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November 30, 2010 AB Calculus Unit Five Review Outline 1. Use calculus to sketch the graph of a function. a) determine the domain, b) determine symmetry, c) find the intercepts, d) use limits to find the vertical asymptotes, e) use limits to find the horizontal/slant asymptote(s) f) use the derivative to find extrema, g) determine the range, h) draw a graph using the information above 2. Analyzing a function using the graph of its derivative. 3. Sketch the graph of a function given information about the first and second derivatives. 4. Solve optimization problems. 5. Find a linear approximation of a funtion. 6. Find a differential. 1 November 30, 2010 Test for Increasing and Decreasing Functions Let f be a function that is continuous on [a, b] and differentiable on (a, b). 1. If f '(x) > 0 for all x in (a, b), f is increasing on [a, b]. 2. If f '(x) < 0 for all x in (a, b), f is decreasing on [a, b]. 3. If f '(x) = 0 for all x in (a, b), f is constant on [a, b]. The First Derivative Test Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows. 1. If f '(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)). 2. If f '(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)). 3. If f '(x) is positive on both sides of c or negative on both sides of c, then f(c) is neither a relative minimum nor a relative maximum. 2 November 30, 2010 3 November 30, 2010 Definition of Concavity Let f be differentiable on an open interval I. The graph of f is concave upward on I if f ' is increasing on the interval and concave downward on I if f ' is decreasing on the interval. Test for Concavity Let f be a function whose second derivative exists on an open interval I. 1. If f "(x) > 0 for all x in I, then the graph of f is concave upward in I. 2. If f "(x) < 0 for all x in I, then the graph of f is concave downward in I. The Second Derivative Test Let f be a function such that f '(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f "(c) > 0, then f has a relative min at (c, f (c)). 2. If f "(c) < 0, then f has a relative max at (c, f (c)). If f "(c) = 0, the test fails. Revert to using the First Derivative Test in this case. 4 November 30, 2010 Guidelines to solving applied min/max problems a) Identify all quantities either given or to be found. Draw a figure if possible. b) Write a function for the quantity to be minimized or maximized. c) Determine the feasible domain. d) Use the derivative(s) to find the desired min/ max. Linear Approximation The linear approximation (tangent line) of f at c is given by Definition of Differential 5