Summer 07 Expectations for Summer Assignment This packet is to be submitted to your Calculus BC teacher on the first day of the school year. All work must be shown in the packet OR on separate paper attached to the packet. Completion of this packet is worth four homework grades. Trust me, this is significant. You should expect to spend between 8 and 0 hours working on this assignment over the course of the entire summer. Two big mistakes to avoid: () completing the assignment this weekend (o.k., that s probably not going to happen) and () completing the entire assignment the weekend before its due (much more likely to happen AND will lead to a disappointing start to our time together). The assignment is meant to be done over the course of the summer. For optimal results, work on the assignment in 0-minute blocks of time two to three times per week for the summer. Support for this Assignment For support, please email your teacher prior to the submission of this assignment. For further support, try the Khan Academy: https://www.youtube.com/user/khanacademy. They have videos for absolutely everything! Recommended Timeline for Completion Round I: August 9 th I. Parent Functions II. Function Notation III. Intercepts IV. Inverses V. Equation of a Line VI. Factoring Polynomials and Solving Polynomial Equations VII. Systems of Equations VIII. Functional Analysis Round I Round II: August rd I. Properties of Exponents II. Radian and Degree Measure III. Reference Triangles IV. Unit Circle V. All Six Trigonometric Ratios VI. Inverse Trig Functions VII. Solving Trig Equations VIII. Functional Analysis Round II Round III: Finish by September 6 th I. Properties of Logarithms II. Solving Exponential Equations III. Conic Sections IV. Limits Graphically V. Limits Numerically VI. Limits Intuitively VII. Limits Analytically P a g e
Summer 07 Start of Round I To Be Completed by July 6 th Section I: Parent Functions Fill in the table below for the characteristics of the given parent functions. Use your calculator or the attached graphs if you are unsure about the graph of the parent function. You must have these memorized for the AP Calculus course. Please use interval notation where appropriate. Even/Odd Function Domain Range Zeroes Increasing Decreasing Symmetry or Neither y = x (, ) (, ) x = 0 (, ) Never Odd y = x y = x y = x y = x About Origin y = x y = x y = ln (x) y = e x y = x y = r x y = sin (x) y = cos (x) y = tan (x) P a g e
Summer 07 Things to Remember About Graphing Functions Domain: the set of all possible inputs (x-values) of a function Range: the set of all possible outputs (y-values) of a function Zeroes: x-values when f(x) = 0, the function crosses the x-axis Increasing: a function is increasing on the interval (a, b) when f(a) < f(b) for all a x b Decreasing: a function is decreasing on the interval (a, b) when f(a) > f(b) for all a x b Concave Up: a function is concave up if it curves upwards (like a cup) Concave Down: a function is concave down if it curves downwards (like a frown) Even: a function y = f(x) is even if f( x) = f(x), example: y = x and y = cos (x) Odd: a function y = f(x) is od if f( x) = f(x), examples: y = x and y = sin (x) Symmetric: a function that is even will be symmetric with respect to the y-axis; a function that is odd will be symmetric with respect to the origin Some Extension Questions Reference the graphs that follow, or your graphing calculator, if you need support in answering these questions.. Is the function f(x) = sec x an even or odd function? Justify your answer.. On what interval(s) is the function f(x) = x + increasing?. Describe the concavity of the function f(x) = e x.. The maximum number of horizontal asymptotes that a function can have is horizontal asymptotes. P a g e
y = x AP Calculus Summer Assignment Summer 07 Parent Functions y = x - - - - 5 - - - - - 5 - - - - - - - y = x y = x - - - - 5 - - - - 5 - - - - - - - - y = x x y = x - - - - 5 - - - - 5 - - - - - - - - y = x y = x - - - - 5 - - - - 5 - - - - - - - - P a g e
y = x AP Calculus Summer Assignment Summer 07 y = ln x - - - - 5 - - - - - - - 5 - - - - - y = e kx, k > 0 y = e kx, k < 0 - - - - 5 - - - - 5 - - - - - - - - y = x + y = x - - - - 5 - - - - 5 - - - - - - - - x + y = 6 y = 6 x - - - - 5 - - - - 5 - - - - - - - - -5 5 P a g e
