(1) The function is one-to-one on the interval . Find a formula for the inverse function on that domain. Sketch the graphs of and for on the same set of x,y-axes.
(2) Show is one-to-one on the interval . (Just saying that it passes the horizontal line test is not sufficient. Explain how you can tell it will pass the horizontal line test. Hint: One way to show a function is one-to-one is to show it is strictly increasing or strictly decreasing.)
For the next three problems, use the formula , where g is the inverse function for f, to compute the requested derivative. You can assume in each case that f is one-to-one on the domain given. Do not attempt to find formulas for the inverse functions!
(3) Let on the interval . Calculate .
(4) Let on the interval . Determine an equation for the line tangent to the graph of at .
(5) (Bonus question): Let on the interval . Calculate .
(1) Calculate the derivative of
(2) Let .
(a) Find the critical numbers for f.
(b) Determine the intervals where f is increasing or decreasing.
(c) Find the intervals where f is concave up or concave down.
(d) Find the points of inflection for f. (It might be a good idea to review the definition of inflection point; it might not be what you think it is!)
(e) Using (a) – (d), and plotting a few points, draw the graph of .
(3) Evaluate the indefinite integral . Don’t forget the +C that is part of the indefinite integral. You will probably want to use a substitution.
(4) Evaluate the definite integral .
(5) (Bonus question): Evaluate the definite integral . If you use a substitution, be sure to change the limits of integration to the correct values for the new variable.
