CALCULUS AB SUMMER ASSIGNMENT

Transcription

1 Dear Prospective Calculus Students, CALCULUS AB SUMMER ASSGNMENT Welcome to AP Calculus. This is a rigorous, yet rewarding, math course. Most of the students who have taken Calculus in the past are amazed at how much they have to rely on prior knowledge from Algebra and to complete a problem. Many times they find that it is not the Calculus steps that trip them up, but the embedded Algebra that needs to be done. To help you prepare for this course and make it through the "tedious algebra", feel that it would be beneficial to show you some of the skills you will come across and have you practice them. You will also have to develop your critical thinking skills in this class. For some of you, this is a natural skill, for others, not so natural. There are many problems in this packet that don't "look" like something you know how to do, but are not that bad once you figure out what it's asking. This is typical of the type of problems that you will see in Calculus next year and want to give you a chance to start practicing your problem solving skills early. n addition to reviewing some of the past material learned, we need to get a jump start on the new material. n order to get everything taught in time, would like to have you start to leprn the first topic in Calculus- Limits. We have shown you examples and given you a few to try on your own. You were also introduced to the topic in PreAP Pre-Cal, although it may not have been called "limits". You should do this packet without a calculator unless indicated. f you are asked to use a calculator, round everything to three decimal places. The answers are given so that you may check you work, and check your UNDERSTANDNG of the problems. This assignment is not to memorize answers or to avoid work; instead it is to help you recognize the steps, connections and abstract ideas of algebra that are imbedded within the topics of Calculus. Please be sure to spend adequate time on these problems so that you will be prepared for next year! f you find that you need to ask any question at all, please feel free to me at my district smcclelen@mesguiteisd.org may not answer immediately, but WLL answer!! n order to ensure that you complete the packet, you will be tested over this material sometime during the first two weeks of school. You will have a chance to get your questions answered the first week of school. Have fun this summer, but do a little bit of studying. Believe me, a little studying now will save a lot of time this coming school year. look forward to meeting all of you in the fall. Mr. Mac

2 Part - A\gebra SkiUs Simplify the following. 5(x+hY -5x 3 1. h (x-ly (3x-l)-(x-l)=B. 4 (x-1) (a/b)-a 3. a+(afb) x(x+ly -3(x+lY 4 8x x + 6x Solve for x. 5. 3xt )=0 (x-1) x (x+l)(x-1) +(x+5)(x+1) =0 B. Which of the following lines have a greater slope? a. b. b a Write the expression as a sum of terms. 9 _ u+l u ul/ + ulf Fu 5x J;:.fx-4

3 n- FunstjON.x ~'). g(x) l 4 1 s ~ <: l s 3.~ s 4.-l 4 1 s 1 s 4 1.Fi... (_1~-)). 1. Find -l(.ro)) 1.Given )=- /(x+l)- + find ) r questions given the groph of = f (x) below. graph the following transformations. 15./{ ) 16./(.!.x) 17. /(-x) 18. Jt(x~ 19. (Calc) L~t f(x) = x and l~t g(x) be a translation of /(z). f the range of g(x) is [ ] on the domain [-1. 5]. then what is g(x) in terms of f(x)? The graph of y=f(x) shown is ~ngent to ~he x-oxisot x=-3. hosa zero at x=3 and o relative minimum at x=l. Use thegroph of y=f(x) shown to on.swer, ~ questions Write a transformation of f(x) that will mak~ the relative. minimum ~ coordinates (5. -35)? ~ 1 1. Choose all of th~ following ~uation(s) that would have a relotiv~ minimum at (1. -8). g(x) =! /(x) lt(x) = -f(x -4)-8 lft(x) = f(x)- 8.Choose the transformation that would mo.ke /(x) have no real zeros? ' t /~zl>+ f(x)+33 -/(x)+33. /(x+3)+33