y = sin x AP Calculus Summer Assignment Summer 07 y = cos x -π/ -π -π/ π/ π π/ π - - - -π/ -π -π/ π/ π π/ π - - - - - y = tan x y = tan x -π/ -π -π/ π/ π π/ π -π/ -π -π/ π/ π π/ π - - - - - - - - y = csc x y = cot x -π/ -π -π/ π/ π π/ π -π/ -π -π/ π/ π π/ π - - - - - - - - 6 P a g e
Summer 07 Directions Using your knowledge and skills about transformations of parent functions, fill in the following information for the given function. y = (x ) Sketch a graph: Domain Range Zeroes Vertical Asymptotes Horizontal Asymptotes Increasing Decreasing y = (x + ) + Sketch a graph: Domain Range Zeroes Vertical Asymptotes Horizontal Asymptotes Increasing Decreasing 7 P a g e
Summer 07 y = x Sketch a graph: Domain Range Zeroes Vertical Asymptotes Horizontal Asymptotes Increasing Decreasing y = ln(x ) + Sketch a graph: Domain Range Zeroes Vertical Asymptotes Horizontal Asymptotes Increasing Decreasing 8 P a g e
Summer 07 y = e x Sketch a graph: Domain Range Zeroes Vertical Asymptotes Horizontal Asymptotes Increasing Decreasing 9 P a g e
Summer 07 Section II: Function Notation (Calculator Inactive) To evaluate a function for a given value, simply plug the value into the function for x. Recall: (f g)(x) = f(g(x)) read f of g of x means to plug the inside function (in this case, g(x) in for x in the outside function (in this case, f(x)). Example: Given f (x) x and g(x) x find f(g(x)). f (g(x)) f (x ) (x ) (x 8x 6) x 6x f (g(x)) x 6x Let f(x) = x + and g(x) = x. Find each of the following. Be sure to simplify as much as possible.. f (). g( ). f (t ). f g( ) 5. g f (m ) 6. f (x h) f (x) h Let f(x) = x, g(x) = x + 5, and h(x) = x. Find each. 7. h f ( ) 8. f g(x ) 9. g h(x ) Find f(x+h) f(x) h for the given function f. Be sure to simplify as much as possible. This expression should look familiar from PreCalculus and is a critical step in finding the derivative of a function. 0. f (x) 9x. f ( x) 5 x. f( x) x 0 P a g e
Summer 07 Section III: Intercepts (Calculator Inactive) To find the x intercepts, let y = 0 in your equation and solve. To find the y intercepts, let x = 0 in your equation and solve. Example: y x x x int. (Let y 0) 0 x x 0 (x )(x ) x or x x intercepts (,0) and (,0) y int. (Let x 0) y 0 (0) y y intercept (0, ) Find the x and y intercepts for each function or implicitly defined curve shown below.. y x 5. y x x. y x 6 x. y x x P a g e
Summer 07 Section IV: Inverses (Calculator Inactive) To find the inverse of a function, simply switch the x and the y and solve for the new y value. Example: f(x) = x + Rewrite f(x) as y y = x + Switch x and y Solve the equation for y x = y + x = ( y + ) x = y + y = x f (x) = x Cube both sides Simplify Solve for y Rewrite using inverse notation Find the inverse for each function.. f(x) = x +. g(x) = x. h(x) = 6e x+. j(x) = ln 5 x P a g e
Summer 07 Section V: Equation of a Line (Calculator Inactive) Slope intercept form: y mx b Vertical line: x = c (slope is undefined) Point-slope form: y = m(x x ) + y Horizontal line: y = c (slope is 0). Find the equation of a line passing through the point (, 8) and parallel to the line y = 5 6 x. Find the equation of a line perpendicular to the y axis passing through the point (,7). Find the equation of a line passing through the points (,6) and (,).. Find the equation of a line with an x intercept (, 0) and a y intercept (0, 5). 5. Find an equation for the line that passes through the origin and the point (a, a). 6. The three points (5,), (,) and (, 5) are vertices of a triangle. Determine the equations which can be used to model each side of the triangle. P a g e
Summer 07 Section VI: Factoring Polynomials and Solving Polynomial Equations When factoring polynomials, you should always check to see if the problem you are tackling involves one of the formulas below. If it does, your life is easy! Use the appropriate formula and call it a day. Important Factoring Formulas Difference of Squares a b = (a + b)(a b) Difference of Cubes a b = (a b)(a + ab + b ) Sum of Cubes a + b = (a + b)(a ab + b ) If none of the formulas above fit your given problem, you will need to try other strategies like finding a greatest common factor (GCF), using the x-box method or factoring by grouping. The example shown below showcases one of the most common factoring challenges you will see in AP Calculus problems the x-box. Review this process, and complete the practice problems that follow. Example: Factor completely: x + 6x 08. Step : If possible, factor out any common terms: x + 6x 08 = (x + x 5) Step : Use the magic x to identify the correct factors. 