II Please Show Work!

1 tex 1. Calculate the derivative of f (x) = 1 + e2x' 2. Find an equation for the line tangent to the graph of f(x) = x2 In x at x = e3. 3. Evaluate the indefinite integral / xe dx. 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 / ettax. 1 5. Evaluate the definite integral dx. Je3 xInx You will probably want to use the method of substitution (aka change of variables) to do this problem. If you do, be sure to remember that when substitution is used for a definite integral, the limits of integra-tion have to be changed to the correct ones for the new variable. 6. Let f (x) = xe-x. (a) Find the critical numbers for f. (b) Determine the intervals where f in increasing or decreasing. (c) Find the intervals where f is concave up or concave down. (d) Find the points of inflection for f. (Warning: review the definition of inflection point; it might not be what you think it is!) (e) Using (1)-(4), and plotting a few points, draw the graph of f (x) = xe-*. 7. Suppose f(x) = In(x2 + 1). (a) Calculate the first and second derivatives of f. (b) Determine the intervals where f is increasing or decreasing. (c) Determine all local maxima and minima for f. (d) Determine the intervals where f is concave up or concave down. (e) Determine all points of inflection for f. (f) Using (1)-(5), and plotting two or three points on the graph, sketch a graph of f