Unit 11 Prerequisites for Next Year (Calculus) Name Period The following Study Guide is the required INDEPENDENT review for you to work through for your final unit. You WILL have a test that covers this material after AP exams have concluded. REQUIREMENTS: 1. Complete this ENTIRE Study Guide. You should complete all work (on separate paper if necessary). While you may work on any part at any time, your final submission should be in order. This is not only your homework for this unit, but also your guide as to what topics to review (from Algebra 1 through Precalculus). 2. You are to spend class days working on these problems. You may work at home as well, but when in class you should take advantage of being able to ask each other questions as well as myself. You will be allowed to use your mobile devices to look up things that you do not remember. 3. The completed guide is due the day of the test which will be announced later.
AP Calculus & Dual Enrollment Assignment: Prerequisites for Calculus Name:!!
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Skill G Factoring Complex Algebraic Expressions You should be able to factor like it is second nature. This includes problems with more difficult exponents. The following is an example with a difficult GCF that is factorable further. Example Solution Factor: 3 1 1 3 2 2 2 2 2x 2x 6x 6x Factor the GCF, then factor by grouping. 3 1 1 3 2 2 2 2 2x 2x 6x 6x 3 2 3 2 = 2x x x 3x 3 3 2 3 2 = 2x x x 3x 3 3 2 2 = 2x x x 1 3x 1 3 2 2 = 2x x 3x 1 Factor each expression completely. 1. 1 2 3 x 3x 2x 2. 2 9x 3x 3xy y 3. 6 64x 1 4. 5 3 1 2 2 2 15x 2x 24x 5. 1 1 2 3 2 3 2 3 2 5x (2x 3) (3x 2) 8x(2x 3) (3x 2)
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Skill Skill I I Solving Algebraic Inequalities You should be able to solve all types of inequalities (using a number line comparison). Example. Solve: 2 3x 11x 4 Solution. 1. Find all critical values for the related equation. 2 3x 11x 4 2 3x 11x 4 0 3x 1 x 4 0 1 x or x 4 3 2. Set critical values on a number line and check versus the factored form: 3x 1x 4 0? Yes No Yes -5 0 1-4 1/3 3. Write the solution. 1 x 4 or x 3 1 OR x (, 4], 3 *Remember more complex functions might have other critical values (asymptotes, holes, endpoints, etc.) Solve each inequality. Check your answers by graphing. 1. 2 x 2 6x 5x 50 2. 15x 8 3 3. 2 x 4 1
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Skill N Knowledge of General Parent Functions (common) You should have a general idea of the shape of any function based on the characteristics of its parent function. While you might not be sure of specific points, you should know characteristics such as end behaviors, asymptotes, certain critical values, etc. In order to analyze functions graphically (as patterns) without the help of a graphing utility, you need to know the basic shapes of the most general versions of most functions. You should also be able to determine the domain and range of most basic functions. The objective of the Skill N practice set is to sketch the parent functions of the most common functions. Sketch the graph of the parent function indicated. Then give its domain in the space provided. 1. y x 2. y 2 x 3. y x 3 Domain: Domain: Domain: 4. y 4 x 5. y 5 x 6. y x Domain: Domain: Domain:
7. y x 8. y x e 9. y e x Domain: Domain: Domain: 10. y ln(x) 11. 1 1 y 12. y 2 x x Domain: Domain: Domain:
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Skill S UNIT CIRCLE TRIGONOMETRY You should be able to evaluate a trigonometric expression for any given standard angle (or co-terminal angle) in degrees or radians. Remember co-terminal angles are those angles that are positioned in the same place. Thus, they are multiples of 360 degrees or 2π radians apart from each other.
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Skill U Graphing Trigonometric Functions Recall: The sine and cosine are periodic functions with a period of 360 o or 2π radians. The tangent function has a period of 180 o or π radians. For the functions: y asinb(x c) d, y acosb(x c) d, and y atanb(x c) d, Recall: The coefficient of the trig function (a) changes the amplitude of the sine and cosine functions. Recall: The coefficient of the variable (b) changes the period of the trig function. Recall: A constant added to the variable (c) changes the phase (creates a horizontal shift) of the function. Recall: A constant added to the function (d) changes the sinusoidal axis (creates a vertical shift) of the function. Graph each trigonometric function. 1. y sinx 2. y cosx 3. y tanx 4. y 2sinx 2 5. y 3cos(2x) 6. y sin2x 1 2
Skill V 1-6
Skill W Rationalizing Denominators (and Numerators) Recall: To rationalize a denominator with an isolated square root, multiply the fraction by the radical over itself (1). Recall: To rationalize a denominator with a sum or difference including a square root, multiply by one in the form of the conjugate of the denominator over itself. Note: To rationalize a numerator (which is helpful in calculus), perform the same operations as rationalizing a denominator. Example: Rationalize the numerator. 1 2 3 Solution: 1 2 1 2 1 2 1 1 3 1 2 3 3 2 3 3 2 3 3 2 Rationalize the Numerator or Denominator as indicated by the problem. 1. 9 3 3 2. 5 2 12 3. 1 2 2 (denominator) 4. 1 2 1 3 (denominator) 5. 2 3 4 6. 1 1 x 2 (numerator)
Skill X Simplifying (+/-) Rational Expressions Recall: To simplify a single rational expression, factor and cancel common factors. Recall: To add or subtract rational functions, factor, get a common denominator, simplify the numerators, and add the numerators together. Note: You should check the result to determine if the numerator can be factored and so that the fraction can be further simplified. Example: Solution: 1 4 x 1 x 2 1 x 2 4 x 1 x 2 4x 4 x 2 4x 4 5x 6 x 1 x 2 x 2 x 1 (x 2)(x 1) (x 2)(x 1) (x 2)(x 1) (x 2)(x 1) Add or subtract, and/or simplify the expression. 1. (x + 1) 3 (x - 2) + 3(x + 1 ) 2 (x + 1) 4 2. 5 3 x 1 x 1 3. 2x 6 3x 1 x 4 2x 3 x 2x 3 x 6x 5 4. 2 2 6x 5 5 5. 2 2x 3x 2x 6. 2 2 6x 11x 7 x 6x 8 2 2 5x 11x 2 3x 5x 28