Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you show all necessary work to receive full credit for your assignments. Your packet is due on the first day of class, Tuesday, August th. We will take an exam covering these topics the first week of school. Please feel free to e-mail me at Kristi.burks@misd.org if you have any questions about the notes or homework assignments. Remember it is summer, so it may take me a day or two to respond, but I will be checking my e-mail throughout the summer. Also, there are many videos online that may be useful. A good place to start is www.rootmath.org or kahnacademy.org. The unit circle and important trigonometric identities are included in your packet. You will be responsible for memorizing this material so you may want to take some time to review the last few pages of the packet. Thank you in advance for your extra effort in preparing for a successful year in AP Calculus at Montgomery High School. I look forward to meeting each of you face to face in the fall. Have a fun and safe summer! Kristi Burks 1
Calculus Summer Packet Topics Covered: 1. Function Graphs. Exponents, Exponential and Logarithmic Functions 3. Rational Functions 4. Trigonometry Work through each set of notes and complete the problem set following. No calculator is permitted unless the problem is labeled (Calculator).
1. Function Graphs Fundamentally, the graph of an equation y = f(x) is the set of all points (x, y) whose coordinates satisfy the equation y = f(x). Linear Functions Equation of a Line: Slope-Intercept Form y = mx + b Point-Slope Form y y 1 = m(x x 1 ) Lines are used extensively in calculus and as such they play a fundamental role in the study of functions. In particular we will use the Point-Slope Form in many different scenarios so you should make this your main form for writing the equation of a line. Example 1 Find the equation of the line containing the point ( 3,5) perpendicular to the line 3x + y = 10. Graph the line and algebraically determine whether or not the point ( 4,5) lies on the line. Solution To determine the equation of a line, we will use Point-Slope Form thus we need a point on the line (which we are given) and the slope of the line (which we will find). To find the slope, we must determine the slop of the given line and use the opposite reciprocal to determine the slope perpendicular. We were given 3x + y = 10, so solving for y yields y = 3 x + 5. The slope of this line is m = 3 so the slope of a line perpendicular is. Plugging the following 3 into point slope formula we have: x 1 = 3, y 1 = 5, m = 3 y 5 = 3 (x 3) y 5 = (x + 3) 3 The graph for this equation is shown to the right. To determine if the point ( 4,5) lies on the line y 5 = (x + 3), plug the point in and see if it results in 3 a true statement. 5 5 = ( 4 + 3) 3 0 = 3 ( 1) 0 3 So the point ( 4,5) is not on the line y 5 = (x + 3). 3 The skills required in Example 1 are required in calculus when we solve differential equations, determine the equation of tangent lines, and find linear approximations. 3
Polynomial Functions The majority of the functions that we will work with in calculus will be polynomials. To be successful in calculus, you must be an expert in the following areas when it comes to all functions. Identifying a function based on its graph Determining solutions graphically and analytically (algebraically) Determining the sign of a function and its meaning (f(x) > 0 the graph is above the x axis, f(x) < 0 the graph is below the x axis, f(x) = 0 the graph intersects the x axis) Determining slope behavior (increasing, decreasing, zero-horizontal, undefined-vertical) Determining extrema (local/absolute maximums and minimums) Determining end behavior (what are the y-values approaching as you move towards a certain x-value) A polynomial function of degree n is a finite sum of nonnegative integer powers of x: p(x) = a n x n + a n 1 x n 1 + +a x + a 1 x + a 0 Recall: x 1 = x x 0 = 1 for all real numbers x Zero Factor Property: ab = 0 if and only if either a = 0 or b = 0 Key Characteristics of Graphs of Polynomial Functions The domain is the set of all real numbers (i.e. polynomials are continuous for all real numbers) There are at most n x-intercepts A polynomial is even if f( x) = f(x) for all values in the domain. o The graph is symmetric with respect to the y-axis o The ends of the graph point in the same direction (Left and right end behavior is the same) A polynomial is odd if f( x) = f(x) for all values in the domain. o The graph is symmetric with respect to the origin o The ends of the graph point in the opposite directions Functions that are nonlinear, such as polynomials, absolute-value, square root, exponential, and rational, do not have a constant rate of change. Nonlinear functions have a variable rate of change. This means that the rise over run y y 1 formula that x x 1 you learned in algebra does cannot be used to find the exact slope at any point on a nonlinear function. A large portion of our calculus course will be devoted to how we actually determine the exact slope at any point on nonlinear equations. The study of this variable rate of change is called Differential Calculus. 4
