West Essex Regional School District. AP Calculus AB. Summer Packet

Transcription

1 West Esse Regional School District AP Calculus AB Summer Packet Calculus AB Calculus AB covers the equivalent of a one semester college calculus course. Our focus will be on differential and integral calculus of a single variable and applications. The AP eam is given the first week of May. You are epected to take the eam to receive AP credit for the course. Prerequisites Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, eponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions at the numbers 0, π/6, π/4, π/3, π/, and their multiples. 05 The College Board.

2 West Esse High School AP Calculus AB Summer Assignment Goals Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations. Students should understand the meaning of the derivative in terms of a rate of change and local linear approimation, and should be able to use derivatives to solve a variety of problems. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, and should be able to use integrals to solve a variety of problems. Students should understand the relationship between the derivative and the definite integral as epressed in both parts of the Fundamental Theorem of Calculus. Students should be able to communicate mathematics and eplain solutions to problems both verbally and in written sentences. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to help solve problems, eperiment, interpret results, and support conclusions. Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment. The summer assignments are designed to help you review topics from algebra, geometry, and pre-calculus so that when you arrive in September, you are ready to begin with the first main topic in calculus: Limits and Continuity. All assignments will be collected the first day of school. These topics will also be included in the first eam of the year. Please complete ALL work in pencil. If you don t know or remember how to do a problem, there are plenty of resources online, such as Khan Academy, Youtube and Purplemath. It is your responsibility to be prepared for the first day of class. Try to only use a calculator when stated in the instructions. About 60% of the AP Eam does not allow use of a calculator. If decimal answers are given, round to the nearest thousandth. 05 The College Board.

3 West Esse High School AP Calculus AB Summer Assignment NAME DATE PERIOD. Graph each function. Clearly indicate the units on the aes provided. f() = f() = 3 f() = f() = sin f() = cos f() = tan f() = sec f() = f() = log f() = f() = f() = a

4 West Esse High School AP Calculus AB Summer Assignment. Answer the following questions about the indicated functions. Function Domain Range Zeros (find when f() = 0) Symmetry with respect to y-ais or origin Even or Odd function? f( ) = f() or f( ) = f() Is the function periodic? If so, state the period. Is f() a one-toone function? (for each f() only one eists) f() = f() = 3 f() = f() = sin f() = cos f() = tan f() = sec f() = f() = log f() = f() = f() = a

5 West Esse High School AP Calculus AB Summer Assignment 3. Convert interval notation to inequality notation or vice versa. a. [-, 3] b. (-, 0) c d. < < 5 e. -3 < 5 f. [ 3, ) g. h. < 3 4. Given the equation: + y + 4 6y + = 0 a. Solve for y b. Sketch the graph. 5. Create an equation whose graph has intercepts of (-5, 0), (0, 0) and (5, 0) 6. Use a calculator to graph y = a. Identify the intercepts of the graph b. Test for symmetry. 7. Identify the type(s) of symmetry: + y + y = 0 8. Find all the points of intersection of the graphs of y = 6 and y = 4 9. Find the slope and y-intercept for the equation 4 y + 8 = 0

6 West Esse High School AP Calculus AB Summer Assignment, < 0. Let f() = { 3,. Evaluate: a. f( 3) b. f( ) c. f(0) d.f(). Let f() = + + a. Graph f() in your calculator b. Estimate the domain and range from the graph c. Find the coordinates of any intercepts of the graph. Use your calculator to find the zeros of f() = Identify the types of symmetry for 4 y + y = 0 4. Find all points of intersection of the graphs of + 3 y = 3 and + y =. 5. Let f() = and g() = + 3. Find the domain of (f o g)(). 6. Find the zero(s) of the function f() = Given f() = 3 + 5, find f( + ) f().

