Supplemental Instruction - Math 165 ABCD 10/18/07 Graphing Functions Being able to find a function’s local extrema (maxima and minima), concavity, horizontal asymptotes, and vertical asymptotes allows us to be able to graph the function. The first two sets of information are useful for graphing polynomial functions and the latter two are useful for graphing rational functions (quotients of polynomials). Consider the following simple case. 1. Sketch the graph of a function F(x) that has the following properties A. F is continuous everywhere B. F(2) = -3, f(6) = 1 C. F’(2) = 0, f’(x) > 0 for x2, f’(6) = 3 D. F’’(6) = 0, f’’(x) > 0 for 2 < x < 6, f’’(x) < 0 for x > 6. Hint : draw horizontal line graphs of f’(x) and f’’(x) and determine where F is increasing, decreasing, concave up, and concave down. 2. Graph f(x) = 2x3x 12x 3. 3. When graphing rational functions, it is often easier using asymptotes than using first and second derivatives. Graph 4. Being able to draw a function based on its derivative means that we can draw a parent function when we are given its derivative. Below is f’(x). Graph f(x) onto it. 5. Bonus question for all engineering majors. The following graph shows the displacement of air molecules in the x-direction during sound propagation. Graph pressure vs. x and tell at which position the pressure has the largest value. Hint : low pressure is created where air molecules have vacated; high pressure is created where air molecules have gathered. (+) air molecule displacement means air molecules moving to the right. (–) air molecule displacement means air molecules moving to the
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