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Chapter 4 | Applications of Derivatives
141. A ball is thrown into the air and its position is given by h ( t ) =−4.9 t 2 +60 t +5m. Find the height at which the ball stops ascending. How long after it is thrown does this happen? For the following exercises, consider the production of gold during the California gold rush (1848–1888). The production of gold can be modeled by G ( t ) = (25 t ) ⎛ ⎝ t 2 +16 ⎞ ⎠ , where t is the number of years since the rush began (0≤ t ≤40) and G is ounces of gold produced (in millions). A summary of the data is shown in the following figure.
142. Find when the maximum (local and global) gold production occurred, and the amount of gold produced during that maximum. 143. Find when the minimum (local and global) gold production occurred. What was the amount of gold produced during this minimum? Find the critical points, maxima, and minima for the following piecewise functions.
⎧ ⎩ ⎨ x 2 −4 x 0≤ x ≤1 x 2 −4 1< x ≤2 ⎨ x 2 +1 x ≤1 x 2 −4 x +5 x >1 ⎧ ⎩
144. y =
145. y =
For the following exercises, find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions. 146. y = ax 2 + bx + c , given that a >0 147. y = ( x −1) a , given that a >1 and a is an integer.
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