Martin Flashman's Courses  Math 115 Fall, '02
Algebra and Elementary Functions
Cooperative Problems

Due 5 pm. Tuesday, October 8: Linear
interpolation: roots and logarithms.

Consider the function f (x) = x^{2 }
5. Use linear interpolation with f and x=
2 and x= 3 to estimate the square root of 5. Call this estimate
x_{1}.
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 x_{2}. Draw a graph showing how x_{1 }
and x_{2} 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.

Consider the function g (x) = x^{2 } 6. Use
linear interpolation with g and x= 2 and
x=
3 to estimate the square root of 6. Call this estimate x_{1}.
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 x_{2}. Draw a graph showing how x_{1 }
and x_{2} 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.

Consider the function f (x) = x^{3 } 10.
Use linear interpolation with f and x=
2 and x= 3 to estimate the cube root of 10. Call this estimate x_{1}.
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 x_{2}. Draw a graph showing how x_{1 }
and x_{2} 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.

Consider the function f (x) = 2^{x } 5. Use
linear interpolation with f and x= 2 and
x=
3 to estimate the log_{2} (5). Call this estimate x_{1}.
Take this estimate and use linear interpolation again with either x
= 2 or x = 3 to make a second estimate for the log_{2} (5).
Call this estimate x_{2}. Draw a graph showing how x_{1 }
and x_{2} are related to the graph of
f.
Discuss how you would continue this process further to obtain a better
estimate for the log_{2} (5). Compare your estimate with a calculator
estimate [using the change of basis method] for the log_{2} (5).

Consider the function f (x) = 2^{x } 6. Use
linear interpolation with f and x= 2 and
x=
3 to estimate the log_{2} (6). Call this estimate x_{1}.
Take this estimate and use linear interpolation again with either x
= 2 or x = 3 to make a second estimate for the log_{2} (6).
Call this estimate x_{2}. Draw a graph showing how x_{1 }
and x_{2} are related to the graph of
f.
Discuss how you would continue this process further to obtain a better
estimate for the log_{2} (6). Compare your estimate log_{2}
(6). for the log_{2} (6).

Use your estimates in the previous two problems together with the properties
of logarithms to estimate the following log_{2} (100) and log_{2}
(3). Compare your estimate with a calculator estimate [using the change
of basis method] .

Use linear interpolation applied to the function log_{2} (x)
with your estimates of log_{2} (5) and log_{2} (6) in the
previous problems to estimate log_{2} (5.3). Draw a graph
showing how your estimate is related to the graph of log_{2}
(x) .