7 West Esse High School AP Calculus AB Summer Assignment, < 8. Let f() = {. Find f(5)., 9. An open bo is to be made from a rectangular piece of material 9 inches by inches by cutting equal squares from each corner and turning up the sides. Let be the length of each side of the square cut out of each corner. Write the volume of the bo as a function of. 0. In which of the following is y a function of? a. y = 3 9 b. + y = 7 c. y = d. 3 + y = 5 e. = y. Given f() = 3 6, find f(0) f(3). Find g(+ ) g() for g() = Find an equation of a line, in point slope form, that passes through the point (3, ) and is perpendicular to the line determined by the points (8, 9) and (0, 6). 4. For the function f() = and g() = +5, find g(f()) 3

8 West Esse High School AP Calculus AB Summer Assignment 5. Factor and simplify. Epress the answer as a fraction without negative eponents. 4(3 ) 4/3 + 4(3 ) /3 6. Epress as a simple fraction + h h 7. Multiply ( 4/3 3 ) 8. Solve for using the quadratic formula. = 5 9. Find the value(s) of that satisfies the equation. 3 = Write the point-slope form of the equation of the line passing through the (, 4) with slope of Solve for t. d t + r = 5 t

9 3. Solve for. West Esse High School AP Calculus AB Summer Assignment = Find the domain of f. f() = Solve for. e ln (+) = Solve for. e + e = Solve for. (e ) 3 = e e Solve for. e [3 ln ln(3 +4)] = 38. Solve for. 5 5 (+3) = Find the -intercept for the graph of the function. f() = ln

10 West Esse High School AP Calculus AB Summer Assignment 40. Use the properties of logarithms to epand the epression. ln [ 3+( 3 7) 9] 4. Solve for. ln ln( + ) = 4. Find the domain of the function. f() = ln( + ) 43. Factor as indicated. a = ( ) b / = ( ) c. e e + e = e ( ) d. + = ( ) e. 6 = ( ) f. sin + tan = sin ( ) g. +4 = ( )

11 44. Factor as indicated. West Esse High School AP Calculus AB Summer Assignment a. ( ) () ( ) = ( )( ) b. 3( + 4)( + ) + 6( + 4) = 3( + 4)( ) c. + + = + ( ) d. ( 3) 3 ( + ) ( 3) ( + ) = ( 3) ( + )( ) e. ( + ) 3/ ( / ) + ( + ) 5/ ( / ) = ( + ) 3/ ( / )( ) 45. Factor as indicated. a. 3 + = ( )( ) b. 9 = ( )( ) c = ( )( ) d = ( )( ) e = ( )( ) f. e + + e = ( ) g = ( )( )( ) h. sin = ( )( ) 46. Reduce each epression to lowest terms. a b. / 3 ( ) ( ) 3( ) c. 4 ( ) / d. / 6 / 3

12 West Esse High School AP Calculus AB Summer Assignment e. ( ) 3/ f. (sin cos ) sin 47. Factor by grouping. a b c. 5 cos 5 sin + sin + cos d. cos + 4 cos + 4 tan 48. Factor completely. a. 3 7 b c d Simplify a. ( )( 3) ( ) b.

13 West Esse High School AP Calculus AB Summer Assignment c d. e. ) ( 50. Rationalize by multiplying by the conjugate of the denominator a. cos b. c.

14 West Esse High School AP Calculus AB Summer Assignment 5. Use division to factor as indicated a = ( )( ) b = ( + )( ) c = ( )( ) 5. Find asymptotes (vertical, horizontal and slant). a. 8 ) ( f b. 3 3 ) ( f c. 3 ) ( f

15 West Esse High School AP Calculus AB Summer Assignment 53. Evaluate the sine, cosine and tangent of each angle without using a calculator: a. π 4 b. π 6 c. 5π 4 d. π e. 5π 3 f. π 54. Find two solutions of each equation. Give your answers in radians 0 θ π. a. cos θ = b. cos θ = c. sec θ = d. sec θ = e. tan θ = f. tan θ = 3 g. sin θ = 3 h. sin θ = 3

16 55. If csc θ = 3 5 West Esse High School AP Calculus AB Summer Assignment and θ is in the second quadrant, find sec θ. 56. If cot θ = 4 and θ is in the third quadrant, find cos θ Simplify: a. b. cot csc sec tan 3 c. (sin sin ) csc 58. Find all θ in the interval [0, π) that satisfy the equation: a. cos tan tan 0 b. sin sin cot 0 c. cos tan 3 cos 0 d. cot sin cot 0