Math 165 Section A Professor Lieberman PRACTICE THIRD IN-CLASS EXAM Carry out the solution of each problem: show steps of any required calculations and state reasons that justify any conclusions. A short sentence is usually enough but answers without any justification will receive no credit. This test merely shows the number of questions and level of difficulty to expect on Wednesday’s exam. Questions on other topics (from the first four sections of Chapter 4) may appear. 1. (15 points) Sketch a curve so that the slope is always positive and decreasing. 2. (25 points) Find the absolute maximum and minimum values of the function f (x) = x − 2 cos x on the interval [−π, π]. 3. (25 points) Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabolic y = 8 − x2 . 4. (35 points) For the function f (x) = 2x + cot x, 0 < x < π, (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f . (c) Find the intervals on which f is concave up or concave down, and find the inflection points.