Summer Packet for Students Taking Introduction to Calculus in the Fall

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1 Summer Packet for Students Taking Introduction to Calculus in the Fall Algebra 2 Topics Needed for Introduction to Calculus Need to know: à Solve Equations Linear Quadratic Absolute Value Polynomial Rational Radical Rational Logarithmic Exponential à Graph Functions and functions with transformations (non- calculator) Linear Quadratic Absolute Value Cubic Quartic Exponential Rational Piecewise à Other Algebra Skills Simplify radicals. (Simple Radical Form/rationalize) Add, subtract, multiply, and divide fractions. Simplify complex fractions. Use properties of exponents to simplify exponential expressions. Factor quadratics, difference and sum of two cubes, and factor by grouping. Find and Write equations of lines. Simplify exponents and solve exponential equations Expand and Condense Logarithms Determine Domain and Range of functions (non- calculator) Graph and analyze parent functions without a calculator 1

2 Calculator Skills Needed Sketch graphs in the appropriate window Adjust the window Find x and y- intercepts Find maximums/minimums * zeros, roots, solutions Find intersection(s) of 2 curves * Increasing/Decreasing, Constant Intervals Solve systems of equations. * Modeling Functions Practice Problems I. Properties of Exponents Simplify the following using the properties of exponents. Assume all variables in the denominator are non- zero. (6x 2 ) 2 y 3 2w 2 z y 1 xy a 3 b ab 2 3b 3 8a 4 b 3 2

3 ii. Linear Equations and Inequalities 1. Solve each of the following equations: 1. y + 6 y = 5 2. x 1 2x 4 + x + 2 3x = x 5 2 x + 5 = 4 x Solve the following inequalities algebraically. State solution in interval notation. a. 1 2 x + 2 ( ) + 4 x 3 ( ) < 1 3 x 3 ( ) b. 4x > 5.6 b. 3x x 4 3 x

4 III. Lines in the Plane 1. Find the equation of the line in both standard and slope- intercept form which passes through the points (- 2, - 8) and (3, - 1). 2. Find the equation of two lines passing through (5, 2), one which is parallel to 4x 5y =20 and one which is perpendicular to this line. IV. Factoring Factor completely. (GCF, Factor by Grouping, Difference of Cubes, Quadratic Formula) 1. 2x 2 15x x 2 y b x 3 22x 2 8x 5. 4x 3 + 5x 2 16x 20 4

5 V. Simplifying Radicals Simplify the following: ( ) VI. Solving quadratics, radical and inequalities 1. Solve the following equation using the quadratic formula: 2x 2 + 6x 7 = 0 2. Solve the following radical equation. Check for extraneous solutions. x x + 3 = 0 5

6 3. Solve each inequality and graph. Use sign analysis (non- calculator)to determine your solution set and state in interval notation. State zeros and asymptotes where applicable. a. 2x +1 x 3 ( )( x 2) < 0 b. ( x + 2) ( x 1) ( x + 4) > 0 VII. Simplify and state any restrictions. 3x 2 17x x 2 20x 24 2x2 x 3 6x 2 7x 3 x 4 +8x x 2 4 x2 + 2x + 4 x x 2 4 x x 2 4 4x x x 3 3x2 36 x 2 9 6

7 VIII. Applications 1. The rate at which a cricket chirps is said to be a linear function of temperature. Suppose at 54 F, a cricket makes 75 chirps per minute while at 68 F, it makes 103 chirps per minute. a. Write an equation expressing the chirping rate in terms of temperature. b. Predict the chirping rate for 85 F. c. Sketch the graph of this function in a reasonable domain. Be sure to label your axes and use the appropriate domain. 7

8 2. Suppose you are trying to predict the likelihood of an automobile accident as a function of the driver s age. From previous accident records, you note the following data: Age (in years) Accidents (per 100 million km driven) You assume a quadratic model is reasonable because you believe the number of accidents should reach a minimum and then go up again for older drivers. a. Write the particular equation expressing accidents per 100 million km driven in terms of age. b. Sketch the graph of this function in a reasonable domain. Be sure to label your axes and use the appropriate domain. c. Based on your model, explain whether a 16- year old driver is safer than a 70- year old driver. d. What age driver appears to be the safest? e. Your company decides to insure licensed drivers up to the age where the accident rate reaches 420 per million km driven. What then is the domain of this quadratic function. 8

