Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 221HIGHER-ORDER DERIVATIVES163From (0), we may substitute 9 for x2 + y2,9y” - -Y3AccelerationLet an object move along a coordinate axis according to the equation s =f(t), where s is thecoordinate of the object and t is the time. From Chapter 18, the object’s velocity is given bydsU=--- -f ‘@)dtThe rate at which the velocity changes is called the acceleration a.dv d2sDefinition: a = - dt = - dt2 -f’’(t)-EXAMPLES(a) For an object in free fall, s = so + oo t - 16t2, where s, measured in feet, is positive in the upward direction andt is measured in seconds. Recall that so and oo denote the initial position and velocity; that is, the values of sand o when t = 0. Hence,dsU=-- - UO - 32tdtdoa=-= - 32dtThus, the velocity decreases by 32 feet per second every second. This is sometimes expressed by saying that the(downward) acceleration due to gravity is 32 feet per second per second, which is abbreviated as 32 ft/sec2.(b) An object moves along a straight line according to the equation s = 2t3 - 3t2 + t - 1. Then,doa=-=l2t-6dtIn this case, the acceleration is not constant. Notice that a > 0 when 12t - 6 > 0, or t > 4. This implies (byTheorem 17.3) that the velocity is increasing for t > 4.Solved Problems22.1 Describe all the derivatives (first, second, etc.) of the following functions:1 YA(a) f(x) = ; x4 - 5x - a (b) f(x) = x+l