Martin Flashman's Courses - Math 115 Fall, '02
Algebra and Elementary Functions
Cooperative Problems 
  1. Due 5 pm. Tuesday, October 8: Linear interpolation: roots and logarithms.
    1.  Consider the function f (x) = x2 - 5. Use linear interpolation  with f  and  x= 2 and x= 3 to estimate the square root of 5. Call this estimate x1. Take this estimate and use linear interpolation again with either x  = 2 or x = 3 to make a second estimate for the square root of 5. Call this estimate x2. Draw a graph showing how x1  and x2   are related to the graph of  f. Discuss how you would continue this process further to obtain a better estimate for the square root of 5. Compare your estimate with your calculator's estimate for the square root of 5.
    2. Consider the function g (x) = x2 - 6. Use linear interpolation  with g  and  x= 2 and x= 3 to estimate the square root of 6. Call this estimate x1. Take this estimate and use linear interpolation again with either x  = 2 or x = 3 to make a second estimate for the square root of 6. Call this estimate x2. Draw a graph showing how x1  and x2   are related to the graph of f. Discuss how you would continue this process further to obtain a better estimate for the square root of 6. Compare your estimate with your calculator's estimate for the square root of 6.
    3. Consider the function f (x) = x3 - 10. Use linear interpolation  with f  and  x= 2 and x= 3 to estimate the cube root of 10. Call this estimate x1. Take this estimate and use linear interpolation again with either x  = 2 or x = 3 to make a second estimate for the  cube root of 10. Call this estimate x2. Draw a graph showing how x1  and x2   are related to the graph of f. Discuss how you would continue this process further to obtain a better estimate for the  cube root of 10. Compare your estimate with your calculator's estimate for  cube root of 10.
    4. Consider the function f (x) = 2x - 5. Use linear interpolation  with f  and  x= 2 and x= 3 to estimate the log2 (5). Call this estimate x1. Take this estimate and use linear interpolation again with either x  = 2 or x = 3 to make a second estimate for the log2 (5). Call this estimate x2. Draw a graph showing how x1  and x2   are related to the graph of f. Discuss how you would continue this process further to obtain a better estimate for the log2 (5). Compare your estimate with a calculator estimate [using the change of basis method] for the log2 (5).
    5. Consider the function f (x) = 2x - 6. Use linear interpolation  with f  and  x= 2 and x= 3 to estimate the log2 (6). Call this estimate x1. Take this estimate and use linear interpolation again with either x  = 2 or x = 3 to make a second estimate for the log2 (6). Call this estimate x2. Draw a graph showing how x1  and x2   are related to the graph of f. Discuss how you would continue this process further to obtain a better estimate for the log2 (6). Compare your estimate log2 (6). for the log2 (6).
    6. Use your estimates in the previous two problems together with the properties of logarithms to estimate the following log2 (100) and log2 (3). Compare your estimate with a calculator estimate [using the change of basis method] .
    7. Use linear interpolation applied to the function log2 (x) with your estimates of log2 (5) and log2 (6) in the previous problems  to estimate log2 (5.3). Draw a graph showing how your estimate is related to the graph of  log2 (x) .