9x 5x x 6x You should place the product, ax c at the top of your X, and the bx term at the bottom of your X. Your job is to come up with two factors that will multiply to equal what you see at the top of your X, and add up to what you see in the bottom of your X. Step : From here, we will finish factoring by using the box method: x +9 x x 9x 6 6x 5 Steph : Finalize your answer! x + 6x 08 = (x 6)(x + 9) P a g e
Practice Problems Factor completely AP Calculus Summer Assignment Summer 07 8x + 7 x x 8x + x 7x + 5 x + x 9 6 9x + 66x + 7 x 5 Solving Polynomial Equations: Once factoring has been mastered, often times you will need to take things one step further by actually solving a polynomial equation. Luckily, you begin this process in exactly the same way you would a regular factoring problem, using Steps from above. The difference here is your final answer will consist of x values, rather than just an expression. Example: Solve the following equation: 5x 0x + 0 = 0 Step : Verify that all terms are on the same side of the equal sign, and if possible, factor out any common terms: 5x 0x + 0 = 5(x 6x + 8) = 0 Step : Using the strategies outlined above, completely factor your equation: 5(x 6x + 8) = 5(x )(x ) = 0 5 P a g e
Summer 07 Step : Take your factored equation and set each individual factor equal to zero to finish solving: 5 = 0 x = 0 x = 0 This is silly, and clearly This leads us to an answer This leads us to an results in no solution. of x =. answer of x =. Practice Problems Solve each polynomial equation: x 7x + 6x = 0 x + x x = 0 x x = 90 8x 6 = 0 5 x = 5x 6 P a g e
Summer 07 Section VII: Systems of Equations (Calculator Inactive) Use substitution or elimination method to solve the system of equations. Example: x y 6x 9 0 x y 9 0 Elimination Method x 6x 0 0 x 8x 5 0 (x )(x 5) 0 x and x 5 Plug x = and x 5 into one original y 9 0 5 y 9 0 y 0 6 y y 0 y Points of Intersection (5, ), (5, ) and (,0) Substitution Method Solve one equation for one variable. y x 6x 9 (st equation solved for y) x ( x 6x 9) 9 0 Plug what y is equal to into second equation. x 6x 0 0 (The rest is the same as x 8x 5 0 previous example) (x )(x 5) 0 x or x 5 Find the point(s) of intersection of the graphs for the given equations. x + y = 8. { x y = 7. { x + y = 6 x + y =. { x + 7x + y = x + y = 6. { y x + x + = x 6x + y = 8 7 P a g e
Summer 07 Section VIII: Functional Analysis - Round I (Calculator Inactive) Answer the questions that follow using the graph and the equation of the piecewise-defined function, f(x) shown at the right, y (x + ), x < f(x) = { x, x x +, < x x. Determine the value of f (f ( )). Determine all values of x such that f(x) =.. On what interval(s) will the graph of f be decreasing and positive?. On what interval(s) will the graph of f be concave up and decreasing? 5. At what point(s) will the graph of f(x) and the line x y + 6 = 0 intersect? 6. For x 0, f (x) exists. Express f (x) as a piecewise-defined function with appropriate domain restrictions. 8 P a g e End of Round I To Be Completed by August 9 th
Summer 07 Start of Round II To Be Completed by August rd Section I: Properties of Exponents and Solving Exponential/Logarithmic Equations In our AP Calculus course, we see exponent rules incorporated at many different instances. Particularly, behind our Calculus lies tons of Algebra that we need to ensure is mastered. Below are the exponent rules and some examples that will illustrate the rules you should know, be able to use, and recognize: Properties for Simplifying Exponents Property Rule Example Multiplication property for exponents x a x b = x a+b x 5 x 7 = x Division property for x a x = xa b exponents xb x = x9 The power property for (x a ) b = x ab (x 5 ) 7 = x 5 exponents The distributive property for exponents (xy) a = x a y a (x) = x = 8x The distributive property ( x m for exponents y ) = xm y m ( ) = = 8 7 Zero exponent property x 0 = 5 0 = Negative exponent x property = x n 5 = 5 = 5 The n th root property Rational Exponent Property n = x x n m n x n = x m 7 = 7 n = ( x) m 7 = 7 Cancellation of bases If x m = x n, then m = n Solve for m OR = = 79 = 9 7 = ( 7) = () = 9 5 m = 5 5 m = 5 m = m = 5 Example #: Simplify x y x y = x x y y = x +( ) y + = x y 0 = x Example #: Simplify ( ) 5 = () 5 () 5 = ( 5) ( 5) since 5 = and 5 =, = = 8 6 9 P a g e