Example ( 1.47,8.55) (1.47, 8.55) Solution: The domain is D: {, } There are three x-intercepts or solutions to f(x) = 0. These solutions are x =, x = 0, and x =. (Notice there are no real solutions to 0 = x + 1.) The function is an odd function, with a degree of n = 5. The function is increasing (positive slope) on < x < 1.47 and 1.47 < x <. The function is decreasing (negative slope) on 1.47 < x < 1.47. The function has a local maximum at x = 1.47 and a local minimum at x = 1.47. 5
Example 3 6
Example 4 The skills required in Examples -4 are required in calculus when we do graphical analysis, which is one of the densest subjects we will cover. 7
Practice Problems 1: (1) Write the equation of the line that passes through the point (, 5) and is perpendicular to 6x 5y = 18. Sketch a graph of this line. Algebraically determine if the point (, 5) lies on the graph. () Write the equation of the line that passes through the points (, 4) and (4, 8). (3) Line A passes through the points (k + 3, 4k 6) and (, 16). Find the value of k if line A has a slope of 0. (4) What are the zeros of the function? What are their multiplicities? (5) (Calculator) What is the relative maximum and minimum of the function? 8
(6) (Calculator) What are the minimum and maximum values of the function on the interval? (7) Find the x and y intercepts of the function f(x) = x 3 9x. (8) Identify the x-intercepts for each of the following functions a. y = (x + ) 5 b. f(x) = x + 13 7 (9) Describe the graph of f(x) = x (x + )(x 1)(x + 1). (.749,.604) ( 1.604, 9.479) 9
(9) 10
(10) 11
Parent Functions In calculus you must be able to recognize the following parent functions and know their behaviors. Exponential Function f(x) = e x Natural Log f(x) = ln(x) Circle Centered about Origin x + y = r Sine Function f(x) = sin(x) Cosine Function f(x) = cos(x) Tangent Function f(x) = tan(x) = sin(x)/cos(x) 1
Practice Problems : 13
. Exponents, Exponential and Logarithmic Functions The majority of your time in calculus will be without a calculator so it is important that you find ways to work smart rather than work hard. For example, the order in which you apply rational exponents can simplify or complicate a problem. Example 1 Evaluate 8 /3. Solution: The solution to this problem can be either of the following. (8 ) 1 3 = (64) 1 3 3 = 64 = 4 Or (8 1 3) 3 = ( 8) = () = 4 The solution on the left requires you to know the cube root of 64, while the solution on the right only requires you know the cube root of 8. 14
Practice Problems 3: (1) Multiply and simplify if possible. () What is the simplest form of the expression? (3) What is the simplest form of the quotient? (4) What is the simplest form of the quotient? (5) Simplify 4x 16 (6) Expand (x 4) (7) What is the solution of the equation? (8) Simplify (9) Simplify 15
Exponential Functions 16
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Example 1 Solve for x. log 9 (x 10) = 1 9 log 9(x 10) = 9 1 x 10 = 9 +10 + 10 x = 19 x = ± 19 x = ± 19 Example Solve for x. e 3x = 3 3 ln(e 3x ) = ln(3 3 ) 3x = 3 ln(3) 3 3 x = ln(3) Example 3 Write in log form: 9 3 = 79 log 9 (79) = 3 Think: 9 raised to 3 is 79. Example 4 Write in exponential form: log 8 (0.15) = 1 8 1 =.15 Note:.15 = 1 8 18
Practice 4: (1) Write the equation in logarithmic form. () Solve the exponential equation. (3) Graph the function f(x) = e x. Then determine the domain, range, and y-intercept. Sketch the graph. (4) Graph the function f(x) = ln(x). Then determine the domain, range, and x-intercept. Sketch the graph. (5) Solve e x e x 3 = 0 for x. (6) Use the properties of logarithms to expand ln ( xy5 (z+1) 4). (7) Graph the following exponential functions together by plotting points. State your observations. a. y = x and y = ( 1 )x b. y = 4 x and y = ( 1 4 )x (8) Evaluate log 4 ( 1 3 ) 16 19
Modeling with Exponential Functions and Solving Equations In Pre Calculus and Algebra II you have worked with problems like those discussed below. In calculus we will extend upon these modeling problems, so knowledge of exponential and logarithmic functions is essential. 0
Practice Problems 5: (Calculator approved for final evaluation step, not for solving.) (1) A bacteria culture contains 100 bacteria and doubles every day. How many hours will it take the culture to reach 10000 bacteria? () A bacteria colony starts off with 100 million bacteria at :00pm and grows exponentially. At 4:00pm there are 110 million bacteria. How many bacteria are there at 10:00pm? At any time t? (3) The time required for a radioactive substance (uranium, kryptonite, radium, etc.) to decay exponentially to half its original amount is called the half-life of the element. A sample of the element stuffium with mass 100 grams will decay exponentially to 70 grams in 11 days. a. Find the formula for N(t) = the mass of stuffium t days from now. b. What is the half-life of stuffium? c. How long does it take for the original sample to lose 65% of its mass? 1
3. Rational Functions and Difference Quotient In calculus you will determine vertical asymptotes by simplifying rational functions and setting the denominator equal to zero. You must have the skills required to solve the following problems in order to be successful when working with limits and graphing functions in calculus.
