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The First and Second Derivatives

The Meaning of the First Derivative

At the end of the last lecture, we knew how to diﬀerentiate any polynomial function. Polynomial functions are the ﬁrst functions we studied for which we did not talk about the shape of their graphs in detail. To rectify this situation, in today’s lecture, we are going to formally discuss the information that the ﬁrst and second derivatives give us about the shape of the graph of a function. The ﬁrst derivative of the function f (x), which we write as f (x) or as df , is the slope of the tangent line dx to the function at the point x. To put this in non-graphical terms, the ﬁrst derivative tells us how whether a function ...view middle of the document...

The Meaning of the Second Derivative

The second derivative of a function is the derivative of the derivative of that function. We write it as f (x) or 2 as d f . While the ﬁrst derivative can tell us if the function is increasing or decreasing, the second derivative dx2 tells us if the ﬁrst derivative is increasing or decreasing. If the second derivative is positive, then the ﬁrst derivative is increasing, so that the slope of the tangent line to the function is increasing as x increases. We see this phenomenon graphically as the curve of the graph being concave up, that is, shaped like a parabola open upward. Likewise, if the second derivative is negative, then the ﬁrst derivative is decreasing, so that the slope of the tangent line to the function is decreasing as x increases. Graphically, we see this as the curve of the graph being concave down, that is, shaped like a parabola open downward. At the points where the second derivative is zero, we do not learn anything about the shape of the graph: it may be concave up or concave down, or it may be changing from concave up to concave down or changing from concave down to concave up. So, to summarize: • if • if • if

d2 f dx2 (p) d2 f dx2 (p) d f dx2 (p)

2

> 0 at x = p, then f (x) is concave up at x = p. < 0 at x = p, then f (x) is concave down at x = p. = 0 at x = p, then we do not know anything new about the behavior of f (x) at x = p.

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For an example of ﬁnding and using the second derivative of a function, take f (x) = 3x3 − 6x2 + 2x − 1 as above. Then f (x) = 9x2 − 12x + 2, and f (x) = 18x − 12. So at x = 0, the second derivative of f (x) is −12, so we know that the graph of f (x) is concave down at x = 0. Likewise, at x = 1, the second derivative of f (x) is f (1) = 18 · 1 − 12 = 18 − 12 = 6, so the graph of f (x) is concave up at x = 1.

Critical Points and the Second Derivative Test

We learned before that, when x is a critical point of the function f (x), we do not learn anything new about the function at that point: it could increasing, decreasing, a local maximum, or a local minimum. We can often use the second derivative of the function, however, to ﬁnd out when x is a local maximum or a local minimum. Recall that x is a critical point of a function when the slope of the function is zero at that point. Now, suppose that x is a critical point and the second derivative of the function at that point is positive. The positive second derivative at x tells us that the derivative of f (x) is increasing at that point and, graphically, that the curve of the graph is concave up at that point. The only way to sketch the graph of a function at a point where the slope of the function is zero but the graph is concave up is to make that point a local minimum of the function. So, if x is a critical point of f (x) and the second derivative of f (x) is positive, then x is a local minimum of f (x). Likewise, if x is a critical point of f (x) and the second derivative of f (x) is...

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