검증된 단계별 안내
05:44
04:50
06:22Motion Along a Coordinate Line
Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.
c. When, if ever, during the interval does the body change direction?
s = 25/(t + 5), −4 ≤ t ≤ 0
Analyzing Motion Using Graphs
[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:
d. When does it speed up and slow down?
s = t³ - 6t² + 7t, 0 ≤ t ≤ 4
Differentiability and Continuity on an Interval
Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be
c. neither continuous nor differentiable?
Give reasons for your answers.
Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.
d. How is dr/dt related to dh/dt if S is constant?
By computing the first few derivatives and looking for a pattern, find the following derivatives.
c. d⁷³/dx⁷³ (x sin x)
Theory and Examples
In Exercises 51–54,
d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)
y = x³/3
