The American School of Marrakesh. AP Calculus AB Summer Preparation Packet

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1 The American School of Marrakesh AP Calculus AB Summer Preparation Packet Summer 2016

2 SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents) *B. Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes, grouping) C. Rationalizing (numerator and denominator) *D. Simplifying rational expressions *E. Solving algebraic equations and inequalities (linear, quadratic, higher order using synthetic division, rational, radical, and absolute value equations) F. Simultaneous equations II. Graphing and Functions *A. Lines (intercepts, slopes, write equations using point-slope and slope intercept, parallel, perpendicular, distance and midpoint formulas) B. Conic Sections (circle, parabola, ellipse, and hyperbola) *C. Functions (definition, notation, domain, range, inverse, composition) *D. Basic shapes and transformations of the following functions (absolute value, rational, root, higher order curves, log, ln, exponential, trigonometric. piece-wise, inverse functions) E. Tests for symmetry: odd, even III. Geometry A. Pythagorean Theorem B. Area Formulas (Circle, polygons, surface area of solids) C. Volume formulas D. Similar Triangles * IV. Logarithmic and Exponential Functions *A. Simplify Expressions (Use laws of logarithms and exponents) *B. Solve exponential and logarithmic equations (include ln as well as log) *C. Sketch graphs *D. Inverses * V. Trigonometry **A. Unit Circle (definition of functions, angles in radians and degrees) B. Use of Pythagorean Identities and formulas to simplify expressions and prove identities *C. Solve equations *D. Inverse Trigonometric functions E. Right triangle trigonometry *F. Graphs * A solid working foundation in these areas is very important.

3 Toolkit of Functions Students should know the basic shape of these functions and be able to graph their transformations without the assistance of a calculator. Constant Cubic f(x) = a f(x) = x 3 Identity Square Root f(x) = x f(x) = x Absolute Value Greatest Integer f(x) = x f(x) = [ x ] Reciprocal Exponential f(x) = 1 x f(x) = x a Quadratic f(x) = x 2 Logarithmic f(x) = lnx

4 Trig Functions f( x) sin x f( x) cos x f( x) tan x Polynomial Functions: n n 1 2 A function P is called a polynomial if P( x) a x a x... a x ax a n n Where n is a nonnegative integer and the numbers a 0, a 1, a 2, a n are constants. Even degree Leading coefficient sign Odd degree Leading coefficient sign Positive Negative Positive Negative Number of roots equals the degree of the polynomial. Number of x intercepts is less than or equal to the degree. Number of turns is less than or equal to (degree 1).

5 Formulas and Identities Trig Formulas: Arc Length of a circle: L = r or L = 1 2 Area of a sector of a circle: Area = d r or Area = d 360 r r 2 Solving parts of a triangle: Law of Sines: a b c sin A sin B sin C Law of Cosines: 2 a 2 b 2 c 2bc cos A 2 b 2 a 2 c 2ac cos B 2 c 2 a 2 b 2ab cos C Area of a Triangle: Area = 1 2 bc sina or Area = 1 2 ac sinb or Area = 1 ab sinc 2 Heron's formula : Area = s( s a)( s b)( s c), where s = semi perimeter Ambiguous Case: is acute is obtuse or right Compute: alt adj sin opp adj No triangle opp < alt No triangle opp > adj 1 triangle opp = alt 1 triangle (right) opp>adj 1 triangle alt<opp<adj 2 triangles Does a triangle exist? Yes - when (difference of 2 sides) < (third side) < (Sum of 2 sides)

6 Formulas and Identities, continued Trig Identities: Reciprocal Identities: csc A = 1 sin A sec A 1 cosa cot A = 1 tan A Quotient Identities: Pythagorean Identities: tan A sin A cosa cot A = cos A sin A sin 2 A + cos 2 A = 1 tan 2 A + 1 = sec 2 A 1 + cot 2 A = csc 2 A Sum and Difference Identities: sin(a + B) = sina cosb+cosa sinb cos(a + B) = cosa cosb sina sinb sin(a B) = sina cosb - cosa sinb cos(a B) = cosa cosb + sina sinb tan (A + B) = tan A tan B 1 tan A tan B tan (A B) = tan A tan B 1 tan A tan B Double Angle Identities: sin(2a) = 2sinA cosa tan(2a) = 1 2 tan A tan 2 A cos(2a) = cos 2 A - sin 2 A cos(2a) = 2cos 2 A 1 cos(2a) = 1 2sin 2 A Half Angle Identities: A 1 cos A sin 2 2 A 1 cos cos 2 2 A A 1 cosa tan 2 1 cos A Polar Formulas: x y r x rcos y r sin y x 1 1 tan x 0, tan x 0 y x Geometric Formulas: 1 1 Area of a trapezoid: A h b1 b 2 Area of a triangle: A bh Area of an equilateral triangle: A s 4 2 Area if a circle: A r Circumference of a circle: C 2 r or C d

7 Unit Circle Degrees and Radians Place degree measures in the circles. tan = Place radian measure in the squares. cot = Place (cos, sin ) in parenthesis outside the square. csc = Place tan outside the parenthesis. sec =

8 AP Calculus Summer Assignment Part 1 of 3: Prerequisites for Calculus Name: Due Date: June 25 th, 2014

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10 "Skill C

11 "Skill D

12 "Skill E

13 "Skill F "Skill G

14 "Skill H

15 "Skill I

16 "Skill J

17 "Skill K

18 "Skill L

19 "Skill M

20 "Skill N

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22 "Skill P

23 "Skill Q

24 "Skill RS "Skill T

25 "Skill U

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27 "Skill V

28 "Skill W

29 "Skill X

30 "Skill Y

31 "Skill Z

32 "Skill AA

33 "Skill BB

34 "Skill CC

35 "Skill DD

36 "Skill EE

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