3.4 Approximation and Related Rates

1-3. Use a differential to approximate the followings.

1.

http://matrix.skku.ac.kr/cal-lab/cal-3-3-1.html 

  In order to find an approximation of , we define a function . Then

  .

(x+32)^(1/5)

Now we use differential to approximate. Set and at . Find an approximation of 

 by using

1/5*dx/(x+32)^(4/5)

2.00062500000000

There are Sage built in command  .n()  for finding such approximations.

(32.05)^(1/5).n()

2.00062460974081

2.

http://matrix.skku.ac.kr/cal-lab/cal-3-3-2.html

1/2*dx/sqrt(x + 25)

5.20000000000000 

3.

http://matrix.skku.ac.kr/cal-lab/cal-3-3-3.html

1/3*dx/(x + 60)^(2/3)

3.93649718310217

4. The height of a circular cone is the same as the radius of its circular bottom. The height and radius were measured and found to be 5 cm with a possible error in measurement of at most 0.02cm. What is the relative error in using this value to compute the volume?

5. Find an approximation of the difference between surface areas of two spheres whose radii are 4cm and 4.05cm, respectively.

6. The period of a pendulum is given by the formula , where L is the length of the pendulum measured in meters and is the gravitational constant. If the length of the pendulum is measured to be 3m with a possible error in mea-surement 1cm. What is the approximate percentage error in calculating the period ?

    ,  ,

        The approximate percentage error = (relatively error) 100%=.

7. A ladder 10 meter long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3m/s, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 6 meter away from the wall?

       Since so

8. A ladder 10 meter long is leaning against a wall. If the top of the ladder is sliding down the wall at 3m/s, how fast is the foot of the ladder being pulled away from the wall when the foot of the ladder is 6 meter away from the wall?

      Since , .

9. A ladder 10 meter long is leaning against a wall. If the top of the ladder is sliding down the wall at 3m/s, how fast is the angle between the top of the ladder and the wall changing when the foot of the ladder is 6 meter away from the wall?

       Since , 

10. Two cars start moving from the same point. One travels south at km/hour and the other travels west at km/hour. How fast is the distance changing between the two cars?

 Distance of travel south is , distance of travel west is , and the between the distance of two travelers is at time

11. Water is being pumped at a rate of 20 liters per minute into a tank shaped like a frustrum of a right circular cone. The tank has an altitude of 8 meters and lower and upper radii of 2 and 4 meters, respectively. How fast is the water level rising when the depth of the water is 3 meters?

     .

          Let .

            (Because )

           .

12. Water is being pumped at a rate of 20 liters per minute into a tank shaped like a hemisphere. The tank has a radius of 8 meters. How fast is the water level rising when the depth of the water is 3 meters?

13. A snowball melts at a rate proportional to its surface area. Does the radius shrink at a constant rate? If it melts to 1/2 its original volume in one hour, how long does it take to melt completely?

 . The rate of melting is described by .

      Notice that  so is constant .

      Let the volume of first time , after an hour, .