Original AP Calculus problem:
Similar AP problems:
2006 #1, 2005B #1, and 2004B #1
Algebra I Suggestion:
Let R be the region bounded by the graphs f(x) and g(x).
a)Find the area of R. Explain the reasoning you used to get your answer.
b)Is your area exact or an estimate? Justify your answer.
c)Find the domain and range of R.
d)Find the equation of the line g(x). Express this equation in slope-intercept form and in point-slope form.
Teacher notes:
a) Students should complete the grid and count squares to get the area (Teacher may want to draw in the grid prior to making copies). In calculus the integral is used to find the area between curves.
b) Answer is an estimate because there was no exact formula that was used. The actual area is 4.5.
c) The lines intersect at (0, 0) and (9, 3). Have students use interval notation to get domain [0, 9] and range [0, 3].
d) Use points of intersection (0, 0) and (9, 3).
Original AP Calculus problem:
Similar AP problems:
2008 #2, 2005 #3, and 2001 #2
Algebra I Suggestion:
Distance from the river’s edge (feet) / 0 / 8 / 14 / 22 / 24Depth of the water (feet) / 0 / 7 / 8 / 2 / 0
A scientist measures the depth of Doe River at Picnic Point. The river is 24 feet wide at this location. The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above.
a) Plot the data on a coordinate plane. Is this graph an exact representation of the depth across the entire river? Why or why not?
b) Use the data points to draw trapezoids and find the area under the graph.
Teacher notes:
Students need to be able to read and understand complex situations in word problems.
a) Students should realize that the data does not provide information for the depth 5 feet from the edge, 19 feet from the edge, etc. Even though the given measurements are exact, the data does not represent the depth for the entire river.
b) This is the trapezoidal sum that will be used in calculus to estimate the integral.
Original AP Calculus problem:
Similar AP problems:
2007B #4, 2006 #3, and 2004 #5
Algebra I Suggestion:
a) Find the area between the graph and the x-axis.
b) Determine the domain and range of f(x).
c)Write the equation for each of the three line segments. Write two of them in point-slope form and one in slope-intercept form.
d) What is the value of the function when x = –3, x = 0.5, and x = 2.5? Show the computations that lead to your answer.
f) On what interval(s) is f(x) > 0? On what interval(s) is f(x) < 0?
g) On what interval(s) is f(x) increasing? Decreasing?
h) Describe a situation that the graph could represent. You may use the entire graph or only selected sections. If you choose to not use the entire graph, be sure to explain your reasoning.
Teacher notes:
a) When looking for area between curve and x-axis any area under the x-axis is negative and therefore gets subtracted. Use standard formulas for shapes (trapezoid, triangle, etc.) to find the area.
b) Self explanatory.
c) Self explanatory.
d) Get students used to plugging values into the appropriate part of a multi-part (piece-wise) function.
f) Students should be comfortable with interval notation, [0, 2], and with inequality notation, 0 ≤ x ≤ 2. Students should also be familiar with multiple intervals in an answer – there are two intervals for which f(x) is negative.
g) Students should be comfortable with interval notation.
h) Answers will vary. Common x values include time, distance, weight, etc. If a student wants to use this type of value for the x variable, they need to realize that x cannot be negative so they should only use the right side of the y-axis.
Original AP Calculus problem:
Similar AP problems:
2007 #4, 2006B #2, and 2003 #2 – use interval 0 < x < 3
Algebra I Suggestion:
The graph v(t) is shown above where t represents time in seconds and v(t) represents the velocity, in feet per second, of a particle.
a) Estimate the domain and range of the velocity of the particle.
b) Estimate the intervals for which the velocity of the particle is increasing and decreasing.
c)At what time(s) does the velocity appear to be zero? Justify your answer.
d) At what time(s) does the velocity appear to have a minimum? Justify your answer.
e) Describe the concavity of the graph at t = 1.5 seconds.
Teacher notes:
a) Actual domain is [0, 5] and actual range is [-0.2877, 2.565]. Students can use the given graph to come up with their own estimates.
b) Actual decreasing is on the interval [0, 1.5) or 0 ≤ t < 1.5 and actual increasing is on the interval (1.5, 5] or 1.5 < t ≤ 5. Important concept here is to get students familiar with interval and inequality notation.
c) The graph of velocity crosses the x-axis at t = 1 and t = 2.
d) The minimum occurs at t = 1.5 because this x value has a lower y value than any other point on the graph (later students will need to justify a min as the graph is changing from decreasing to increasing).
e) The graph is concave up at t = 1.5.
Original AP Calculus problem:
Similar AP problems:
2008 #4, 2002B #4, and 2000 #3
Algebra I Suggestion:
The graph of f(x) is shown above.
a) Find the domain and range of f(x).
b) Determine the interval(s) for which f(x) > 0 and for which f(x) < 0.
c) Determine the intervals of increasing and decreasing of f(x).
d) Find the equation of a line that has the same y-intercept as f(x) and shares one the x-intercepts of f(x). You may choose either slope-intercept form or point-slope form for your answer.
Teacher notes:
a) Self explanatory
b) Students should be able to look at a graph and determine where the graph is positive (f(x) > 0) and negative (f(x) < 0) and should be comfortable with interval and inequality notation.
c) Students should be able to identify where a graph is increasing and decreasing as well as use interval and/or inequality notation.
d) Self explanatory.
Original AP Calculus problem:
Similar AP problems:
2005 #1, 2004 #2, and 2003 #1
Algebra I Suggestion:
Let y = f(x) be the curve in the graph above and let l be the line in the graph above.
a) Find the area of S, the region between f(x) and l. Explain your reasoning.
b) Let the point (0, 3) and (3.3899, 1.3051) be the point of intersection of the f(x) and l. Find the equation of l.
Teacher notes:
a) Students can count squares or create geometric shapes to find area of S. Area between curves is always positive so do not subtract area below x-axis.
b) Self explanatory