What Does f’ Say About f? Many of the applications of calculus depend on our ability to deduce facts about a function f from information concerning its derivatives. Because f’(x) represents the slope of the curve y = f(x) at the point (x, f(x)), it tells us the direction in which the curve proceeds at each point.

What Does f’ Say About f? In particular, to see how the derivative of f can tell us where a function is increasing or decreasing, look at figure 1 on page 175. The general results are as follows: 1.If f’(x) > 0 on an interval, then f is increasing on that interval. 2.If f’(x) < 0 on an interval, then f is decreasing on that interval.

What Does f’ Say About f? Example 1: a)If it is known that the graph of the derivative f’ of a function is as shown in figure 2, what can we say about f? b)If it is known that f(0) = 0, sketch a possible graph of f.

What Does f’ Say About f? The function f from example 1 has a local maximum at -1 because near x = -1 the values of f(x) are at least as big as the neighboring values. **Note that f’(x) is positive to the left of -1 and negative just to the right of -1. Similarly, f has a local minimum at 1, where the derivative changes from negative to positive.

What Does f’ Say About f? Let’s see how the sign of f”(x) affects the appearance of f. f” tells us what if the slopes of the tangent lines of the curve are increasing or decreasing.

What Does f’ Say About f? Look at figure 4 on page 176. Figure 4 shows the graph of a function whose slopes of the tangent lines increase from left to right. As a result the curve bends upward. Such a curve is called concave upward.

What Does f’ Say About f? Look at figure 5. Figure 5 has slopes of tangent lines that decrease from left to right. This makes the curve bend downward. This curve is called concave downward.

What Does f’ Say About f? In general: 1.If f”(x) > 0 on an interval, the f is concave upward on that interval. 2.If f”(x) < 0 on an interval, then f is concave downward on that interval.

What Does f’ Say About f? Example 2: Sketch a possible graph of a function f that satisfies the following conditions: i) f’(x) > 0 on (-∞, 1), f’(x) < 0 on (1, ∞) ii) f”(x) > 0 on (-∞, -2) and (2, ∞), f”(x) < 0 on (-2, 2) iii),

What Does f’ Say About f? Antiderivatives In many problems in mathematics and its applications, we are given a function f and we are required to find a function F whose derivative is f. If such a function F exists, we call it an antiderivative of f.

What Does f’ Say About f? Antiderivatives In other words, an antiderivative of f is a function F such that F’ = f. Look at example 4 on page 177.

What Does f’ Say About f? ASSIGNMENT P. 178 (2, 3, 6, 16, 20, 27)