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| Section | Objectives |
|---|---|
| Topic 1: Validity of Models | - Examine utility - Determine validity - Determine fit |
| Topic 2: Exponential Functions | - Interpret inputs and outputs for exponential functions - Interpret rates of change for exponential functions - Interpret concavity for exponential functions |
| Topic 3: Graphical Depictions | - Interpret concavity for situations - Interpret inputs and outputs for situations - Interpret maximum and minimum for situations - Interpret asymptotes for situations - Interpret rates of change for situations |
| Topic 4: Functions and Algebra of Functions | - Derive conclusions based on notation - Derive conclusions based on graphs - Derive conclusions based on data |
| Topic 5: Linear Functions | - Interpret rates of change for linear functions - Interpret inputs and outputs of linear functions |
| Topic 6: Polynomial Functions | - Interpret concavity for polynomial functions - Interpret rates of change for polynomial functions - Interpret inputs and outputs for polynomial functions |
| Topic 7: Logistic Functions | - Interpret inputs and outputs for logistic functions - Interpret rates of change for logistic functions - Interpret asymptotes for logistic functions |
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問題 #36
Consider the graph of h(t) shown. The function represents the height, h, in feet, of a ball t seconds after being launched.
What is the point, if any, at which the concavity changes from concave down to concave up?
答案：A
解題說明：
The graph represents the height of a launched ball over time. Such height functions are typically quadratic because gravity causes the ball to rise, slow down, reach a maximum height, and then fall. The graph shown is a downward-opening parabola. A downward-opening parabola is concave down for its entire domain.
Concavity changes only if the curve switches from bending downward to bending upward, or from bending upward to bending downward. Since this graph bends downward the entire time, there is no point where the concavity changes from concave down to concave up. The vertex is a maximum point, but it is not a change in concavity. Therefore, the correct statement is that the graph is always concave down.
問題 #37
The graph shows the value, O, in dollars, of an investment account over time.
What represents the value of the account after 8 years?
答案：A
解題說明：
The graph shows the total value of an investment account as time increases.
The horizontal axis represents:
" Time in years "
The vertical axis represents:
" Total value in dollars "
We need to find the value of the account after:
8 " years "
From the graph, when t=8, the curve is slightly above $2,200. The closest answer choice is:
$2,221.12
So the value of the account after 8 years is approximately:
$2,221.12
Therefore, the correct answer is:
# ( " D " )
問題 #38
The exponential function
f(t)=2900(1.13)
t
represents the size of a bacteria population, where t is the time in hours.
How many bacteria are in the population when t=18?
答案：B
解題說明：
This is an exponential growth problem because the base 1.13 is greater than 1. The function starts with 2,900 bacteria and grows by a factor of 1.13 each hour, which represents a 13% hourly increase. To find the population after 18 hours, substitute t=18 into the function: f(18)=2900(1.13)
18
Evaluating the exponent and multiplying by 2900 gives approximately 26,170.38. Since bacteria are counted as whole organisms, this rounds to about 26,170 bacteria. The other answer choices do not match the exponential model at t=18. Therefore, the correct answer is C.
問題 #39
The graphed function v(t) represents the number of vehicles, v, stopped at a toll booth t hours after 6:00 a.m.
The coordinates of points A and B are (1,49.4) and (4,45.8), respectively.
What is the average rate of change of the number of vehicles from point A to point B?
答案：D
解題說明：
The average rate of change measures how much the output changes per one unit of input. Here, the input is time in hours after 6:00 a.m., and the output is the number of vehicles stopped at the toll booth. The two given points are A=(1,49.4) and B=(4,45.8). Use the average rate of change formula: (y
2
#y
1
)/(x
2
#x
1
). Substituting the coordinates gives (45.8#49.4)/(4#1). The numerator is #3.6, and the denominator is 3, so the average rate of change is #1.2. The negative sign means the number of vehicles decreased over the interval.
問題 #40
A team was assembled at a manufacturing plant in order to boost productivity. The team was tasked with producing as many widgets each day as possible. The results are shown in the graph.
What is the correct interpretation of the average rate of change from day 6 to day 14?
答案：A
解題說明：
The graph models the number of widgets the team can produce each day.
The horizontal axis represents:
" days "
The vertical axis represents:
" widgets "
The graph gives two labeled points:
(6#3.4)
and
(14#23)
The average rate of change is:
" change in widgets " / " change in days "
Substitute the values:
(23-3.4)/(14-6)
=19.6/8
=2.45
Rounded to the nearest tenth:
2.5
This means the team increased production by an average of approximately:
2.5 " widgets each day "
Therefore, the correct answer is:
# ( " B " )
問題 #41
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