ReviewUsingDerivatives.nb 1. As we have seen, the connection between derivatives of a function and the function itself is given by the following:
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1 ReviewUsingDerivatives.nb Calculus Review: Using First and Second Derivatives As we have seen, the connection between derivatives of a function and the function itself is given by the following: à If f ' > 0 on an interval, then f is increasing on that interval. à If f ' < 0 on an interval, then f is decreasing on that interval. à If f '' > 0 on an interval, then the graph of f is concave up on that interval. à If f '' < 0 on an interval, then the graph of f is concave down on that interval. Information given by the first and second derivatives of a function can help identify regions with interesting behavior. First let's look at some definitions. Local Maxima and Minima Suppose p is a point in the domain of f: near p. æ f has a local minimum at p if f(p) is less than or equal to the values of f for points æ f has a local maximum at p if f(p) is greater than or equal to the values of f for points near p. æ For any function f, a point p in the domain of f where f ' (p) = 0 or f ' (p) is undefined is called a critical point of the function. In addition, the point (p, f(p)) on the graph of f is also called a critical point. A critical value of f is the value, f(p), of the function at a critical point, p. Notice that "critical point of f" can refer either to a point in the domain of f or to a point on the graph of f. You will know which meaning is intended from the context. Note: Geometrically, at a critical point where f ' (p) = 0, the line tangent to the graph of f at p is horizontal. At a critical point where f ' (p) is undefined, there is no horizontal tangent to the
2 ReviewUsingDerivatives.nb graph- there's either a vertical tangent or no tangent at all. (For example, x = 0 is a critical point for the absolute value function f(x) = x.) However, most of the functions we will work with will be differentiable everywhere, and therefore most of our critical points will be of the f ' (p) = 0 variety. The critical points divide the domain of f into intervals on which the sign of the derivative remains the same, either positive or negative. Therefore, if f is defined on the interval between two successive critical points, its graph cannot change direction on that interval; it is either incresing or decreasing. Therefore, we have the following result: Theorem: If a continuous function f has a local maximum or minimum at p, and if p is not an endpoint of the domain, then p is a critical point. Warning: The sign of f ' does not have to change at a critical point. In other words, not every critical point is a local maximum or local minimum! Testing for Local Maxima and Minima If f ' has different signs on either side of a critical point, with f ' (p) = 0, then the graph changes direction at p and looks like one of those pictured below. f decreasing f'<0 Local min f'hpl = 0 f increasing f'>0 p Local max f'hpl = 0 f increasing f'>0 f decreasing f'<0 p
3 ReviewUsingDerivatives.nb 3 These graphs suggest the following criterion: The First-Derivative Test for Local Maxima and Minima Suppose p is a critical point of a continuous function f. æ If f ' changes from negative to positive at p, then f has a local minimum at p. æ If f ' changes from positive to negative at p, then f has a local maximum at p. There is another method for finding the local maxima and local minima. This method involves knowing the concavity of a function at the critical points. Notice on the first graph above that the graph is concave up at the critical point and on the second graph the graph is concave down at the critical point. This suggests the following: The Second Derivative Test for Local Maxima and Minima æ If f ' (p) = 0 and f ''(p) > 0 then f has a local minimum at p. æ If f ' (p) = 0 and f ''(p) < 0 then f has a local maximum at p. æ If f ' (p) = 0 and f ''(p) = 0 then the test tells us nothing. As you can see the second-derivative does not always tell us what we need to know. If we get zero for the second derivative at a critical point then we must fall back on the first derivative test. However, the second-derivative test is a much shorter test and it is usually worthwhile to attempt it first. We have also looked at points on the graph of f where the concavity changes. We now define this point and determine a method for locating it analytically. Concavity and Inflection Points Definition: A point at which the graph of a function changes concavity is called an inflection point of f. Note: The words "inflection point" can refer either to a point in the domain of f (just the x
4 ReviewUsingDerivatives.nb 4 value) or to a point on the graph of f (the x and y values listed as coordinates). The context of the problem will tell you which is meant. Since the concavity changes at an inflection point, the sign of f '' changes there. It is positive on one side of the inflection point, and negative on the other; so at the inflection point, f '' is zero or undefined. So to find the points of inflection we first find the second derivative, set it equal to zero, and then construct a sign chart. Example: Given f(x) = x 3 - ÅÅÅÅ 3 x find the intervals where f(x) is increasing and decreasing, the intervals where f(x) is concave up and concave down, the coordinates of any local extrema, and the coordinates of any inflection points. Solution: We will need both the first and second derivatives of f(x) so we start there. f ' (x) = 3 x - 3 x and f '' (x) = 6x - 3 We now find the critical point of f(x) by setting the first derivative equal to zero and solving for x. 3 x - 3 x = 0 3x(x - ) = 0 x = 0 or x = So the critical points are 0 and. We now set up a sign chart. f inc. dec. inc. f' We can see from the sign chart that there must be a local max at x = 0 and a local min at x =. Plugging these x-values into f(x) will give us the maximum and minimum values (remember value means y)
5 ReviewUsingDerivatives.nb 5 f(0) = 0 so the local maximum value is 0. f() = - ÅÅÅÅ so the local minimum value is - ÅÅÅÅ. The intervals where f is increasing are (-, 0) and (, ) and the interval where f is decreasing is (0, ). Now let's find the point of inflection. Since we know this is where the second derivative changes signs we first find where the second derivative is zero and then set up a sign chart. 6x - 3 = 0 3(x - ) = 0 x - = 0 x = x = / f conc. down conc. up f'' ÅÅÅÅÅ So we can see that the point of inflection occurs at x = /. The y-value that corresponds to this x-value is -/4 so the coordinates of the inflection point to the graph of f are (/, -/4). The interval where the graph of f is concave down is (-, /) and the interval where the graph of f is concave up is (/, ). See the graph below for the results of our investigation (the first graph shows the local maximum and minimum and the second graph shows the point of inflection).
6 ReviewUsingDerivatives.nb 6 fhxl local maximum H0, 0L H, - ÅÅÅÅÅ L local minimum x fhxl inflection point - - x - H ÅÅÅÅÅ,- ÅÅÅÅÅ 4 L
7 ReviewUsingDerivatives.nb 7
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