AP Calculus Summer Review Packet School Year. Name

Transcription

1 AP Calculus Summer Review Packet School Year Name

2 Objectives for AP/CP Calculus Summer Packet I. Solving Equations & Inequalities (Problems # 1-6) Using the properties of equality Solving compound inequalities Solving absolute value inequalities Representing the solution set of an absolute value inequality on a number line Solving quadratics by completing the square Solving quadratics using the quadratic formula II. Factoring and Simplifying Fractions (Problems # 7-8) Rewriting the difference of two fractions as one fraction Factoring a polynomial using greatest common factor III. Working in the Coordinate Plane (Problems # 9-13) Using the distance formula: ( ) ( ) Using the midpoint formula: Identifying the properties of triangles Identifying the slope between two points Writing the equation of a line in slopeintercept form IV. Graphs of Functions (Problems # 14-19) Performing transformations of graphs Finding the intercepts of a function Identifying domain & range of a function Finding the symmetry of a function Solving for points of intersection between two functions V. Modeling (Problems # 0-6) Deriving a linear model from data Using the model to predict future estimates Describing the characteristics of the graph in context with the situation VI. Piecewise Functions (Problems # 7-8) Using the domain rules Finding functional values VII. Rational Functions (Problems # 9-30) Identifying the domain (restrictions) of a rational function Finding the roots of a rational function across its domain VII. Composite Functions & Inverse Functions (Problems # 31-3) Determining the composite of two functions Finding the inverse of a function IX. Logarithms (Problems # 33-36) Evaluating logarithms Solving logarithmic equations X. Trigonometry (Problems # 37-50) Solving right triangles Evaluating trigonometric expressions Solving trigonometric equations Graphing trigonometric expressions with transformations Using trigonometric identities

3 Question Hint 1. Solve for x: x + 5x < 0 Factor and test intervals. Solve for x: -3 5x 1 Dividing by negatives 3. Solve for x: x 1 3 Rewrite as two inequalities 4. Draw a number line graph of x for Determine direction and if the solution to #3. graph includes endpoints 5. Solve by completing the square: Move constant to other side, add new term to both sides x 8x Solve using the quadratic formula: Use correct signs for a, b, c x 5x1 0

4 Question Hint 7. Write as one fraction: Find common denominator 4x x x1 x 8. Factor: Use GCF 7 8 1x 4x 5 x 1 4x 4x 5 Given the points A (4, -5), B (6, -), C (3, 0) 9. Find the distance AB Use distance formula 10. Name the midpoint of AB Use the midpoint formula 11. Is ABC scalene, isosceles, or equilateral? Find length of each side Explain. Question Hint 1. Is ABC acute, right, or obtuse? Sketch graph

5 13. Write the equation in slope-intercept form Perpendicular slopes are for the line perpendicular to line AC through A. negative reciprocals, and start with pointslope form Use the graph to the right for # Graph g( x) f ( x) Reflection 15. Graph h( x) f ( x) Nothing below x-axis

6 Question Hint 16. Find the x- and y-intercepts: Set one variable at a time equal to zero. x 1 y x What is the range of the function in #16? Set of all possible y-values ============================================================================== 18. What type of symmetry does the function If f ( x) f ( x), then y-axis x y x 4 symmetry. If f ( x) f ( x), then origin symmetry. 19. Find all points of intersection of: Use substitution or graphing calculator. y x 4x y x For Problems # 0-3 According to company records, in 1980 the company earned $3. million in earnings and in 1985 earned $4.7 million. Assume a linear model for earnings in millions of dollars, E, versus time in years, t. Let 1980 be year t = Write an equation for E as a function of t. Find slope and y-intercept

7 1. Estimate the company s gross profits in 000. Use equation from #0 y y1. What is the slope of the line? m x x 1 3. What information does the slope convey to Look at what the units the company? would be For Problems #4-6 A large piece of paper is cut in half and one of the resulting pieces is placed on top of the other. The stacked pieces are then cut in half and placed on top of each other. Suppose this procedure is repeated several times. 4. How many pieces are in the stack after the Geometric Progression first cut? Second cut? Fourth cut? 5. Use this pattern to write an equation for the Power rules number of pieces in the stack after x cuts. 6. If the thickness of a piece of ordinary paper is Power rules in., write a function for the thickness of the stack after x cuts.

8 For x 7, f( x) x 3, x x 7. Find f ( 5) Determine which rule 8. Find f (4) Determine which rule 3x 5 For the function f x x 3x4 9. Find the domain Where denominator is not Find the roots (zeroes) Set f(x) = 0 Given the functions f ( x) x 4 and g( x) x Find f g x Substitute 3. Find the inverse of g(x) Switch x and g(x), then solve 33. log5 65 Exponent-Log Relationship 34. ln 4.56 = Use Calculator

9 35. Solve this equation: Properties of log, make 1 log log x3 log 5 x log log log 64 Exponent-Log Relationship Find x in each right triangle shown below: x Use trig or inverse trig functions to find sides or angles 17 8 x Name these values. Note all angles must be in radians. Use the Unit Circle or set up rt. triangle 39. sin = 40. tan = 41. sec = cot 5 = 43. sec( ) = 44. csc( ) = arcsin = 46. arctan(-1) = 47. arcsec(-) =

10 48. Solve for x, 0x : Move 3 to the left side and then sin xsin x 3 factor the trinomial 49. Sketch one period of the graph of Amplitude, period, phase, y cos x 1 and vertical shift 50. Prove the following identity Use Pythagorean or Trig Id. for the left side tan x sin x sin x tan x