May 29, 2009
Dear Calculus Student of 2009-2010,
Attached you will find your Summer Review work for AP Calculus. Because we do not have time to review algebra and trigonometry in Calculus, these problems were written to help you review on your own. All of the problems involve concepts that you learned in Algebra II or in Precalculus. I have included some notes and sample problems (with answers) that you may find helpful. You may use any math text, you may work together, and you may get help as needed on the sample problems. Most of the problems should be completed without a calculator, but you may use a calculator to check you work where possible. (Tests in Calculus will usually have sections where calculators are not permitted and sections where calculators are permitted.) The TI-83/84 Series is the calculator that I suggest.
Included in this review work are problems for two BINGO games that we will play on the first two days of school (the BINGO cards will be distributed on the first day of school). There are also separate pages of problems that are to be handed in for a grade. The hand-in section is due onFriday, August 28, 2009. You may hand it in early for a bonus: +3 on Wednesday, August 26 and +1 on Thursday, August 27. There will be a tray in my room to place your work. On Wednesday or Thursday of the first week, we will have a quiz covering this algebra and trig review. On the Algebra BINGO and Trigonometry BINGO problem sheets, you should work any 25 problems (on each sheet) so that you will be ready to make your BINGO cards in class.
We have A LOT of new material to learn next year, so thanks for reviewing all during the summer! It will be a busy (but sometimes fun…) year as we prepare for the AP Calculus Exam in May, so get ready!
Sincerely,
Ms. T. Taylor
Name:DUE: FRIDAY, AUGUST 29, 2008
AP Calculus Summer Review
Show all necessary work; give exact answers unless otherwise indicated. Place a box around your final answers.
SIMPLIFY:
1. 2.3.4.
SOLVE:
5. 6.
7. 8.
9.10.
11. 12.
SOLVE over the interval [0, 2π]; give answers in interval form:
13. 14.
SOLVE for d:
15. 16.
MISCELLANEOUS:
17. Given that f(x) = 1 – 3x and g(x) = , find
a) b) domain of g(x)c) d)
18. Factor: a)b) 2x4 – 32
19. Given . Find f(x) and g(x) such that
20. Evaluate: a) b)
21. Let w = j(x) be the daily quantity of water (in gallons) required by an oak tree of height x feet. Use words to explain the following, including units:
a) What does j(25) = 10 mean?b) What does j-1(5) = 2 mean?
22. Write the equation of a line that goes through the point (-5, -1) and is perpendicular to the line with equation (answer in point-slope form).
WRITE AN EQUATION AND SOLVE.
Sketch a diagram if not given.For angle measures, answer to the nearest degree; for segments answer to the thousandths. (Note: Use a calculator on this section.)
23. A Calculus student has thrown his book onto the roof of a building. He has a 30 ft ladder and the maximum angle of elevation that is safe for the ladder is 68°. If the height of the building is 28 ft, will he be able to reach the book? Explain your answer.
24. To find the distance from a point A to a point B across a creek, Kermit establishes a base line AC that measures 90 m. Miss Piggy measures the angles and finds that. Find the distance from A to B across the creek.
25. To raise money for a convention, the math club sells t-shirts for $8 each and can sell 200 at that price. If they increase the price, they find that for every $1 increase they will sell 16 fewer shirts. What should they charge for a t-shirt to maximize their money?
