3.11 related rates mon nov 10

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3.11 Related Rates Mon Nov 16 • Do Now • Differentiate implicitly in terms of t • 1) • 2)

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3.11 Related Rates Mon Nov 10. Do Now Differentiate implicitly in terms of t 1) 2). Related Rates. When we use implicit differentiation, we obtain dy/dx, or the change of y in terms of x. In many real life situations, each quantity in an equation changes with time (or another variable) - PowerPoint PPT Presentation

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Page 1: 3.11 Related Rates Mon Nov 10

3.11 Related RatesMon Nov 16

• Do Now

• Differentiate implicitly in terms of t

• 1)

• 2)

Page 2: 3.11 Related Rates Mon Nov 10

Related Rates• When we use implicit differentiation, we obtain

dy/dx, or the change of y in terms of x.

• In many real life situations, each quantity in an equation changes with time (or another variable)

• In this case, any derivative we find is called a related rate, since each rate in the derivative is related to each other

Page 3: 3.11 Related Rates Mon Nov 10

Related Rates Steps

• 1) Make a simple sketch, if possible• 2) Identify what rate you are looking for• 3) Set up an equation relating ALL of the

relevant quantities• 4) Differentiate both sides of the equation in

terms of the variable you want – if you want dv/dt, you differentiate in terms of t

• 5) Substitute in values we know• 6) Solve for the remaining rate

Page 4: 3.11 Related Rates Mon Nov 10

Ex 1• A 5-meter ladder leans against a wall. The

bottom of the ladder is 1.5 m from the wall at time t=0 and slides away from the wall at a rate of 0.8m/s. Find the velocity of the top of the ladder at time t=1

Page 5: 3.11 Related Rates Mon Nov 10

Ex 2

• Water pours into a fish tank at a rate of 0.3 m^3 / min. How fast is the water level rising if the base of the tank is a rectangle of dimensions 2 x 3 meters?

Page 6: 3.11 Related Rates Mon Nov 10

Ex 3• Water pours into a conical tank of height 10

m and radius 4 m at a rate of 6 m^3/min• A) At what rate is the water level rising when

the level is 5 m high?• B) As time passes what happens to the rate

at which the water level rises?

Page 7: 3.11 Related Rates Mon Nov 10

Ex 4• A spy uses a telescope to track a rocket

launched vertically from a launching pad 6km away. At a certain moment, the angle between the telescope and ground is equal to pi/3 and is changing at a rate of 0.9 radians/min. What is the rocket’s velocity at that moment?

Page 8: 3.11 Related Rates Mon Nov 10

Ex 5

• See book

Page 9: 3.11 Related Rates Mon Nov 10

Closure

• At what rate is the diagonal of a square increasing if its sides are increasing at a rate of 2 cm/s?

• HW: p.199 #1-37 every other odd

• Ch 3 Test next week Mon-Tues (2 days)

Page 10: 3.11 Related Rates Mon Nov 10

3.11 Related Rates Cont’dTues Nov 17

• Do Now

• Air is being pumped into a spherical balloon at a rate of 5 cm3/min.  Determine the rate at which the radius of the balloon is increasing when the radius of the balloon is 10 cm.

• (hint: Volume = 4/3 pi x r^3)

Page 11: 3.11 Related Rates Mon Nov 10

HW Review p.199 #1-35

• Probably all of them

Page 12: 3.11 Related Rates Mon Nov 10

Closure• Hand in: A 15 foot ladder is resting against

the wall.  The bottom is initially x feet away from the wall and is being pushed towards the wall at a rate of 0.5 ft/sec.  How fast is the top of the ladder moving up the wall when the bottom of the ladder is 4 feet from the wall?? (Hint: Use Pythagorean Theorem)

• HW: p.199 #1-35 all other odd• p.AP3-1 Multiple Choice #1-20 due Thurs• Free Response #1-4 due Fri