Lawrence Woodmere Academy. AP Calculus AB

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1 Lawrence Woodmere Academy AP Calculus AB Dear AP Calculus AB Student, Welcome to the fun and exciting world of AP Calculus AB. In the upcoming school year, we will be using the concepts that you previously learned in integrated algebra, geometry, algebra II, trigonometry and pre-calculus to expand your knowledge into the world Calculus. To be able to move forward in AB Calculus, you must have a strong foundation in trigonometry, concepts involving functions, and be able to create models from word problems. This summer assignment is designed to allow you to continue to practice these skills and concepts throughout the time that school is not in session. The AP Calculus AB Summer Assignment packet will not require a lot of time, but it is lengthy enough that you will want to manage your time appropriately. The whole assignment should not be completed at the end of this school year, but should be worked on all summer to keep the material fresh in your mind. As AP Calculus AB students, you will need to be able to manage your time appropriately. This summer assignment is composed of two sections that review the old material from your years in high school. The first section is a review of trigonometry. Make sure that you make yourself familiar with all the exact values for trigonometric functions on the interval [0,π]. There is a NO calculator section on the AP and you must have these values memorized. The second section focuses on a review of functions and modeling. You are expected to answer all questions on a separate sheet of paper and hand in the assignment on the first day of school. All work must be shown for each of the questions and you must provide explanations for all multiple choice questions. It s not sufficient enough to get the right answer, but you must be able to explain your answer as well. The assignment will be graded for completion and effort. You should also get the required supplies for the course which includes graph paper, notebook, pencil, and a TI-83 or TI-84. You may also want to go to a bookstore this summer and pick up an AP preparation guide for the AP exam (I recommend Baron s). If you have any questions, do not hesitate to me over the summer at dearley@lawrencewoodmere.org Again, welcome to AP Calculus AB! Good luck and I look forward to seeing you in September. Sincerely, Mrs. Danielle Earley dearley@lawrencewoodmere.org

2 TRIGONOMETRY:. Use the diagram on the right to find the exact values of the following: a. tan A cos B. Which of the following is equal to cscθ? a. c. d. sinθ cosθ tanθ secθ 3. Find the exact value of sec Find the reference angle for an angle measuring Which of the following graphs represents the equation y = + sin x over a one-period interval? a. c. d. 6. Change 5 to radian measure. 7. Change 5π 8 radians to degree measure.

3 8. Find the exact value of each of the following: a. sin π 3 tan π c. cos π 4 d. sec π e. csc 3π f. cot π 6 3 g. sin h. cos i. tan 3 **You must know the exact values of trigonometric functions by heart. There is a portion of the AP Calculus exam that does NOT allow calculators and these values are necessary in that section. You should have them memorized by the beginning of the school year. FUNCTIONS:. Given that f ( x )= x + x 5, find f ( 3).. For f(x) = x + 5 and g(x) = 3x +, find the domain of a. (, ) (, 3 ) ( 3, ) c. (, 5) (5, ) d. (, ) (, 5) (5, ) 3 3 f g. 3. Find the domain for f(x) = x + 5 a. (, 5 ) ( 5, ) (, 5 ) (5, ) c. (, 5 ] d. [ 5, ) 4. For f(x) = x 5 and g(x) = x 6 find f(g(x)) a. 4x 0x + 9 x c. x 7 d. x 5. Write the equation of the vertical asymptote of f(x) = 4 x +x. 6. The graph of the equation x +3x 0 x is a line with a hole in it. At what point does the hole occur?

