Math 120R Lab
Comparing the long term behaviors of Exponential, Power & Logarithmic Functions
Power Functions are of the form where k and p are constants. In class, we discussed the general shapes for power functions. Show these graphs and states possible equations for each shape. You should include positive and negative powers and fractional powers.
Part I: Power functions
1) Let How does multiplying by k change the graph of
2) Sketch graphs of and on the same axis.
A. For is larger than Illustrate your conclusion.
B. When do the graphs cross? How many times do they cross? Show these intersection points both graphically and algebraically.
C. We say that one function dominates another function as (as x goes to infinity) if and only if for very large positive values of x, the first function is always greater than the second. Which of the above functions dominates for large, positive values of x ?
D. Is your answer to part C still the same if
E. Is your answer to part C still the same if
3) Repeat parts A through C if
4) In general, if is a degree polynomial and is a degree polynomial, where what can you say about the long-term behavior of vs.
Part II: Comparing Power Functions and Exponential Functions
1) Graph and on the same axis.
A. For do f and g ever intersect? Find any points of intersection.
B. Over what intervals is Over what intervals is When examining this behavior, think about and discuss the growth rate of the exponential function. Which of these functions dominates the other for large positive values for x ? Explain.
2) Graph and What is the growth rate of g? Based on what you know about the patterns of these graphs, how often do they intersect for Find all points of intersection and discuss which function dominates the other for large, positive values of x.
3) If and answer all the questions from #2.
4) What can you say in general about the long term behavior of the exponential functions versus the power functions where the exponent is a positive integer? You should have at least two cases here depending on the value of the base of the exponential function. Do you think your conclusions would hold if the power function had any real number for its base?
Part III: Comparing Logarithmic and Power Functions
1) Let and Graph both functions and show any points of intersection. For which values of x is Which of these functions dominates as
2) Let and Graph both functions and show any points of intersection. For which values of x is Which of these functions dominates for large, positive values of x ?
3) Let and Graph both functions and show any points of intersection. Answer the questions from #2.
4) What can you say in general about the long term behavior of power functions versus log functions?
Remember this is a lab. It is not just a series of answers to questions. Your answers must be in full sentences and the lab must be word processed. Your graphs can be very neatly hand drawn.