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Chapter 3 | Derivatives
101. [T] Construct a table of values for h ′( t ) and graph both h ( t ) and h ′( t ) on the same graph. ( Hint: for interior points , estimate both the left limit and right limit and average them. An interior point of an interval I is an element of I which is not an endpoint of I.) 102. [T] The best linear fit to the data is given by H ( t ) =7.229 t −4.905, where H is the height of the rocket (in meters) and t is the time elapsed since takeoff. From this equation, determine H ′( t ). Graph H ( t ) with the given data and, on a separate coordinate plane, graph H ′( t ). 103. [T] The best quadratic fit to the data is given by G ( t ) =1.429 t 2 +0.0857 t −0.1429, where G is the height of the rocket (in meters) and t is the time elapsed since takeoff. From this equation, determine G ′( t ). Graph G ( t ) with the given data and, on a separate coordinate plane, graph G ′( t ). 104. [T] The best cubic fit to the data is given by F ( t ) =0.2037 t 3 +2.956 t 2 −2.705 t +0.4683, where F is the height of the rocket (in m) and t is the time elapsed since take off. From this equation, determine F ′( t ). Graph F ( t ) with the given data and, on a separate coordinate plane, graph F ′( t ). Does the linear, quadratic, or cubic function fit the data best? 105. Using the best linear, quadratic, and cubic fits to the data, determine what H ″( t ), G ″( t ) and F ″( t ) are.What are the physical meanings of H ″( t ), G ″( t ) and F ″( t ), and what are their units?
99. The graph in the following figure models the number of people N ( t ) who have come down with the flu t weeks after its initial outbreak in a town with a population of 50,000 citizens. a. Describe what N ′( t ) represents and how it behaves as t increases. b. What does the derivative tell us about how this town is affected by the flu outbreak?
For the following exercises, use the following table, which shows the height h of the Saturn V rocket for the Apollo 11 mission t seconds after launch. Time (seconds) Height (meters)
0
0
1
2
2
4
3
13
4
25
5
32
100. What is the physical meaning of h ′( t )? What are the units?
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