Calculus 221 worksheet

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1 Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function has a global minimum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global minima are sometimes also called absolute minima. A function has a local maximum at some a in its domain if there is a small δ > 0 such that f(x) f(a) for all other x with a δ < x < a + δ that lies in the domain of f. A function has a local minimum at some a in its domain if there is a small δ > 0 such that f(x) f(a) for all other x with a δ < x < a + δ that lies in the domain of f. Any x value for which f (x) = 0 is called a stationary point for the function f. An interior point of the domain of a function f where f is zero or undefined is a critical point of f. A function f is convex on some interval a < x < b if the derivative f (x) is increasing on that interval. Sometimes we call concave up. A function f is concave on some interval a < x < b if the derivative f (x) is decreasing on that interval. Sometimes we call concave down. A point on the graph of f where f (x) changes sign is called an inflection point. So at an inflection point (c, f(c)), either f (c) = 0 or f (c) fails to exist. Procedure of Graphing y = f(x): 1. Identify the domain of f and any symmetries the curve may have. 2. Find the derivatives y and y. 3. Find the critical points of f, if any, and identify the function s behavior at each one. 4. Draw a table and find where the curve is increasing or decreasing. 5. Find the points of inflection, if any occur, and determine the concavity of the curve. 6. Identify any asymptotes that may exist. 7. Plot key points, such as the intercepts and the points found in Step 3-5, and sketch the curve together with any asymptotes that exist. Example 1. Sketch the graph of the function f(x) = x 3 (x 4). The derivative is f (x) = 4x 3 12x 2 = 4x 2 (x 3). The critical points are where 4x 2 (x 3) = 0, so x = 0, 3. On the interval (, 0), f < 0, so f is decreasing. On the interval (0, 3), f < 0, so f is decreasing. On the interval (3, ), f > 0, so f is increasing. The point x = 0 is neither a local maximum nor a local minimum. The point x = 3 is a local minimum. The second derivative is f = 12x 2 24x = 12x(x 2). The inflection points are where 12x(x 2) = 0, so x = 0, 2. 1

2 2 On the interval (, 0), f > 0, so f is convex. On the interval (0, 2), f < 0, so f is concave. On the interval (2, ), f > 0, so f is convex. Example 2. Sketch the graph of the function f(x) = sin(x). The derivative is f (x) = cos(x). The critical points are where cos(x) = 0, so x = π 2 + πk for each integer ( k. π On the interval 2 + πk, π ) 2 + π(k + 1), f > 0 when k is even and f < 0 when f is odd. Thus f is increasing on the intervals where k is even and decreasing on the intervals where k is odd. The points x = π 2 + 2πk are local maxima and the points x = 3π + 2πk are local minima. 2 The second derivative is f (x) = sin(x). The inflection points are where sin(x) = 0, so x = πk for each integer k. On the interval (πk, π(k + 1)), f > 0 when k is odd and f < 0 when k is even. Thus f is convex on the intervals when k is odd and concave on the intervals when k is even. Example 3. Sketch the graph of the function f(x) = x + 4 x.

3 3 The derivative is f (x) = 1 4 x. The critical points are where 1 4 = 0, so x = 2, 2. 2 x2 On the interval (, 2), f > 0, so f is increasing. On the intervals ( 2, 0), (0, 2), f < 0, so f is decreasing. On the interval (2, ), f > 0, so f is increasing. The point x = 2 is a local maximum. The points x = 2 is a local minimum. The second derivative is f (x) = 8 x. The inflection points are where 8 = 0, which never 3 x3 occurs. Thus, there are no inflection points. On the interval (, 0), f < 0, so f is concave. On the interval (0, ), f > 0, so f is convex. The graph has a vertical asymptote of x = 0. The graph has a horizontal asymptote of y = 0. Example 4. Let f(x) = 1 8 x8 1 2 x6 x 5 + 5x 3. Here is a plot of y = f(x) on the interval [ 3, 3]:

4 4 (1) Using the picture, identify the approximate locations of the local minima and maxima of f. (2) Find the actual x-coordinates of all the critical points, and decide which of them are local minima and local maxima. [Things will factor nicely.] (3) You should find that the results of part (2) are not reflected by the graph. Explain why not. (2) The derivative is f (x) = x 7 3x 5 5x x 2 = x 5 (x 2 3) 5x 2 (x 2 3) = (x 5 5x 2 )(x 2 3) = x 2 (x 3 5)(x 2 3). The critical points are where x 2 (x 3 5)(x 2 3) = 0, so x = 0, 3 5, ± 3. On the intervals (, 3), ( 3 5, 3), f < 0, so f is decreasing. On the intervals ( 3, 0), (0, 3 5), ( 3, ), f > 0, so f is increasing. This show that x = 3, 3 are local maxima and x = 3 5 is a local minimum. Example 5. Find the absolute minimum and maximum values of g(x) = x sin(x) + cos(x) on the intervals [ π, π ] and [0, π]. 3 3 The derivative is g (x) = x cos(x). The critical points are where x cos(x) = 0. On [ π 3, π 3 ], the only critical point is x = 0. On [0, π], the critical points are x = 0, π 2. To find the absolute minimum and maximum values, we must check the function values at each critical point and each endpoint. On the interval [ π 3, π 3 ], g( π 3 ) = π 2, g(0) = 1, g( π) = π The absolute minimum 3 value is 1 and the absolute maximum value is 1 + π On the interval [0, π], g(0) = 1, g( π) = π, g(π) = 1. The absolute minimum value is -1 and 2 2 the absolute maximum value is π. 2 Example 6. For each of the following, give an example of a function with the given property: (1) A function with a local maximum and no local minimum. (2) A differentiable function with a critical point that is neither a maximum nor a minimum. (3) A continuous function with no global maximum or global minimum. (4) A function which has a local maximum but whose derivative is never zero. (1) One example is f(x) = x 2. f(x) has a local maximum at x = 0, but no local minimum, (2) One example is f(x) = x 3. f(x) has a critical point at x = 0, but it is neither a local maximum nor a local minimum. (3) One example is f(x) = x 3. As x, f(x) and as x, f(x). Therefore f has no global maximum or minimum. (4) One example is f(x) = x. f(x) has a local maximum at x = 0, but the function is not differentiable at x = 0.

5 5 Exercise 1. Sketch the graph of the function f(x) = x 3 3x. 2. Find the absolute minimum and maximum values of f(x) = x 3 3x + 2 on the intervals [ 2, 1] and [ 3, 5]. 3. Sketch the graph of the function b(x) = x x Sketch the graph of the function c(x) = x2 x Sketch the graph of the function f(x) = x3 x 2 4.