...value of x, and for every such value of xa corresponding algebraic value of y. In the present case у is positive when x is positive, and negative when x is negative. To every point on the x-axis there corresponds a point P. The points P will therefore constitute a...

...Thus y = x2 is concave upwards everywhere, since 6 = 2 is positive. For y = x3 we find 6 == 6 x, which is positive when x is positive, and negative when x is negative ; hence y = y? is concave upwards at the right, and concave downwards at the left of the origin. A point at...

...written in the form 4 from which it appears that when x = 0, у = oo and when x = oo, у = 0. Also у is positive when x is positive, and negative when x is negative. Hence the graph must lie entirely in the first and third quadrants. It will be convenient in this case to take the...

...curve is not symmetrical with respect to the origin. 3rd. Solving for y, It is seen that the numerator is positive when x is positive and negative when x is negative, also that the denominator is positive when x is greater than 2 and negative when x is less than 2,...

...curve y = x2 rising and for what values falling? Solution. — Given y = x2. Then ^ = 2x. dx Now 2x is positive when x is positive, and negative when x is negative. Hence the curve is rising when x>0, and falling when x<0. Example 2. — For what values of x is the function...

...curve y = xz rising and for what values falling? Solution. Given y — xí. Then ^ = 2x. dx Then Now 2x is positive when x is positive, and negative when x is negative. Hence the curve is rising when x > 0, and falling when x < 0. The curve is as shown in Fig. 18. Example 2. For...

...differentiable for all values of x, and its differential coefficient is 2#; (vi) its differential coefficient is positive when x is positive and negative when x is negative; (vii) it is integrable between 0 and 1, and in fact between any two values a and b; (viii) the area...

...monotonely increasing function of x. From this, in connection with arc tg 0 = 0, it follows that arc tg x is positive when x is positive and negative when x is negative. 5. Formula for arc tg a + arc tg b. We consider the function xb <p (x) — arc tg x — arc tg +bx'...

...function that assigns to each number x the number (xp if x > 0 0 if x = 0 . -(-x)P if x < 0 Note that xP is positive when x is positive, and negative when x is negative. For any numbers y and z, (yz) P = yP. zp, and = y = I If p = 1/2, the pth-power function is the function...

...sinha; = — sinh( — a;). For sinh ( - x) = \(e~x - e-<-x>) = \ ,(e~x - ex) = —sinh a;. Note. Sinh x is positive when x is positive, and negative when x is negative. For example, if a; is positive, then ex is greater than e~x, since e is greater than 1. Hence \(ex...