Pre-Calculus EOC Review 2016 Name ___________________________

The Exam – 50 questions, multiple choice, paper and pencil.

I. Limits – 8 questions

a. (1) decide if a function is continuous at a point

b. (1) understand continuity in terms of limits

c. (1) find limits of sums, differences, products, and quotients.

d. (4) understand the concept of limit and estimate limits from graphs and tables of values

e. (1) find limits by substitution

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II. Functions – 8 questions

a. (1) Rewrite simple rational expressions in different forms

b. (1) Understand and apply the extreme Value Theorem: If f(x) is continuous over a closed interval, then

f has a maximum and a minimum on the interval.

c. (2) find the inverse of a function

d. (1) Write a function that describes a relationship between two quantities.

e. (1) Understand that rational expressions from a system analogous to the rational expression; add,

subtract, multiply and divide rational expressions.

f. (1) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials

g. (1) Understand and use the Intermediate Value Theorem on a function over a closed interval.

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15. Find all the rational zeros.

16. Find all the rational zeros.

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18. Graph the rational function, find the following: x and y intercepts, HA and VA (asymptotes), where the function is

increasing and/or decreasing. Find any local extrema (max/min points).

19. Graph the rational function, find the following: x and y intercepts, HA and VA (asymptotes), where the function is

increasing and/or decreasing. Find any local extrema (max/min points).

20. Find the inverse of the function.

21. Find the inverse of the function.

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23. Describe how to transform the graph of f into the graph of g.

III. Conics – 3 questions

a. (1) Derive the equation of a circle of given center and radius using the Pythagorean Theorem;

complete the square to find the center and radius of a circle given by an equation.

b. (2) Derive the equations of ellipse and hyperbolas given the foci and directrices

24. Find an equation for the circle.

25. Find the center and radius of the circle.

X2 – 14x + y2 + 12y + 60 = 0

26. Find an equation in standard form for the ellipse that satisfies the given conditions.

27. Find an equation in standard form for the hyperbola that satisfies the given conditions

IV. Trigonometry – 22 problems

a. (3) Prove the Pythagorean identity sin2x + cos2x = 1 and use it to calculate trigonometric ratios.

b. (2) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the

angle: convert between degrees and radians.

c. (3) prove the addition and subtraction formulas, half-angle, and double-angle formulas for sine, cosine

and tangent and sue them to solve problems.

d. (5) Use trigonometric ratios and the Pythagorean Theorem to solve right triangle in applied problems.

e. (1) Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right

and non-right triangles.

f. (1) Use special triangle to determine geometrically values of sine, cosine, tangent for 𝜋

3,

𝜋

4, 𝑎𝑛𝑑

𝜋

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g. (1) Use inverse functions to solve trigonometric equations that arise in modeling contexts.

h. (1) Choose trigonometric functions to model periodic phenomena with specified amplitude,

frequency, and midline.

i. (1) Derive the formula A = ½ ab sin C for the area of a triangle by drawing an auxiliary line from a

vertex perpendicular to the opposite side.

j. (3) Explain how the unit circle in the coordinate plane enables the extension of trigonometric

functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise

around the unit circle.

k. (1) Prove the Laws of Sines and Cosines and use them to solve problems.

28. Prove the identity:

29. Convert from degrees to radians:

30. Convert from radians to degrees:

31. Using your knowledge of the Unit circle find the exact value.

32. Solve for x using the trigonometric ratios.

33. Solve for x using the trigonometric ratios.

34. Solve the problem using trig ratios.

35. Solve the problem using trig ratios.

36. Find the exact value using the sum or difference identities

37. Find the exact value using the sum or difference identities

38. Find all solutions to the equation in the interval [0, 2π) using double angle identities

39. Find all solutions to the equation in the interval [0, 2π) using half angle identities

Evaluate the following without using a calculator by using ratios in a reference triangle.

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For the following two problems use Law of Sines to solve the problem.

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For the following two problems use Law of Cosines to solve.

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48. Find the exact value.

49. Find the exact value.

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V. Vectors – 5 problems

a. (1) solve problems involving velocity and other quantities that can be represented by vectors.

b. (1) Multiply a vector by a scalar.

c. (1) Add and subtract vectors

d. (1) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by

directed line segments and sue appropriate symbols for vectors and their magnitudes.

e. (1) find the components of a vector by subtracting the coordinates of an initial point from the

coordinates of a terminal plane.

VI. Complex/polar numbers – 4 problems

a. (1) Represent complex numbers on the complex plane in a rectangular and polar form and explain why

the rectangular and polar forms of a given complex number represent the same number.

b. (1) Represent addition, subtraction, multiplication, and conjugation of complex number geometrically

on the complex plane; use properties of this representation for computation.

c. (2) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex

numbers.

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