My English lesson DEFINITION The primitive of the function f(x) is a function, denoted by P(f(x)), whose derivative is equal to the original function f(x) itself: D(P(f(x)). Thus the primitives of a function, if they exist, are infinite: we therefore refer to the set of primitive functions of a given function. example O In each of the following cases, a function y = f(x) is assigned (on the left) through its graph. To its right, the graphs of two other functions are drawn, one of which is the primitive P(f) of the given function. By analysing the points of f(x) with the x-axis and the intervals in which it is positive or negative, determine which of the two graphs to the right is the graph of its primitive function. y y O y O x 1 a. O x 1 I. II. y y x 1 y 3 1 O 2 1 1 1 1 2 2 1 1 O x 2 1 1 2 2 1 O 1 x 1 2 x 3 b. I. II. Knowing from the definition that D (P(f(x))) = f(x), the exercise requests to determine a primitive by knowing its derivative. As we have just seen for the derivative function, from the positive / negative sign of the given function we can deduce the increase or decrease of the primitive; from the points of intersection with the x-axis of the function we can go back to the stationary points of the primitive. Graph a: the function is negative for x 0 P (f(x)) increasing Q the function has no intersections with the x-axis P (f(x)) has no stationary points Q The graph which has such characteristics is II. Graph b: Q the function is negative for x 2 P (f(x)) decreasing in these intervals Q the function is positive in 2 < x < 1 and for 1 < x < 2 P (f(x)) increasing in these intervals Q the function intersects the x-axis at x = 2 P (f(x)) has stationary points at x = 2 The graph which has such characteristics is I. 229

My English lesson