Simplify each of the following: AP Calculus Summer Assignment Summer 07 x y x y x x 65 5 e x (e x + e x ) ( 8 6 + y 5 ) y 0 P a g e
Summer 06 Section II: Radian and Degree Measure (Calculator Inactive) Multiply the angle by 80o Multiply the angle by π π 80 o to get rid of radians and convert to degrees. to get rid of degrees and convert to radians Convert each radian measure to degrees:. 5π 6. π 5. 7π Convert each degree measure to radians:. 5 o 5. 660 6. 5 o Section III: Reference Triangles (Calculator Inactive) Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. 7. π 8. 5 9. π 0. 0 P a g e
Summer 06 Section IV: Unit Circle (Calculator Inactive) You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x coordinate of the circle is the cosine and the y coordinate is the sine of the angle. Example: sin90 o cos 0 (-,0) (0,) (,0) (0,-) - Evaluate the following. sin 80. cos 70 (0,) (-,0) (,0). sin π. sin π (0,-) - 5. cos π 6. cos π P a g e
Summer 06 Section V: All Six Trigonometric Ratios (Calculator Inactive) The Trigonometric Ratios You Need To Know For any right triangle, sine = opposite hypotenuse cosecant = hypotenuse opposite cosine = adjacent hypotenuse secant = hypotenuse adjacent tangent = opposite adjacent cotangent = adjacent opposite Mnemonic Devices: SOH CAH TOA and CHO SHA - CAO Example Evaluate sec ( 5π ) First, create your reference triangle. Since 5π = 50, the angle is in the third quadrant. See diagram below. Since the secant of an angle involves finding the ratio of the hypotenuse to the adjacent side, we have sec ( 5π ) = = = = = Evaluate the following trigonometric ratios. All answers must be reduced to simplest form.. sec( 0 ) =. csc ( π ) =. tan ( π ) =. sec ( π ) = 5. csc(π) = 6. cot ( π ) = P a g e
7. AP Calculus Summer Assignment Summer 06 tan ( π ) = 8. cot ( 5π 6 ) = 9. csc ( π ) sec (5π ) = 0. cot(5 ) cos ( π 6 ) =. tan ( 0 ) =. sin ( π ) csc ( π ) = Section VI: Inverse Trig Functions (Calculator Inactive) Recall: Inverse trigonometric functions can be written in one of two different ways: arcsin x sin x Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains. cos x < 0 sin x > 0 cos x > 0 tan x > 0 Example: sin x < 0 tan x < 0 Express the value of y in radians. y = arctan ( ) Draw a reference triangle. - This means the reference angle is 0 or π. So, y = π. Since that it falls in the interval from 6 6 y Answer: y = π 6 P a g e
Summer 06 For each of the following, express the final answer in radians. Arcsin ( ). Arccos ( ). Arctan( ) Example: Find the value without a calculator. cos arctan 5 6 6 Draw the reference triangle in the correct quadrant first. 5 θ Find the missing side using Pythagorean Theorem. 6 Find the ratio of the cosine of the reference triangle. cos 6 6 For each of the following give the value without a calculator.. tan (arccos ( )) 5. sec (sin ( )) 6. sin (arctan 5 ) 7. sin (sin ( 7 8 )) 5 P a g e
Summer 06 Section VII: Solving Trigonometric Equations (Calculator Inactive) Example Solve the equation cos x sin x + cos x = 0 for 0 x π. The Math What I m Actually Doing cos x sin x + cos x = 0 Rewriting the original problem. cos x ( sin x + ) = 0 Factoring out the common factor of cos x. cos x = 0 or sin x + = 0 Setting each factor equal to zero. cos x = 0 or sin x = If cos x = 0, then x = π or x = π If sin x =, then x = π or x = 5π x = π, π, π, 5π Solving the second factor for sin x = 0 Using a reference triangle if needed, determining which angles make the trigonometric ratio equal to the needed value. Writing my final answer.. sin x + = 0, [ π, π]. sin x cos x sin x = 0, (, ). sin x =, [0, π]. cos x cos x = 0, all real numbers 5. cos x = 0, (, ) 6. sin x = sin (x), (, ) 6 P a g e