9 IX. Graph each and State the intervals of increasing, decreasing, and constant (if any) for each of the following functions. a. g( x) = 2 x b. h( x) = x x 3 2 9

10 X. Sketching graphs and analyzing. Non- Calculator. 1. Sketch the graph of g(x). 3 if x <-2 g(x) = x 2 +1 if -2 x<2 x 1 if x >3 a. State the domain. b. State the range. c. State any discontinuities. 2. Graph g(x) = 3 4 2x + 5 a. State the transformations: 10

11 XI. Use logarithmic properties to expand each expression as much as possible. 1. log 2 ( x 4 y ) xy 2 log 5 125z 4 3. log 6 ( x 3 y 2 z) x log 7 y 2 3 z 11

12 XII. Use properties of logarithms to condense each logarithmic expression 1. log 4 3+ log log 2x 5 ( ) log(x +1) ( ) 4log( yz) 2ln x 2 y log xy 4. ( ) + 5ln( yz 2 ) log x log y 4 4 ( ) 5ln x 2ln y 3ln z 7. 3ln( x + 4) + 4ln x XIII. Write as a single logarithm: ( ) 1. 3ln x ln x log x 2 ( ) log(x +1) log b x + 2log b 3 5log b y 12

13 Change of Base to Common Logarithms Use common logs to evaluate each: 1. log log Forms of Equations of Lines General Form: Ax +By + C = 0 A and B both cannot equal 0 Slope- Intercept Form: y = mx + b Point- Slope Form: y y 1 = m( x x 1 ) Vertical Line: x = a Horizontal Line: y = b Parallel and Perpendicular Lines 1. Two nonvertical lines are parallel if and only if their slopes are equal. 2. Two nonvertical lines are perpendicular if and only if their slopes, m 1 and m 2 are opposite reciprocals. Midpoint Formula (Coordinate Plane) The midpoint of the line segment with endpoints P(x 1,y 1 ) and Q(x 2,y 2 ) is: " # x 1 + x 2 2, y 1 + y 2 2 $ % Distance Formula (Coordinate Plane) The distance d between points P(x 1,y 1 ) and Q(x 2,y 2 ) in the coordinate plane is: d = ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 13

14 Multiplication Property of Order If x, y, and z are real numbers and x < y, then xz < yz if z is positive xz > yx if z is negative xz = yz if z is zero Properties of Absolute Value Inequalities If c is a non- negative constant, then ax + b > c is equivalent to ax + b >c or ax + b < - c and ax + b < c is equivalent to ax +b < c and ax +b > - c or, in a more compact form, c < ax + b < c. Definition of a Quadratic Equation in x A quadratic equation in x is one that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers and a 0. Solving Quadratic Equations Quadratic Equations may be solved by 1. Factoring and then using the Zero Product Property 2. Using the Quadratic Formula 3. Extracting Square Roots 4. Completing the Square Method 5. Graphing Quadratic Formula The solutions of the quadratic equation ax 2 + bx + c = 0 where a 0 are given by the quadratic formula: x = b ± b2 4ac 2a 14

15 Zero Product Property If a and b are real numbers, and if ab = 0, then a = 0 or b = 0, or both a = 0 and b = 0. Simplest Radical Form An expression is in simplest radical form if: a. The radicand of the n th root contains no n th powers as factors. b. The root index is as low as possible. c. There are no radicals in the denominator. Properties of Exponents Let a and b be real numbers, variables, or algebraic expressions and m and n be real numbers. Product of Powers a m a n = a m +n Power of a Power ( a m ) n = a m n Power of a Product ( a b) m = a m b m a m a = n am n Quotient of Powers, a 0 " a$ Power of a Quotient = an, b 0 # b% b n Zero Power a 0 = 1, a 0 Negative Power a n = 1 1 or, a n a = n an a 0 n 15

16 Properties of Logarithms Logarithms have special algebraic characteristics that help us to solve logarithmic equations; before we can solve equations, we need to have a solid understanding of the properties and how to use them Properties of Logs Let b, R, and S be positive real numbers with b 1, and c any real number 1. Product Rule: log b MN = log b M + log b N M 2. Quotient Rule: log b N = log M log N b b 3. Power Rule: log b M K = K log b M 16

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