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04:27A sliding ladder
A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.
b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?
Quadratic approximations
b. Find the quadratic approximation to f(x) = 1/(1 − x) at x = 0.
In Exercises 47 and 48, find an equation for
(b) the horizontal tangent line to the curve at Q.
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Fruit flies (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.
b. During what days does the population seem to be increasing fastest? Slowest?
The folium of Descartes (See Figure 3.27)
b. At what point other than the origin does the folium have a horizontal tangent line?
Hauling 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.
b. At what rate is the angle θ changing at this instant (see the figure)?