Summer 06 Section VIII: Functional Analysis Part II (Calculator Inactive) Mr. Desrosiers cat Jewels is running along a straight line in the backyard. Her velocity at time t, measured in seconds, cane be expressed using the sinusoidal function: j(t) = sin ( πt ) + for 0 t 5 where j(t) is measured in meters per second, and time t, is measured in seconds. If the cat s velocity is positive, this represents the cat running away from Mr. Desrosiers. Using this information, answer the questions that follow. (a) Sketch a graph of the cat s velocity for the time interval 0 t 5. Be sure to clearly label the maximum, minimum, and midline points for a full period of the graph. (b) Rewrite the cat s velocity as a cosine function. (c) Evaluate j(5). Using correct units, explain the meaning of your answer in the context of this situation. (d) When the cat is at rest, j(t) = 0. For the time interval 0 t 5, determine all of the times at which the cat is at rest. 7 P a g e
Summer 06 (e) Determine the average rate of change of the cat on the time interval [,7]. Simplify your answer as much as possible. Using correct units, explain the meaning of your answer in the context of this situation. (f) Mr. Desrosiers also has a dog named Buttercup who is running along a different straight line in the backyard. Buttercup s velocity can be modeled by the sinusoidal function b(t) = cos (πt) + cos(πt) When Buttercup first comes to rest, how much faster is Jewels running than Buttercup? End of Round II To Be Completed by August rd 8 P a g e
Summer 06 Start of Round III To Be Completed by September 6 th Section I: Properties of Logarithms (Calculator Inactive) Recall: The properties of logarithms can be summarized in the table below. The Property log b x = y b y = x log b (m p ) = p log b m log b (m n) = log b m + log b n log b ( m n ) = log bm log b n m = n log b m = log b n If m > n, then log b m > log b n log b b = log b = 0 Example: Write the following expression in terms of log M and log N: log (MN) Begin by expanding the expression using the properties of logarithms. log(mn) = log (M N ) = log(m ) + log(n ) = log(m) + log(n) Example: Express as a single logarithm or rational number: log 8 log 7. Begin by condensing the expression using the properties of logarithms. log 8 log 7 = log 8 log 7 = log 8 log = log 8 = log 6 = For each of the following, write each expression in terms of log A and log B.. log A B =. log(a B ) =. log ( A B ) Write each expression as a rational number or a single logarithm.. log 69 + log 6 5 5. ln 6 ln 6. log A + log B log C 7. log π + log r 8. (log bm log b N log b P) 9. ln + ln 6 ln 9 9 P a g e
Summer 06 Section II: Solving Exponential Equations Example Solve the equation 7 + 5e x = 0. The Math What I m Actually Doing 7 + 5e x = 0 Rewriting the original problem. 5e x = Subtracting 7 to the right side. Dividing both sides by 5 to isolate the exponential e x = 5 ln(e x ) = ln ( 5 ) term. Taking the natural log of both sides. Choose a logarithm with the same base as the base of the exponential term, as log ( x)ln (e) = ln ( 5 ) Bringing down the exponent to the front of the natural logarithm. x = ln ( 5 ) ln (e) =. x = (ln ( 5 ) ) x 0.869 Isolate the x. This is your exact answer. This is the estimate generated from your calculator. Solve the following exponential equations for x. Find the exact answer and then use your calculator to generate an estimate accurate to places after the decimal.. x =. 6 x+ = x+7. 5 x x+ = 0. e x 5 9 = 0 0 P a g e
Summer 06 5. 7 x+ = ( x )(9 x+ ) 6. e x e x e x + e x = 7. e x e x = 0 8. (e x ) = e x P a g e
Summer 06 Section III: Conic Sections (Calculator Inactive) (x h) + (y k) = r (x h) a (y k) b Minor Axis b a CENTER (h, k) FOCUS (h - c, k) c FOCUS (h + c, k) Major Axis For a circle centered at the origin, the equation is x y r, where r is the radius of the circle. For an ellipse centered at the origin, the equation is x a y b, where a is the distance from the center to the ellipse along the x axis and b is the distance from the center to the ellipse along the y axis. If the larger number is under the y term, the ellipse is elongated along the y-axis. For our purposes in Calculus, you will not need to locate the foci. Graph the circles and ellipses below. When creating your complete graphs, be sure to include the center and the four extreme points.. x y 6. (x+ 7) 6 + y = P a g e
. (x+) + (y ) 9 AP Calculus Summer Assignment Summer 06 =. (y ) (x ) = End of Round III To Be Completed by September 6 th P a g e