Calculus Summer Review Notes: Things you should know and/or be able to do…
TRIGONOMETRY:
- Unit circle – have values memorized
- Fundamental trig identities
- Basic trig identities – double angle and sum/difference for sine and cosine
- Solve a trig equation
- Trig inverse functions – know restrictions and how to find inverses
- Graphs of all 6 trig functions and how to translate and stretch them
ALGEBRA:
- Point-slope form of equation of a line
- Factor algebraic expressions
- Know and use the Quadratic Formula
- Solve absolute value inequalities
- Simplify complex fractions
- Solve equations
- Set up and solve word problems, including max/min problems
- Solve systems of linear and non-linear equations
- Odd/even functions
- Compositions of functions
- Simplify radicals (including appropriate use of absolute value) and expressions with exponents
FUNCTIONS:(Know characteristics of parent functions)
- Six trig functions
- Linear and absolute value
- Quadratic and polynomial
- Rational
- Logarithmic and exponential
- For each type of function: be able to determine and use the following to sketch a graph
- x- and y-intercepts
- asymptotes
- domain and range
- symmetry
- transformations
LOGARITHMS AND EXPONENTS
- Natural exponential function
- Compound Interest Formulas
- Properties of logarithms
- Natural logarithmic function
TRANSFORMATIONS FROM PARENT FUNCTIONS
Consider the parent function y = f(x)
New FunctionReplacementResult
y = f(x – h) x x – hgraph of f(x) moves h to the right
y = f(x + h)x x + hgraph of f(x) moves h to the left
y – k = f(x)
y = f(x) + ky y – kgraph of f(x) moves k up
y + k = f(x)
y = f(x) – ky y + kgraph of f(x) moves k down
y = –f(x)f(x) –f(x)graph of f(x) is reflected about the x-axis
y = f(–x)x –xgraph of f(x) is reflected about the y-axis
y = –f(–x)graph of f(x) is reflected about the origin
y = cf(x)f(x) cf(x)vertical stretch/shrink (dilation)
(each old y-value is multiplied by c)
y = f(cx)x cxhorizontal stretch/shrink (dilation)
(each old x-value is multiplied by c)
y = |f(x)|f(x) |f(x)|all y-values of f(x) are made non-negative
y = f(|x|)x |x|left-hand side of function is erased;
points for non-negative x values are maintained and
are also reflected about y-axis
INTRODUCTION TO GRAPHING CALCULATOR
During the AP Calculus course, we will use a graphing calculator (TI-83/84 Series) to verify solutions, explore mathematical ideas, and visualize mathematical principles. Below are some of the basic features that you should be acquainted with on your calculator (Please consult your owner’s manual for detailed explanations).
Equation Editor (a.k.a. “Y=” screen)
Table feature
Creating a viewing window
Xmin = the smallest value of x
Xmax = the largest value of x
Xscl = the number of units per tick mark on the x-axis
Ymin = the smallest value of y
Ymax = the largest value of y
Yscl = the number of units per tick mark on the y-axis
Sample Problems:
- Graphing – NO CALCULATOR (Use calculator to check)
Sketch the graph of each function as accurately as possible on graph paper. State the x-intercept(s), y-intercept, vertex, asymptote(s), and/or an other important point(s) on the graph.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14.
- Simplify
1. 2. 3. 4.
5. 6. 7. 8.
- Solve
1. 2. 3.
4. 5.6.
7. 8. Solve for x:
Answers for Samples:
II. Simplify:
1. 2. 3. 4. 5. 6. 7. 8.
III. Solve:
1. 22. 1, 1, 1, 23. 44. 5. 6. 7. 8.
Algebra BINGO
Work any 25 of the following problems, then write one answer in each square of the grid (include the problem #) to make your BINGO card.
Solve each equation or inequality. Give exact answers.
1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11. 12.
Simplify:
13. 14. 15. 16.
17. 18. 19. 20.
21.
Miscellaneous:
22. Find the vertex of the parabola
23. Factor:
24. Find the distance between the points (-1, 6) and (11, 0)
25. Given; find f(4)
26. Write the equation (point-slope form) for the line with slope ¾ and containing the point
(-6,9)
27. Find the slope of the line perpendicular to the line that passes through (-3, -1) and (4, 7)
28 Find a so that these lines are parallel, given b =
29. Find a so that the remainder is 3 for
30. Given that f(x) = x + 5 and g(x) = x2 – 2, find f(g(x)).
31. Write as one logarithm
32. Given Find f(x) and g(x) so that
33. Solve
Trigonometry BINGO
Work any 25 of the following problems, then write one answer in each square of the grid (include the problem number) to make your BINGO card.
1. sin 120°2.
3. sin 2θ if and θ is in the first quadrant4. θ if cos2 θ= 1 and
5. cos 120°6.
7. cos 2θ if and θ is in the fourth quadrant8.
9. θ if tan θ = 1 and 10.
11. sin2θ + cos2θ12.
13. 14. the period of
15. sin 300°16.
17. the period of18. the period of
19. arcsin(2)20. 1 – 2cos2θ
21. tan2θ + 122.
23. 24.
25. 26. θ if 2sin θ + 1 = 0 and
27. θ if 2cos θ – 1 = 0 and 28. sin 47° (to the nearest thousandth)
29. (to the nearest thousandth)30. sec 151° (to the nearest thousandth)