( ) - 4(x -3) ( ) 3 (2x -3) - (2x +12) ( x -1) 2 x -1) 2 (3x -1) - 2(x -1) Section 1: Algebra Review. Welcome to AP Calculus!

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1 Welcome to AP Calculus! Successful Calculus students must have a strong foundation in algebra and trigonometry. The following packet was designed to help you review your algebra skills in preparation for Calculus. Please do these problems thoughtfully (on separate paper) and check your answers (they are at the end of this document). If you find you need more practice on particular topics, find extra practice online. The stronger your algebra skills are when you enter Calculus, the better off you will be throughout the year. Section 1: Algebra Review 1. Solve for y: xy -x - = y. Solve for y: ln(y -1)-ln x = x +ln x. Factor completely: 8x Factor completely: x -9x 1 +6x-1 ( ) - 4(x -) 5. Factor completely: x x - 6. Simplify 5( x + h) -5x h 1 7. Simplify + 4 x x x x + ( x + )(x - 4) = 0 Simplify the following. Leave answers in fraction form. ( ( ) (x -) - (x +1) ( x -1) x -1) (x -1) - (x -1) ( x -1) 4 9. x

2 Simplify by rationalizing the numerator: - x x x + h - h x Solve each over the real numbers. Write your answer in interval notation. 1. x x +1 > x x 10 0 Section : Exponentials and Logarithms Solve each equation over the real numbers. Round all decimal answers to the nearest thousandth x = 8 x x +5 = e x +4 = log x = 0. ln x = 4 1. log 6+ log x = log 54. Expand using the laws of logs: log x y z 4 w. Write as 1 log statement: ln Simplify using the fact that a log a x = x x - 1 ln y + ln z - 4ln w

3 Example: Simplify Example: Simplify 5 log 5 5 = 5 e 1 ln = e ln 1 = 1 ( )t Example: e ln 1 4 = e ln 1 4 using the property of exponents: in reverse æ = e ln ( 1 4) ö ç using the property of logs è ø t t ( ) t a b = æ 1 ( using the fact that 4) ö ç a log a x = x è ø = 1 using the property of exponents: ( ) c = a bc alogb = logb a ( 4) t a b ( ) c = a bc 4. e t ln5 5. e 1 Section : Trig review In Calculus, trigonometry shows up randomly so there are things you need to remember. You need to know the Pythagorean Identities, your double angle and power reducing identities and You MUST know your special values. 6. List the Pythagorean identities : 1... ( )t 7. Double angle identities: cosx = or or sinx = 10 ln 4 8. Power reducing identities: sin x = cos x = 9. Leave all answers in radical form: cos( ) sin( ) sec( ) sin( ) 7 a. b. c. d cos ( ) cot( ) cos( ) sin( ) 7 5 e. f. g. h. 4 ( ) 5 i. cos j. sin -1 k. l. 6 ( ) cos -1-1 ( ) ( ) sin -1 -

4 Section 4: Graphing review. 0. Determine algebraically whether each of the following is even or odd. ( ) = 4x ( ) = 4x a. f x b. f x c. x - x - f ( x) = 4x x 4-1. You should know the graphs of the basic functions. Graph the following: ( ) = x f ( x) = x f ( x) = x a. f x b. c. ( ) = sin x f ( x) = cos x f ( x) = tan x d. f x e. f. ( ) = tan -1 x f (x) = e x f ( x) = ln x g. f x h. i. Section 5: linear functions. Write the equation of the line through (, 5). Write the equation of the line through with slope. Write your answer in (, -) and (-4, 1) in point-slope form. - point-slope form. 4. The slope of a line is - 1 and the line passes through the points (, 5) and (-4, y). Find y.

5 Section 6: Average Rate of Change 5. Find the average speed of a car that has traveled 50 miles in 7 hours. 6. Suppose f ( 1) = and the average rat of change of f between 1 and 5 is. Find f ( 5). Section 7: Parametric Equations Sketch the graph, indicating orientation. 7. x = t -5, y = 4t +, -1 t 4 8. x = 4sint, y = cost, 0 t p Section 8: Inverse Functions. 9. Algebraically find the inverse of y = x If ( a,b) lies on the graph of y = f (x), what points lies on the graph of y = f -1 (x) 41. Given the function f ( x) = x + x +1, find f -1 ( 4). Section 9: Graphing Calculator Skills You should be able to do the following using the appropriate functions on your graphing calculator. Find an appropriate viewing rectangle for any function so that you can see a complete graph. Find x-intercepts (zeros) of a function. These are points. Find the intersection of functions These are points. Find the relative maximum or minimum value of a function. Remember that relative max/min value is the y-coordinate of the point. 4. Given y = x -15x + a) Find the appropriate viewing window to see a complete graph. b) Find the x-intercepts, accurate to decimal places. c) Find the relative maximum and minimum values, accurate to decimal places 4. Find the coordinates of the intersection points for the functions f ( x) = x +, g(x) = -x - x +7

6 Section 1: Algebra Review 1. y = x+ x. y = x e x + 1. y = (x + )(4x 6x + 9) 4. y = x(x + 6 x ) 5. y = (x )(x + )(x ) 6. 10x + 5h x+4 7. x(x 1) Answers 8. x = 9. (x 1) (x 7x 9) 10. x 4x 1 (x 1) x x+h+ x 1. [1,5] 14. (, 5 ) (, ) 15. (, 5) Section : Exponentials and Logarithms log + log x + log y 4 log z 1 log w. ln xz w 4 y 4. 5t t 5. ( ) 10 4 Section : Trig Review 6. sin + cos = 1 ; tan + 1 = sec ; cot + 1 = csc 7. cos x = (cos x) (sin x) or (cos x) 1 or 1 (sin x) sin x = sin x cos x 8. (sin x) 1 cos x = ; (cos x) 1+cos x = 9. (a) 1 (b) 1 (c) (d) 1 (e) 1 (f) 1 (g) 1 (h) (i) (j) π (k) π (l) π 4 0. (a) odd (b) neither (c) even

7 1. a b. c. d. e. f. g.. y 5 = (x ). y + = (x ) or y 1 = (x + 4) 4. y = mph #7 9. y = 1 x (b, a) 41. x = 1 4. (a) x = [ 6,6], y = [ 0,0] (b) (.98,0), (0.1,0), (.805,0) (c) rel max: 4.61, rel min: (.6, 0.6), (1.6, 4.6) #8