AP Calculus AB Summer Assignment

Transcription

1 AP Calculus AB Summer Assignment As Advanced placement students, our irst assignment or the school ear is to come to class the ver irst da in top mathematical orm. Calculus is a world o change. While words like it, continuit, derivative and integral ma seem oreign and daunting to ou, I assure ou that at this time net ear at this time the will seem as commonplace to ou as graduation and senioritis! In order to begin our journe, ou must do some mathercise to keep our mind in shape through the summer months. So, I have prepared our assignments to help ou burn our mathematical muscles! Make sure all o our work is presented in a logical order showing ALL o our work indicating our answers clearl. IF I CAN T READ YOUR HANDWRITING, IT IS WRONG!! This packet is due the periodicall throughout the summer. Below is a list o due dates. It will be graded. You need to get o to a good start so spend some qualit time on this packet this summer. Do not rel on the calculator; use the calculator onl on the problems where calculator use is indicated. Hal o the AP eam net ear is taken without a calculator. All assignments need to be to me b the indicated date. You can drop it o at the main oice or mail it to me at m home; I MUST have an original. Summer Assignment Part : Due b noon Jul th Summer Assignment Part : Due b noon Jul 9 th Summer Assignment Part : Due b noon Jul th Summer Assignment Part : Due b noon August 9 th Summer Assignment Part : Due the irst Da o Class There will be a quiz on the unit circle on the irst da o school, be prepared! I ou have an questions about an o these problems or techniques used in solving them, ou ma contact me at the school website/ address: bquick@barnegatschools.com Have a great summer!

2 Below are listed topics in the review. You can certainl do Google searches or an o these topics. But we have given ou several sites that will cover prett much all o these topics. Here is a good site or most algebra topics: Beginning algebra topics Eponents Negative and ractional eponents Intermediate algebra topics Domain Solving inequalities: absolute value Solving inequalities: quadratic Special Factoring ormulas Function transormation Factor theorem (p over q method) Even and odd unctions Solving quadratic equations and quadratic ormula Advanced algebra topics Asmptotes Comple ractions Composition o unctions Solving Rational (ractional) equations Trig Inormation Once in the site, go to lecture notes. Basic right angle trig Trig equations Limits This packet is a review o some Pre-calculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet o paper. Use our discretion as to whether ou should use a calculator or not. When in doubt, think about i I would use one that should guide ou! Points will be awarded onl i the correct work is shown, and that work leads to the correct answer. Have a great summer and I am looking orward to seeing ou in September.

3 Summer Assignment Part : Due Jul, 07 Name: Cell: . Solve each equation or. Give eact answers onl. a. b c. 7 0 d. e. 0. ln e g. 8 0 h. sin 0 and 0 i. ln ln ln j. cos sin and 0 k. e l. ln ln e. Equation o a line Find an equation o a line passing through (, ) and having the ollowing characteristics. a. Slope o. b. Perpendicular to the line 0. c. Passing through the point (, ). d. Parallel to the -ais.. Zeros o Polnomial Functions a. Given the act that one root o the polnomial unction ( ) the other zeros o the unction. is, ind b. I is a actor o g ( ) 9 other two actors., ind the c. Find all the zeros o ( ) using the actor theorem. d. Find all the zeros o h( ) 8 9 using the actor theorem.

4 . Tangent Lines Consider the circle 0. a. Find the center and the radius o the circle (write the equation o the circle in standard b. Find an equation o the tangent line to the circle at the point (0, 0). orm ( h k r center and the radius is r), where (h, k) is the c. Find an equation o the tangent line to the circle at the point (, 0). d. Find the coordinates o the point where the two tangent lines intersect.. Intercepts and Points o Intersection Find the and intercepts or each. a. b. c. d.

5 . Solve the Sstem o Equations a. 7 g b. g c. g d. e g h. 7. Inverses Find the inverse o each unction a. ) ( b. c. 7 ) ( d. 7 ) ( e. ) ( 8. Simpli the Rational Epression a. b. c.

6 9. Comple Fractions Eample: Simpli each o the ollowing a. b. c. d. a a a e..

7 Summer Assignment Part : Due Jul 9, 07 Name: Cell: . Functions: Answer the ollowing questions about each unction.

8 Transormations o Functions. The unction H () is deined: H, 0, 0 a. b. c. H d. H H H. The unction g() is deined: g, 0, 0 a. g b. g c. g d. g

9 Operations with Functions ind. Given ( ). Given ind. Find or 7. Find or 8 8. Let ( ) and g ( ). Find each o the ollowing. a. () b. g() c. ( t ) d. g d. g m e. h h 9. Let ( ) and g( ) and h ( ). Find each o the ollowing. a. h b. g c. g h h 0. Find or the given unction. h a. ( ) 9 b. ( ) c.. For ( ) and g( ), ind the ollowing: a. g b. g c. g d. g * e.. g h g. g h g h. g h g( ) h

10 Piecewise Functions. Graph the unction. Then ind the indicated values., 0 ( ), 0 a. b. 0 c. d. 0. Graph the unction. Then ind the indicated values., ( ) ln, a. b. c. 0 d.