4 q 3.Given the graphs of f(x) and g(x) shown above, if h(x) = g(f(x)), what is the range of h(x)? ~1 ( ftx) g(x) 51!'.<!} ~ 1 l+ 3) 3 4.(Calc)f f(x) = x -1, find the solution to the equation f(x) = f - 1 (x). 5.Given the function f(x) = { x+s -x+5 equivalent to f(x). x < 0, write an absolute value equation that is x~o U th f. h. {x -4x+3 x<3 se e unctton (x) = to answer questions 6-7 x-9 x~3 6.What is the minimum value of h(x)? 7.For what values of x is h(x) =8? 8.A swimming pool can hold a maximum of 360 gallons of water. The full pool develops a leak and is losing water at a constant rate. After 3 hours the pool has 354 gallons of water in it. a. Write a function g(t) for the total number of gallons of water that is in the pool in terms of the time, t, the number of hours since the pool developed the leak. b. Find g(0). Explain the meaning of the answer in the context of the problem. c. f the leak is fixed after 0 hours and the owner immediately begins to fill the pool at a rate of 4 gallons of water per hour, write a piecewise function, f(t), for the total number of gallons of water that is in the pool in terms of time, t. 9.f(x) =-lx+3j- a. Graph f(x) b. Domain: Range: c. /(3) d. f f(x) = -3, then x =---- e. Rewrite without absolute values.

5 X ' f/ X~~ 4 30.g(x).fx. ({0 <x< 4 x, tf x< O a. Graph g(x) b. g(- 3) ---- c. s g(x) contl nuou? go)!j (O) Given t he graph of g(x) on the right, a. Estimate 8( 6 )- g(o) 6-0 b. The ratio in part (a) is the slope of a line 'egment joining two points on the graph. Sketch n thlt line segment, c. Estimate the slope of the graph at the point (0, 0), Draw in this line. 3, The rote at which water is entering a tank (t > 0) ;, repre1ented by the given graph. A 1 negative rate means that water is leaving the tonk. State the interval( ) on which each of the following holds true: a. The volume of water is constant. b. The volume of water is decreasing. c. The volume of water is increasing. d. The volume of water is increasing f~tut. Pgrt - Bevrew of Polynomrgl god Batrongl Eunctrons 3x 33.Q(x) - x+l a. Where is this function discontinuous? b. State the equation of the vertical Cll)'mptote. e. State the equation of the horizontal asymptote. d. Sketch. 34.Find a value of p in the inequality p+x- x ~ 4 if the solution is - l ~xs.!.

6 _. mptotes are x = -3 and x = -1, Write an equation of the function below 1f the vertlca asy horizontal asymptote y = -1 and x- intercept of 1. \!Vj : : i--r Given f(x) = x -3x+, find the value of x that gives a slope of zero for the secant line passing through the two points (x,f(x)) and (x+3,f(x+3)). x -a 37.Let f(x) = ( )( ). Find values of a,b,c and d such that f(x) would have b X-C x-d. 1 x- intercepts at ±4, vertical asymptotes x = 9 and x = -, and end behavior asymptote y = \ 1!- : Use the given the rational function f(x) = -+ x +x-s to answer x- 38.Write the equation of the slant asymptote on the graph of f(x). 39. (Calc) At what point on the graph does f(x) have a relative minimum? 40. f g(x) = f(x)(x-), find the zero(s) of g(x). 41. f h(x) = f(x) + c, find the value of c needed to make the relative minimum value of h(x) equal to the relative minimum of g(x)... (ax-)(x+b) 4. Gtven the functton, h(x) = ( )( ). Which of the following statement(s) are true? x+b x - c. The function has a vertical asymptote at x =c.. the function has a vertical asymptote at x = -b.. The function has a horizontal asymptote at y =a... 3x 3 +4x -x+ 43. What S the slant asymptote of the functton g(x) =? x -x-l

