DERIVATIVES

Review- 4 Rates of Change

Rates of Change

There are two types of rates of change

Average abafbf

Instantaneous xxfxxf

x

)()(

0lim

1) Temperature readings in degrees Celsius were recorded every hour starting at midnight. The time, x, is measured in hours.

a) Find the average rate of change from noon to 3 pm

B) from 3 am to 5 am

x T

0 6.5

1 6.1

2 5.6

3 4.9

4 4.2

5 4.0

6 4.0

7 4.8

8 6.1

9 8.3

10 10.0

11 12.1

12 14.3

13 16.0

14 17.3

15 18.2

16 18.8

2) A bug’s position (in inches) along a straight path is given by the equation

on the interval [0,4] minutes. Find the instantaneous velocity at t= 3 minutes

tttx 22)(

xxfxxf

x

)()(

0lim

Related Rates

When two or more related variables are changing with respect to time they are called related rates

3) A pebble is dropped into a pond, causing ripples in the form of concentric circles.

The radius of the outer ripple is increasing at the rate of 2 feet per second. At what rate is the area of the water changing when r = 5 ft

4) How fast does the radius of a spherical soap bubble change when

air is blown into it at the rate of

sec10 3cm

When

a) r = 1 cm

b) r= 3 cm

5) A ladder 30 ft long rests against a wall. The foot of the ladder is pulled away from the wall at the rate of

How fast is the top sliding down the wall when the foot is 10 feet from the wall?

sec4 ft

6) A windlass is used to tow a boat to a dock. The rope is attached to the boat at a point 15 feet below the level of the windlass. If the windlass pulls in rope at the rate of 30 feet per minute, at what rate is the boat approaching the dock when there is 25 feet of rope out?

rope z15 ft.

x

windlass

7) A street light is at the top of a 18 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 4ft/sect along a straight path. How fast is the tip of his shadow moving when he is 45 ft from the base of the pole?

8) A hot-air balloon rising straight up is 500 feet from the lift-off point. At that moment the angle of elevation is π/4 and the angle is increasing at the rate of .14 rad/min. How fast is the balloon rising?

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