Approximation of Instantaneous Rate of Change in Various Contexts, Assignments of Mathematics

Download Approximation of Instantaneous Rate of Change in Various Contexts and more Assignments Mathematics in PDF only on Docsity! MAT 294 Fall 2005 h=0 h=2 h=1 h=3 At this Rate Instructions: You will approximate the instantaneous rate of change for one of the situations below by answering each of the following questions algebraically, numerically, and by representing each answer in your diagram: 1. Draw a large picture of the physical situation for the value of the variable given. 2. Imagine how things are changing in this situation. List all of the quantities that you think are changing. Describe how they are changing. 3. On the same picture, draw several “snapshots” of the situation. 4. Label the changing and constant quantities in your drawing. 5. Describe in more detail what you have been asked to approximate. 6. What can you use for approximations? 7. What are the errors? 8. Find an approximation and a bound for the error. What is the resulting range of possible values for your instantaneous rate? 9. How can you find an approximation with error smaller than a predetermined bound? Context 1: An object is falling according to the equation 2( ) 100 16h t t= − feet (with t measured in seconds). Approximate the speed when 2t = seconds. Context 2: Approximate the instantaneous rate of change of the area of a circle with respect to its radius when the radius is 3 cm. Context 3: The force of gravity between two objects is inversely proportional to the square of the distance separating them. Approximate the instantaneous rate of change of the gravitational force with respect to distance when two objects are 230 km apart. (Note that all of your answers will involve the constant of proportionality.) Context 4: Approximate the rate of change of the height of water in this bottle with respect to the volume of water when the height is 1.5. (Note that your answers will involve the size of the spherical portion of the bottle.) Context 5: The half-life of Iodine-123, used in some medical radiation treatments, is about 13.2 hours. Thus a sample that originally has 6.4 µg of Iodine-123 will decay so that the amount left after t hours will be roughly ( ) /13.212( ) 6.4 t I t = µg. Approximate the instantaneous rate at which the Iodine-123 is decaying after 5 hours.