# The function is one-to-one on the interval - Expert Answers

(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.