Problem of the Day

The graph of the function f is shown in the figure above. Which of the following statements about f is true?

b) lim f(x) = 2 x a

c) lim f(x) = 2 x b

d) lim f(x) = 1 x b

e) lim f(x) does not exist x a

a) lim f(x) = lim f(x) x a x b

Problem of the Day

The graph of the function f is shown in the figure above. Which of the following statements about f is true?

b) lim f(x) = 2 x a

c) lim f(x) = 2 x b

d) lim f(x) = 1 x b

e) lim f(x) does not exist x a

a) lim f(x) = lim f(x) x a x b

A Rising Balloon

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift-off point. At the moment the range finder's elevation angle is Π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

range finder

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift-off point. At the moment the range finder's elevation angle is Π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?Draw a picture and name the variables and constants.

θ500 ft.

y

Write down the numerical information in terms of the variables you have chosen.dθ = 0.14 rad/min when θ =

Π/4dt

Write down what you are asked to find using the variables you have chosen.Find dy when θ =

Π/4 dt

range finder

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift-off point. At the moment the range finder's elevation angle is Π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

θ500 ft.

ydθ = 0.14 rad/min when θ = Π/4dt

Find dy when θ = Π/4 dt

Write an equation that relates the variables y and θ.

tan θ = y or y = 500 tanθ 500

range finder

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift-off point. At the moment the range finder's elevation angle is Π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

θ500 ft.

ydθ = 0.14 rad/min when θ = Π/4dtFind dy when θ = Π/4 dt

y = 500 tanθDifferentiate with respect to t. dy = 500(sec2θ)

dθdt dt

range finder

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift-off point. At the moment the range finder's elevation angle is Π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?

θ500 ft.

ydθ = 0.14 rad/min when θ = Π/4dt

dy = 500(sec2θ) dθdt dtEvaluate dy = 500(sec2 Π)

(0.14)dt 4 dy = 500(√2 )2 (0.14) = 140dt

A Highway Chase

A 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 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?

A Highway Chase

A 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 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?

dy = -60dt

dx/dt = ?

ds = 20dt

Find when y = 0.6 mi and x = 0.8mi

(police should be slowing down to make turn at intersection thus negative)

What will s be then?

A Highway Chase

dy = -60dt

dx/dt = ?

ds = 20dt

Find when y = 0.6 mi, x = 0.8mi and s = 1mi

How are the variables related?

A Highway Chase

dy = -60dt

dx/dt = ?

ds = 20dt

s2 = x2 + y2

2s ds = 2x dx + 2y dy dt dt dt

Find when y = 0.6 mi, x = 0.8mi and s = 1mi

A Highway Chase

dy = -60dt

dx/dt = ?

ds = 20dt

2s ds = 2x dx + 2y dy dt dt dt

2(1)(20) = 2(0.8) dx + 2(0.6)(-60) dt

Find when y = 0.6 mi, x = 0.8mi and s = 1mi

70 mph = dx/dt

Filling a Conical Tank

Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

Filling a Conical Tank

Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

h10 ft

r

5 dv = 9 ft3/mindt

Find dh when h = 6 ft dt

Filling a Conical TankWater runs into a conical tank at the rate of 9 ft3/min. The

tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

h10 ft

r

5 dv = 9 ft3/mindtFind dh when h =

6 ft dt V = 1/3 Π r2h

There are too many variables. How can r be eliminated?

Filling a Conical TankWater runs into a conical tank at the rate of 9 ft3/min. The

tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

h10 ft

r

5 dv = 9 ft3/mindtFind dh when h =

6 ft dt V = 1/3 Π r2h

There are too many variables. How can r be eliminated?

5 = r thus r = h 10 h 2

Filling a Conical TankWater runs into a conical tank at the rate of 9 ft3/min. The

tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

h10 ft

r

5 dv = 9 ft3/mindt

Find dh when h = 6 ft dt V = 1/3 Π r2h

V = 1/3 Π h 2 h 2)(V = 1/3 Π h3

4

Filling a Conical TankWater runs into a conical tank at the rate of 9 ft3/min. The

tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

h10 ft

r

5dv = 9 ft3/mindt

Find dh when h = 6 ft dt V = 1/3 Π h3

4dv = 1/3 Π (3) h2 dhdt 4 dt9 = Π(62) dh 4 dt 0.34 ≈dh

dt

Find The ErrorA ladder 10 ft long rests against a vertical

wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?wall

groundx

y 10

find dy/dt

x2 + y2 = 102

62 + y2 = 100 y2 = 64 2y dy = 0 dt dy = 0 dt

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