AP CALCULUS AB SUMMER PACKET

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1 AP CALCULUS AB SUMMER PACKET As I write this I know that you are one of 72 darling students are enrolled in AP Calculus AB for the school year. I anticipate that you signed up for this course because you were told that you needed to take a math course, this was the next course in your mathematics progression and you didn t want to do the really challenging work of Calc BC. Regardless of the circumstances that brought you to calculus, I am glad you are enrolled in this course. This year we will study the equivalence of one semester of college calculus in the span of the school year. My goal is that you master calculus at a level that earns you a 5 on the AP Calculus AB exam. To excel at that level there will be many facts that you must have memorized and we will frequently write out an entire sheet (front and back) of Stuff You MUST Know Cold. I recognize that at this moment you have not learned the calculus on the SYMKC pages, but you will need to memorize all this information so I want you to write it once as part of your summer assignment. In addition to memorizing the SYMKC to be a calculus superstar you must have a conceptual understanding of the course content and you must have superb algebra skills. I will do all I can to ensure that you gain the conceptual understanding of calculus. You must do all you can to have superb algebra skills. Most of the calculus problems you will engage in solving have only 1 or 2 steps of calculus followed by many steps of algebra. Before the first day in calculus, you must have the algebra know-how to complete the problems in this summer packet. In this summer packet I have included problems from these categories: 1. Writing equations of lines 2. Factoring 3. Solving equations 4. Solving literal equations 5. Adding/Subtracting Rational expressions 6. Multiplying/Dividing Rational expressions 7. Exponents 8. Multiplying by conjugate 9. Logarithmic and exponential equations 10. Function notation 11. Functions (graphing including piecewise functions) 12. Graph analysis 13. Trigonometry 14. Using a calculator Please practice all these problems. Work each problem without a calculator (until the 14th section, which is about using a calculator). Review these problems in the week before school begins. If you need help with any of this packet, find Ms. Riley at school on Friday, August 19 th from 1 pm to 4 pm. You need to come to the first day of calculus with this summer math packet completed and the skills to solve these problems already in your mathematics repertoire.

2 Writing equations of lines Write the point-slope equation of a line with a slope of ¾ passing through the point (-4, 7). Write the point-slope equation of the line perpendicular to 2x+y = 4 and passing through (5, 8). Write the point-slope equation of a line passing through the points (2, -3) and (-5, 4). Write the equation of a line with a slope of 0 containing the point (3, 7). A line has a slope given by the formula m = 2xy, where x = 3 and y = -4. Write the point-slope equation of this line that passes through (3, -4) The slope of a line passing through the point (6, 2) is m = x. Write the point-slope equation of this line. 9y A line passing through the point (2, 3) has a slope that is defined by m = x2, write the point-slope equation of this y 1 line. A line passing through the point (-2, 1) has a slope that is defined by m = 2x y, write the point-slope equation of this line. Factoring Factor: 3x 2 + 7x + 4 Factor: 3x x + 4 Factor: 3x 2 + x 4 Factor: 3x 2 8x + 4 Factor: 3x 2 4x 4 Factor: 3x 4 y + 9x 2 y Factor: (x + 6) 4 + 9(x + 6) 2 Factor: 6x 4 + x 2 15 Factor: x 2x 2 Factor: 9x 2 64 Factor: 4x Factor: x 6 y 6

3 Solving equations Solve algebraically: 3x 4 12x 2 = 0 Solve algebraically: 4x 2 21x 18 = 0 Solve algebraically: 2x 2 3x + 3 = 0 Solve algebraically: x 4 9x = 0 Solve algebraically: (x 3) 2 = 49 Solve algebraically: 3 x 4 = 15 Solve algebraically: x + 1 3x = 1 Solve algebraically: 5 + 2x 2 = 8x Solve algebraically: lne 4 x = 0 Solve algebraically: 5x 4 5x+4 = 2 3 Solve algebraically: 4 lne x = 0 Solve algebraically: e x = 10

4 Solving literal equations Solve for m: xm + y = 1 + m Solve for m: 3x 2 ym + 2xy 2 = 2ym Solve for m: 3y 2 m + 2ym = 5m + 2x Solve for m: 3xy + 5ym 2xm = 1 Solve for m: x 3 y + my 2 2m + 3y = 0 Solve for m: 3x m = 2ym Simplifying and Adding/Subtracting Rational expressions Write as a single fraction with denominator in factored form: 7x2 +5x 5x x 2 +1 x 2 6 Write as a single fraction with denominator in factored 2 form: 3 x+1 x Write as a single fraction: x(1 2x) (1 2x) 1 2 Write as a single fraction: (3x 2) x(3x 2) 1 2

5 Simplify using only positive exponents: 4x 16 (x 4) 3 Simplify using only positive exponents: ( x 2 y 1 x) 1 2 Simplify using only positive exponents: ( 1 x x 1 y y 2) 1 2 Write as a single fraction: x + x+7 x 3 x 2 9 Multiplying/Dividing Rational expressions Eliminate the complex fraction: Eliminate the complex fraction: Eliminate the complex fraction: Eliminate the complex fraction: Simplify: 2 2 ( 2 2 x (2 x) 2) Simplify: h x+h h

6 Eliminate the complex fraction: 2 x x 1 Eliminate the complex fraction: Exponents Evaluate without a calculator: 64 5/6 Evaluate without a calculator: (16) 3 2 Evaluate without a calculator: (8) 2 3 Evaluate without a calculator: 27 5/3 Evaluate without a calculator: ( 1 1 ) 2 9 Evaluate without a calculator: 3 ( 1 9 ) 1 2 Evaluate without a calculator: ( 16 9 ) 3 2 Evaluate without a calculator: (32) 7 5 Evaluate without a calculator: 1 3 (81)1 2 Multiplying by conjugate Multiply by the conjugate: x Multiply by the conjugate: 8 x + 3 Multiply by the conjugate: 4x + 3 Multiply by the conjugate: 9x + 3

7 Multiply by the conjugate: 64 x Multiply by the conjugate: 3 x 2 9 Logarithmic and exponential equations Solve for x: ln e = x Solve for x: Solve for x: Solve for x: Solve for x: Solve for x: Solve for x: Solve for x:

8 Solve for x: Solve for x: Function notation Given f(x) = x 3 5 find f(5) f(1). Given f(x) = x 2 3x + 4 find f(5) f(1). Given f(x) = (x 2) 2 5 find f(5) f(1). Given f(x) = x 2 3x + 4 find f(x + 2) f(2). Given f(x) = (x 2) 2 5 find f(x + 2) f(2). Given f(x) = x 2 3x + 4 find f(x + h) f(x). Given f(x) = (x 2) 2 5 find f(x + h) f(x). Given f(x) = x + 3 and g(x) = x, find f(g(x)) and g(f(x)) Given f(x) = 2 x 5 and g(x) = x2 5, find g(f(x)). Given f(x) = 2 x 5 and g(x) = x2 5, find f(g(x)).

9 Functions (graphing including piecewise functions)

10 Graph both functions on the graph. f(x) = 16 x 2 g(x) = 4 x Graph both functions on the graph. f(x) = e x f 1 (x) = lnx Graph analysis: Use the graph of g(x) to answer the questions.

11 Trigonometry

12 Using a calculator

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14 ON the next pages you will find the SYMKC for Calculus AB. Please print out the blank copy of SYMKC and write in the answers from the Key. It will do your brain good to copy down this information (even though you will not understand any of it yet). We will write this information over and over and over again this year. It is what STUFF YOU MUST KNOW to perform well in calculus.

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