4 7. Identify the possible formula for the graph shown on the right. a. y = (t + )(t 4)(t 3) y = (t )(t 4)(t + 3) c. y = (t + )(t + 4)(t + 3) d. y = (t + )(t + 4)(t 3) 8. Which formula best matches the graph shown on the right. a. y = (x 3) + y = (x ) c. y = (x 3 + 3) d. y = (x 3) For the function g(x) = 8 + 4x + 3x x 3, what is the leading coefficient? 0. Find the roots of the following equation. Give the values in exact form. x 3 + 4x = 5x. Find the vertex for the following equation. x 8x y + 8 = 0. Find a formula for the inverse of a. f(x) = 4x + 3. f(x) = x 3x+ 3. Suppose $8000 is invested at a 4% interest rate, compounded monthly. How much will the investment be worth after 9 years? 4. Evaluate the logarithm: log 5. Simplify using the rules of logarithms: log 5 + log 3 a. log 4 8 log 5 3 c. log 75 d. log Expand the following as sums and/or differences of simpler logarithmic expressions. ln 3x x (x + ) a. ln 3x + ln x ln(x + ) ln(x + ) ln 3x + ln x c. 3 ln x + ln x ln(x + ) d. ln 3x + ln x ln(x + )

5 7. Convert to an exponential equation: log x = 5 a. e 5 = x 0 5 = x c. 5 0 = x d. x 5 = 0 8. The population of bacterial culture doubled in 8 hours. What was the exponential growth rate? a. 3.8% 4.% c. 5.5% d. 8.7% 9. Find the indicated term of the geometric sequence 00, 80, 64,..., a 6 a. 6,384 65,04 5 c. 4,096 5 d Find the sum of the first 36 terms in the arithmetic series: a c d. 34. Find the x and y intercepts for each graph. a. y = x 5 y = x + x c. y = x 6 x d. y = x 3 4x. Find the intersection points of the graphs for the given equations a. x + y = 8 x + y = 6 4x y = 7 x + y = 4 c. x = 3 y y = x d. y = x y = x 3x + 3. If a and h are real numbers, find and simplify f(a), f(a + h), f(a+h) f(a) h a. f(x) = x x + 3 when: f(x) = x

6 4. Divide by using long division. a. (0x 3x ) (4x ) ( x x 3) ( x 5) c. 3 ( x x x ) ( x ) d. (6x 7x 5) (3x 5) 5. Divide by using synthetic division. a. (7x 3x 6) ( x 3) 4 ( x 5x 0) ( x 3) c. (x 3x 8) ( x ) d. 4 3 ( x 6x 6 x ) ( x 5) 6. a. 7. Solve the following equations for y ' a. x + xy = y + 3y y x y +y = 5 y 8. Find the zeroes of the function (algebraically): f(x) = x e x + xe x 9. Triangle ABC has vertices A(0,0), B(4,8), C(0,0) a. Find the coordinates of M, the midpoint of segment AB Find the equation of the line that contains M and is parallel to segment BC c. Find an equation of the line through points C and M. Is this line perpendicular to bisector BC 30. Use points (-, 4) and (6,) a. Find the slope of the line containing these points. Find the length of the segment that connects these two points c. Find the midpoint of the line segment that connects the points

7 3. Find the line that passes through (-,4) and the point of intersection of the lines x + 3y = and x y = Describe how each of the following graphs compare to its base graph a. y = (x + ) 4 y = x c. y = x d. y = 3 x Find the vertex and the axis of symmetry for the parabola y = x + 8x + 5 a. Rewrite the graph in vertex form y = z(x h) + k Suppose this graph is shifted 3 units left and units up rewrite the new graph in both vertex form and standard form. 34. You need a Lear Jet for one day. Knowing that Swissair rents a Lear jet with a pilot for $3000 a day and $.5 per mile, while Air France rents a Lear jet with a pilot for $500 a day and $.0 per mile, find the following a. For each company, write a formula giving the cost as a function of the distance traveled At what Mileage is the price going to be the same for both Swissair and Air France c. What can you conclude from this? 35. In 984, the Fizzy Cola Co. sold 3 million gallons of soda. By 003, the company was selling 7 million gallons of soda. What is the average rate of change in number of gallons of soda per year? 36. The sides of a rectangle a x and 3 x. Express the rectangle s area as a function of x. Express the rectangle s perimeter as a function of x. Why can x not equal? 37. The height and the diameter of a cylinder are equal. Express the volume of the cylinder as a function of its radius. 38. Each leg of an isosceles triangle is twice as long as its base. Express the perimeter of the triangle in terms of the length of the base (b).