Avon High School Name AP Calculus AB Summer Review Packet Score Period f 4, find:.) If a.) f 4 f 4 b.) Topic A: Functions f c.) f h f h 4 V r r a.) V 4.) If, find: b.) V r V r c.) V r V r.) If f and g are given in the graph, find: a.) f g b.) f g, 0 f, 0, find:, 4.) If a.) f 0 f b.) 5 f 4 c.) f f
Topic B: Domain and Range Find the domain of the following functions using interval notation:.) f.) y.) y 4 4.) y 6 f 5.) 4 4 6.) y 9 5 7.) f t t 8.) f 9.) y 4 5 4.) y 49 0.) y log 0.) y 5 log Find the range of the following functions:.) y 4 4.) 00 y 5.) y Find the domain and range of the following functions using interval notation: 6.) 7.) 8.)
Topic C: Graphs of Common Functions Sketch each of the following as accurately as possible. You will need to be VERY familiar with each of these graphs throughout the year. You may use a graphing calculator for some of them if you have access to one over the summer. If you plan on renting a TI-Nspire and thus do not have one for the summer, I strongly recommend you use try www.desmos.com. There is an app for desmos as well that is free that you can install on your phones. Again, these are VERY important graphs to know. Be very accurate with regards to open circles and closed circles as those features may not be revealed on a graphing utility. For students who have not taken Trigonometry yet, do your best with # s 9-4.. y. y. y 4. y 5. y 6. y
7. y 8. y 9. y sin 0. y cos. y tan. y cot. y sec 4. y csc
5. y e 6. y ln 7. y 8. y 9. y 0. y. y 4. sin y
Topic D: Even/Odd Functions and Symmetry Show work to determine if the relation is even, odd, or neither. You may want to research how to determine evenness and oddness..) f 7.) f 4.) f 4.) f 5.) f 6.) f 8 7.) y 8 8.) f 8 8 9.) 8 8 f Show work to determine if the graphs of these equations are symmetric to the -ais, y-ais, or the origin. 4 0.) 4.) y 6.) 4y.) y 4.) y 5.) y y
Topic E: Function Transformations If f, describe in words what the following would do to the graph of f :.) f 4.) f 4.) f 4.) 5f 5.) f 6.) f Here is a graph of y f : Sketch the following graphs: y f 7.) 8.) y f 9.) y f 0.) y f.) y f.) y f
Topic F: Special Factorization Factor completely..) 8.) 8.) 7 5y 4.) 80 5.) ac cd ab bd 6.) 4 50 0 y y 7.) 6 9y 8.) y y y 9.)
Topic G: Linear Functions.) Find the equation of the line in point-slope form, with the given slope, passing through the given point. a.) m 7,, 7 b.) m,, 8 c.) m, 6,.) Find the equation of the line in point-slope form, passing through the given points. a.), 6,, b.) 7,,, 4 c.),,,.) Find the equations of the lines through the given point that are a.) parallel and b.) normal to the given line. a.) 5,, y 4 b.) 6,, 5 y 7 c.), 4, y 4.) Find the equation of the line in general form, containing the point 4, and parallel to the line containing the points, 4 and,. 5.) Find k if the lines 5y9and ky are a.) parallel and b.) perpendicular.
Topic H: Solving Quadratic Equations.) Solve each equation for over the real number system. a.) 78 0 b.) 0 c.) 4 7 0 d.) 5 e.) 0 56 5 0 f.) 8 76 0 g.) 0 7 h.) 4 5 i.) 7 7 0 j.) 7 k.) 4 5 5 5 0 l.) 4 5 8 0.) If y k k, for what values of k will the quadratic have two real solutions?
Topic I: Asymptotes For each function, find the equations of both the vertical asymptote(s) and horizontal asymptote (if it eists) and the location of any holes. 8 6.) y.) y.) y 5 8 4.) y 6 56 5.) y 5 6.) y 5 7.) y 4 8.) y 5 9.) 5 y 0.) y 4.) y 4 4 8 0 0.) y 4 8.) y (Hint: epress with a common denominator)
Simplify and write with positive eponents. 5.) 5.) Topic J: Negative and Fractional Eponents.) 4 4.) 4 4 5.) 5 y 6.) 8 7.) 8.) 4 8 9.) 5 5 0.) y.).) 6 4.) 4 4.) 5.)
Topic K: Comple Fractions Eliminate the comple fractions: 5.) 8.) 4 9 4.) 7 5 5 4 4.) 5.) 6.) y y 7.) 8.) 4 4 4 9.)
Topic L: Inverses.) Find the inverse of each of the following functions and show graphically that its inverse is a function. a.) 6y b.) y a b c.) y 9, 0 d.) y e.) 9 y f.) y.) Find the inverse of each of the following functions and show that a.) 4 f 5 f f b.) f 4 c.) f.) Without finding the inverse, find the domain and range of the inverse to f
Topic M: Adding Fractions and Solving Rational Equations.) Combine the following fractions: a.) b.) c.) 5 5 5 d.).) Solve the equation for. 5 a.) b.) 0 6 9 c.) 5 5 5 5 d.)
Topic N: Absolute Value Equations Solve the following equations:.) 4 8 0.) 7.) 8 40 4.) 45 5 0 5.) 7 6.)