11 Open Ended Let be a unction deined on the closed interval [0, 7]. The graph o, consisting o our line segments, is shown above.. Write the piecewise unction or ().. Find the values o or which () =. 7. For what interval(s) o does have a rate o change o? Eplain our answer. 8. On which interval o is the rate o change o the greatest? Justi our answer. The unction g is a transormation o the above unction. Write an equation or g in terms o. 9. The maimum value o g is located at the point (,). 0. The maimum value o g is located at the point (,).. The minimum value o g is located at the point (, -).. The range o g is [0,].

12 Summer Assignment Part : Due Jul, 07 Name: Cell: . Without a calculator, determine the eact value o each epression. Please note: You MUST be asble to do this in our sleep b the time ou get back to school WITHOUT A CALCULATOR!! This is one o the single most important sills necessar or ou to be successul in AP Calculus! a. sin 0 b. 7 cos c. tan d. cossin e. sin. cos g. tan h. 7 sin sin i. sin j. 7 tan k. tan l. sin m. cos n. cos o. tan. I sin and, ind the values o the other ive trigonometric unctions.. Evaluate each o the ollowing. a. 7 b. coscsc tansin c. d. cos cot tan. I. I csc and is in the second quadrant, ind the remaining trigonometric ratios. cot and is in the third quadrant, ind the remaining trigonometric ratios.

13 Unit Circle Trigonometr. You will be epected to know the values o all trigonometric unctions at radian measures 0,,,,, and, and all radian measures that are multiples o these -- and ou must know these cold, without a calculator. These can best be done b reerencing the unit circle, a circle o radius centered at the origin. Ever point on that circle can be interpreted as (cos,sin ), where is the angle (, ) ormed with the positive -ais. It s onl necessar to memorize the important irst-quadrant points -- the rest can be reckoned b relecting appropriatel. (See diagram below.) The remaining trigonometric unctions can be computed rom sines and cosines. You will have a quiz on these values on the irst da o class, and our score on that quiz will either be 00% or 0.

14 Summer Assignment Part : Due August 9, 07 Name: Cell: Evaluate each it using the graph below.. Evaluate the Limit a. c. b. d e g. h. 7 i. j

15 . Use the graph below to answer the ollowing questions. a. b. c. For what value o a is a noneistent? k. Consider the unction ( ), sin In order or the unction to be continuous at =, the value o k must be sin 0. Consider the unction ( ), k 0 In order or the unction to be continuous at = 0, the value o k must be

16 Summer Assignment Part : Due First Da o Class Name: Cell: Use the inormation given about the region R to answer questions -. The region R is the area bounded b the graphs o 0. Sketch the region R. Find the area o R. I the region r is revolved about the -ais, what is the volume? (perhaps the ormula or volume o a cone will help!). Graph a. Eplain wh this graph is a linear unction instead o a rational unction having a vertical asmptote at. b. I ou had to ill in a -value when =, what would it be?. The graph below shows the time versus distance plot or two runners (Runner A and Runner B). a. Which runner had a head start? How ar did runner A go b the end o the race? b. Which runner ran astest? How much aster? c. How much longer did it take or the loser to cross the inish line? d. How ar did the winner win b?

17 . The table below shows the growth o a certain pupp. Use a linear regression equation or the data to estimate the weight o the pupp at age 8 weeks to the nearest pound. Age (weeks) Weight (pounds) A utilit compan serves two dierent cities. Let P be the requirements in megawatts (m) or cit and let P be the requirement or cit. Both P and P are unctions o t, the hours elapsed since midnight. Suppose and P P are given b the ollowing ormulas: P 0 cos t and P 0 0sin t a. Describe the power requirements o each cit in words. b. What is the maimum total power the utilit compan must be prepared to provide. 8. A 00 Leus costs $,0 and the car depreciates a total o % during its irst 7 ears. a. Suppose the depreciation is eponential. Find a ormula or the value the car at time t. b. Suppose instead that the depreciation is linear. Find a ormula or the value the car at time t. c. I this were our car and ou were trading it in ater ears, which depreciation model would ou preer (eponential or linear) 9. Water is lowing at a rate o m /s into the conical tank shown at the right. Find the volume V o the water as a unction o the water level h.

18 0. A light m above the ground causes a bo.8 m tall to cast a shadow s meters long measured along the ground, as shown. Epress s as a unction o d, the bo s distance in meters rom the light.