finder 700 feet from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.16 radians per minute. How fast is the balloon rising at that moment?

6

feet 700

h

dtdh :find want toWe

rad/min 16.0 :know We dtd

700tan h

Example 2 A Rising Balloon

feet 700

h

dtdh :find want toWe rad/min 16.0 :know We

dtd

700tan h

tan700h

tan700dtd

dtdh

dtd

dtdh 2sec700

16.06

sec700 2

dtdh

33.149 ft/min

Example 3 A Highway ChaseA police cruiser approaching a right-angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.70 mile north of the intersection and the car is 0.80 mile to the east, the police determine with radar that the distance between them and the car is increasing at 15 mph. If the cruiser is moving at 45 mph at the instant of measurement, what is the speed of the car?

Example 3 A Highway ChaseA police cruiser approaching a right-angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.70 mile north of the intersection and the car is 0.80 mile to the east, the police determine with radar that the distance between them and the car is increasing at 15 mph. If the cruiser is moving at 45 mph at the instant of measurement, what is the speed of the car?

Example 3 A Highway ChaseA police cruiser approaching a right-angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.70 mile north of the intersection and the car is 0.80 mile to the east, the police determine with radar that the distance between them and the car is increasing at 15 mph. If the cruiser is moving at 45 mph at the instant of measurement, what is the speed of the car?

Let x = distance of the speeding car from the intersectionLet y = distance of the police cruiser from the intersectionLet z = distance between the car and the cruiser

Example 3 A Highway ChaseWe want to know:

We know:

x

y

z

dtdx

mph 15dtdz

mph 45dtdy

222 zyx

222 zdtdyx

dtd

dtdzz

dtdyy

dtdxx 222

dtdzz

dtdyy

dtdxx

Example 3 A Highway Chasex = .8 and y = .6

x

y

z

157.08.0457.08.0 22 dtdx

9.155.318.0 dtdx 4.478.0

dtdx

25.59dtdx

At that moment, the car’s speed was about 59.3 mph.

mph 15dtdz mph 45

dtdy

We can now substitute, remembering that z2 = x2 + y2 ; so z =(x2 + y2)1/2

dtdzz

dtdyy

dtdxx

Example 4 Filling a Conical TankWater runs into a conical tank at the rate of 12 ft3 /min. The tank stands point down and has a height of 12 ft and a base radius of 9 ft. How fast is the water level rising when the water is 8 ft deep?

Example 4 Filling a Conical TankWater runs into a conical tank at the rate of 12 ft3/min. The tank stands point down and has a height of 12 ft and a base radius of 9 ft. How fast is the water level rising when the water is 8 ft deep?

rdtdh :find want toWe

h

min/ft 12 :know We 3dtdV

hrV 2

31

129

hr

hr43

hhV2

43

31

Example 4 Filling a Conical Tank

r

dtdh :find want toWe

h

min/ft 12 :know We 3dtdV

129

hr

hr43

hhV2

43

31

3

163 hV

3

163 h

dtdV

dtd

dtdhh

dtdV 2

169

Example 4 Filling a Conical Tank

r

dtdh :find want toWe

h

min/ft 12 :know We 3dtdV

129

hr

hr43

dtdhh

dtdV 2

169

dtdh28

16912

dtdh3612

dtdh

31 106.0 ft/min

Homework

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