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Example: 4
Practice Problems 6: (1) Rationalize the denominator. 14 3+ () Find the difference indicated. Write your answer in simplest form. x + a 3 x + 6 x + a x (3) Factor and reduce to simplest form. 6x +11xy 10y 3x +10xy 8y 5
(4) Find the domain of the function. x 3x 4 6x 54 (5) Find the domain of the function below. (6) Find the x-intercept(s) of the function below, if any exist. f(x) = 6x 7x 5 4x 1x 7 (7) Find the vertical and horizontal asymptote(s) of the function below, if any exist. f(x) = 6x 7x 5 4x 1x 7 6
(8) Find the domain of the following functions: a. f(x) = x 4x + 5 b. g(t) = ln (4t 3) 1 c. h(x) = x 3 +3x x 3 (9) Find f g (also denoted f(g(x))) if f(x) = x x+1 and g(x) = x. Simplify. (10) Perform the indicated operations and simplify. 8 x + 1 ( y z + y 4 w ) (11) Simplify 1 x+8 1 8 x (1) Use (a 3 ± b 3 ) = (a ± b) (a ab + b ) to factor the following. a. (x 3 64) b. (8x 3 + 7) 7
4. Trigonometric Functions We will work with trigonometric functions in every section of calculus. Therefore, your understanding of these functions and their properties must be absolute. The intent of this section is to remind you of some of the more important topics from your previous courses. This by no means covers all required knowledge and some outside studying on your part may be required for you to be successful. First let s start with the six trig functions and how they relate to each other. Basic Trig Functions sin(x) cos(x) tan(x) = sin(x) cos(x) Reciprocal Trig Functions (Not to be confused with Inverse Trig Functions) csc(x) = 1 sin(x) sec(x) = 1 cos(x) cot(x) = 1 tan(x) = cos(x) sin(x) Inverse Trig Functions sin 1 (x) = θ Think: At what angle θ, is sin(θ) = x? cos 1 (x) = θ tan 1 (x) = θ Note the superscript 1 does NOT mean 1 over the function! It is the sign for inverse of a function. The inverse trig functions can be written as arcsin(x), arccos(x), and arctan(x) to avoid this misconception. Recall as well that all trig functions can be defined in terms of a right triangle. The definitions below create the unit circle when applied to right triangles with hypotenuse = 1. The values of the trig functions that you know on the unit circle come from applying the following definitions to 30 60 90 and 45 45 90 special right triangles with hypotenuse = 1. 8
Special Right Triangles Special Right Triangles with hypotenuse = 1 1 π 3 1 1 π 4 1 π 6 π 3 Applying the trig ratio definitions to the triangles above, we get: π 4 1 π sin(30 ) = sin(60 ) = opposite hypotenuse = opposite hypotenuse = 1 1 = 1 tan(30 ) = 3 1 sin (x) cos (x) = = 3 cos(30 ) = 1 3 = 1 3 = 3 3 cos(60 ) = adjacent hypotenuse = adjacent hypotenuse = 3 1 = 3 1 1 = 1 tan(30 ) = sin (x) cos (x) = 3 1 = 3 1 = 3 sin(45 ) = cos (45 ) = side hypotenuse = 1 1 = tan(30 ) = sin (x) cos (x) = = 1 9
Connecting Special Right Triangles and the Unit Circle Unit Circle Above is the unit circle. The way the unit circle works is to draw a line from the center of the circle outwards corresponding to a given angle. Then look at the coordinates of the point where the line and the circle intersect. The first coordinate is the cosine of that angle and the second coordinate is the sine of that angle. We ve put some of the basic angles along with the coordinates of their intersections on the unit circle. 30
Unit Circle and Triangles 31
Reciprocal Identities sin u = 1 csc u csc u = 1 sin u cos u = 1 sec u sec u = 1 cos u tan u = 1 cot u cot u = 1 tan u Pythagorean Identities sin u + cos u = 1 1 + tan u = sec u 1 + cot u = csc u Quotient Identities tan u = sin u cos u cot u = cos u sin u Double Angle Formulas sin(u) = sin u cos u cos(u) = cos u sin u = cos u 1 = 1 sin u 3
To be successful in our calculus course you should have the content on the previous pages memorized. Practice Problems 7: (1) Solve the equation sin θ + sin θ = 0 for 0 θ π. () Find all solutions of the equation cos θ = 1 for 0 θ π. (3) Find all solutions of the equation sin θ = cos θ for 0 θ π. (4) Find cos ( 4π 3 ). (5) Find sec ( 5π 4 ). 33
(6) Find sin 1 ( 1 ). (7) Find cos 1 (1). (8) Find sec 1 (1). (9) Find sin 1 ( 1 ). (10) Find sin (tan 1 ( 4 3 )). Hint: Draw a triangle that fits the property tan(θ) = 4 3. 34