Chapter 2 - Derivatives - 2.7 Rates of Change in the Natural and Social Sciences - 2.7 Exercises - Page 178: 5

(a) From the figure, the velocity $v$ is positive on the interval (0, 2) and negative on the interval (2,3). The acceleration $a$ is positive (negative) when the slope of the tangent line is positive (negative), so the acceleration is positive on the interval (0,1), and negative on the interval (1,3). The particle is speeding up when $v$ and $a$ have the same sign, that is, on the interval (0,1) when $v$ $\gt$ 0 and $a$ $\gt$ 0, and on the interval (2,3) when $v$ $\lt$ 0 and $a$ $\lt$ 0. The particle is slowing down when $v$ and $a$ have opposite signs, that is, on the interval (1,2) when $v$ $\gt$ 0 and $a$ $\lt$ 0 (b) $v$ $\gt$ 0 on (0,3) and $v$ $\lt$ 0 on (3,4) $a$ $\gt$ 0 on (1,2) and $a$ $\lt$ 0 on (0,1) and (2,4) The particle is speeding up on (1,2) [$v$ $\gt$ 0, $a$ $\gt$ 0] and (3,4) [$v$ $\lt$ 0, $a$ $\lt$ 0] The particle is speeding down on (0,1) and (2,3) [$v$ $\gt$ 0, $a$ $\lt$ 0]

(a) From the figure, the velocity $v$ is positive on the interval (0, 2) and negative on the interval (2,3). The acceleration $a$ is positive (negative) when the slope of the tangent line is positive (negative), so the acceleration is positive on the interval (0,1), and negative on the interval (1,3). The particle is speeding up when $v$ and $a$ have the same sign, that is, on the interval (0,1) when $v$ $\gt$ 0 and $a$ $\gt$ 0, and on the interval (2,3) when $v$ $\lt$ 0 and $a$ $\lt$ 0. The particle is slowing down when $v$ and $a$ have opposite signs, that is, on the interval (1,2) when $v$ $\gt$ 0 and $a$ $\lt$ 0 (b) $v$ $\gt$ 0 on (0,3) and $v$ $\lt$ 0 on (3,4) $a$ $\gt$ 0 on (1,2) and $a$ $\lt$ 0 on (0,1) and (2,4) The particle is speeding up on (1,2) [$v$ $\gt$ 0, $a$ $\gt$ 0] and (3,4) [$v$ $\lt$ 0, $a$ $\lt$ 0] The particle is speeding down on (0,1) and (2,3) [$v$ $\gt$ 0, $a$ $\lt$ 0]