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05:25Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.
a. How fast is the boat approaching the dock when 10 ft of rope are out?
Motion 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.
a. Find the body’s displacement and average velocity for the given time interval.
s = (t⁴/4) − t³ + t², 0 ≤ t ≤ 3
Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.
a. f(x) = (1 − x)⁶
Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.
a. How is dA/dt related to dθ/dt if a and b are constant?
a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).
Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
a. Assuming that x, y, and z are differentiable functions of t, how is ds/dt related to dx/dt, dy/dt, and dz/dt?