Topic O: Solving Inequalities Solve the following inequalities:.) 5 8 5.) 5 4.) 4.) 7 5 4 5.) 5 6.) 4 7.) 5 6 8.) Find the domain of: 6 4
Topic P: Eponential Functions and Logarithms Simplify the following:.) log.) log8 4.) 4 ln e log5 40 4.) 5 5.) ln e 6.) log log 9 log 8 7.) 4 log log 8.) log log 9.) log 5 Solve the following: log 8 0.) 5.) log 9.) log log5.) log log 5 4.) log log 5.) 5 5 ln ln 6.) 8 7.) e 0 8.) 8 5
Topic Q: Basic Right Angle Trigonometry Solve the following: If point P is on the terminal side of θ, find all 6 trigonometric functions of θ. (Answers need not be rationalized.).),4 P.) P 5, 5 0.) If cos, θ in quadrant II, 4.) If cot, θ in quadrant III, find sin and tan. find sin and cos. 5.) State the quadrant in which each of the following is true. a.) sin 0 and cos 0 b.) csc 0 and cot 0 c.) tan 0 and sec 0
Topic R: Special Angles Evaluate each of the following..) sin 0 cos 0.) tan 00 sin 50 cos 80.) 4.) cot 0 tan 600 csc450 cot 5 sin 0 5cos 5 5.) cos tan 4 6.) 5 5 sin tan sin tan 6 6 6 6 Determine whether each of the following statements is true or false. 5 5 cos cos 7.) sin sin sin 8.) 6 6 5 5 tan sec 9.) sin cos 0 0.) 4 4 cos sin 0 4 cos
Topic S: Trigonometric Identities Verify the following identities:.) sin sin cos.) sec tan 4.) sec sec cos 4.) tan cot 5.) cos cos y sin sin y 0 sin sin y cos cos y 6.) sin cos sin cos sin cos 7.) csc csc 8.) cos cos cos 4sin
Topic T: Solving Trigonometric Equations Solve each equation on the interval 0,..) sin sin.) tan tan.) sin cos 4.) cos sin tan 5.) sin cos 6.) cos sin 0
Topic U. Graphical Solutions to Equations and Inequalities You have a shiny new TI-Nspire graphing calculator. So when are we going to use it? So far, no mention has been made of it. Yet, a graphing calculator is a tool that is required on the AP Calculus eam. For about 5% of the eam, a calculator is permitted. So it is vital you are comfortable using it. There are several settings of the calculator you should make. First, so you don t get into rounding difficulties, it is suggested you set your calculator to show at least three decimal places (and preferably more). This is sandard on the AP Calculs eam, so it s best we start getting used to it. To do this, press the button, followed by 5 Settings, then : Document Settings Be sure the Display Digits option is set to Float 6. This will ensure that you always see 6 digits across the screen. (There may be times that this can be a problem i.e. when you have a decimal answer with four or more digits to the left of the decimal. We ll deal with this later.) Also, be sure that your calculator s Angle Setting is in Radian mode throughout the year. To make these changes stick select Make Default at the bottom. You must know how to graph functions on your TI-Nspire. The best way to graph a function is to press the Calculator key (located in the upper left corner of your handheld just below ) twice on the left side. Notice how each time you press this button, your screen toggles between a calculator page and a graphing page. While on a graphing page, select e to bring up the function entry line. Net input the epression you wish to graph and press Enter. How to find zeros (solutions or -intercepts) of a function. Step : Enter the left side of the equation that s alreay set equal to zero (for eample, 9 0) into the f() entry line. Step : Select followed by 6: Analyze Graph and : Zero Step : Move the cursor to a position you feel is left of the zero you wish to find first and press Enter. Then move the cursor to the right of the desired zero you wish to find and press Enter. You will notice that the ordered pair for the zero will show up automatically once it falls within the range of your lower and upper bound. How to find the intersection of two functions. Step : Enter each side of the equation (for eample, f() entry lines. into the f() and Step : Select followed by 6: Analyze Graph and 4: Intersection Step : Repeat Step above. This problem could also be solved by setting the above equation equal to zero, and using the following procedure 6: Analyze Graph and : Zero instead. Note: You can always move things around on your screen and place them in positions that may make them easier to read. To do this, rub your finger over the Touchpad until the cursor appears. Move the cursor to the item (i.e. intersection ordered pair from the eample above) you want to move. An open hand should appear. Press Enter to close the hand. By rubbing the Touchpad, the ordered pair should move. Press Enter when you have found a suitable place to put it. We will learn much more about the functionality of this calculator in the first several days of class. While many of these problems on this page can be done with other graphing utilities (desmos, etc.), I highly recommned that you wait to do these problems when you have a TI-Nspire.
Topic U: Using the TI-Nspire (Continued) Use your TI-Nspire to find the zeros of each of the following functions. Make sure each equation is set equal to zero first..) 5 0.).) ln( ) 5cos on 0, Use your TI-Nspire to find the solution (intersection) of the given system of equations. 4 f ( ) 6.5 6 4.) g( ) e Use your TI-Nspire to find both the absolute maimum and absolute minimum values of the given function. 5.) h 5 4 ( ) 4