1 - x + In x / 1 + COS pi x x / tan 1(4x) xa - ax + a - 1 / (x - 1)2 ex - e-x - 2x / x - sin x cos x ln (x - a) / ln(ez - ea) x sin( pi / x) x2ex cot 2x sin 6x sin x In x x3e-x2 x tan(1/x) (x / x - 1 - 1 / In x) (csc x - cot x) (x - In x) (xe1/x - x) xx2 (tan 2 x)x (1 - 2x)1/x (1+a / x)bx x1/x x(In2)/(l + In x) (4x + l)cot x (2 - x)tan( pi x/2) Graph the function. Use I'Hospital's Rule to explain the behavior as x rightarrow 0. Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. f(x) = x2 In x f(x) = xe1/x Graph the function. Explain the shape of the graph by computing the limit as x rightarrow 0+ or as x rightarrow infinity . Estimate the maximum and minimum values and then use calculus to find the exact values. Use a graph of f" to estimate the x-coordinates of the inflection points. f(x) = x1/x f(x) = (sin x)sin x Investigate the family of curves given by f(x) = xe-cx, where c is a real number. Start by computing the limits as x rightarrow plusminus infinity. Identify any transitional values of c where the basic shape changes. What happens to the maximum or minimum points and inflection points as c changes? Illustrate by graphing several members of the family. Investigate the family of curves given by fix) = xne-x, where n is a positive integer. What features do these curves have in common? How do they differ from one another? In particular, what happens to the maximum and minimum points and inflection points as n increases? Illustrate by graphing several members of the family. What happens if you try to use 1'Hospital's Rule to evaluate