7 Part V - Review of Exponential and Logarithmic Functions 44. Sketch a graph of y =ex and y = ln X. 45. Evaluate the following. a. log 16 d. lnl c. loglo f. lne Write an equation for the inverse of f(x) = e :r:- 47. Which of the following would be equivalent to ~n ( 4~ ) for x > 0?. ln+ lnx T. ln3+ n~ T. 5(1n+ lnx) 48. f f(x) = 3:r: then what is f (b+3). 49. Graph g(x) =31n(x) using transformations (without a calculator). 50. (Calc) Let g(x) =4xe 4 x what are the coordinates of the absolute minimum point of g(x)? Part V - Rates of Change 51. Consider the curve x + 4y = 7 + 3xy. The slope of the curve at any point on the graph can 3 be found using m y- x. Find a point on the curve at which the slope is 0. 8y-3x 5. for the curve 3y -x+ y = 1, the slope function m for any point on the curve is given by 1 m. Find the point on the curve where the slope is undefined. 6y+ 53.A particle moving in a straight line has a velocity given by v(t) = sin(t) -1. What is the rate of change of velocity, called the average acceleration, of the particle over the interval ( 1! 1!)? 6'4 Part V- Review of Trig 54. Evaluate without the use of a calculator or chart. a. tan(:) b. ros( -;) c. sin(1r) d. sin- 1 ( ~) e. ros-t) f. arctan ( -J3)

8 55. Determine the domain, range, and period for six parent trig functions. 56. Of the six trig functions, which ones are even? Odd? 57. What is the domain of the function f(x) = arcsin(x+)? 58. The equation v(t)=sint+1 represents the velocity (rate of change of position) in feet per seco~d at any time t ~ 0 of a particle's movement along a horizontal line. At which time(s) would the particle be at rest (a particle is at rest when it's velocity is zero). 59.(Calc) A particle moves along a horizontal axis so that its velocity in feet per second at any time t ;, 0 is given by v(t) = cosg). When the velocity is negative, the particle is moving to the left. Find the intervals of t, 0::; t ::;Q, when the particle is moving left. 60. For what value of a is sin(x+a}=cosx? 61. Graph the following f(x) =3cos4( x- ~ )-. Use the functions below to answer questions f( ) ( ) x =cos x g x =-smx f /(1.5) = 0.005, what is the value of f( -1.5). h(x) = cos(x) What is the maximum value attained by g(x)? f the domain of g is restricted to [0, n-], when will g(x) obtain the maximum value? 64. Find all values of x in [0, 7r] such that f(x)=g(x)? 65. Simplify h(x) to show that h(x) = f(x) for all real-vafues of x.

9 part V - An ntroduction to Limits The limit of a function is they-value that you are getting close to as x gets close to some number in the domain. We write limf(x), which is read "the limit of f(x) as x approaches a". X-Ta The limit must be the same as x approaches "a" on both the left and the right. There are many ways to find a limit. We will focus on three main ways for now: from a graph, from a table, and by direct substitution. Study the following examples of each type and then try the sample exercises on your own. To find the limit from a table, look at they-values as.the x values get closer and closer to your "a'' value. See if there is one number that all y-va/ues seem to be going towards. Example: Find im(x ). X-7 X y Solution: We are looking for they-value as our x-values get closer and closer to. Looking at the chart from both the left and right sides, we can see that our y-valu are getting closer and closer to 4. Therefore, lim(x ) = 4. X-7 Your turn: Find the following limits using the charts. 1. lim ( x -1). x-73 X y :. lim(x). X-T-J l ~.999 ~' l-1.01 l-1.1 Solution: 1. 8;. -

10 To find a limit from a graph, we follow the graph on either side of our "a" value towards ~ that "a" value. The answer will be whatever y-value our graph is approaching. t is important to A.. o note that the graph does not have to actually "hit" that y-va/ue. A limit is simply "what the y v~lue should be." f there is no clear y-value that your graph is approaching, or if there are two dtfferent y-values that your graph is approaching, then the limit does not exist (DNE). Example: Find the limit as x approaches for each of the graphs below. a. 7 b. limf(x)=7.x-+ c. d. does not exist ~ limf(x) does not exist.x-+ Your turn: The graphs of some functions a~e pictured below. Do you think that limf(x) exists? f so, state its value..x-+l l Solution: l Z; Z. Z; 3. DNE; 4. DNE

11 L find a limit algebraically, you can Jie direct substitution. Simply plug the a value into the 'e~uatlon and simplify. As long as you do not get 0 on both the tcp and bottom (if it i1 a fraction), the number you get s the limit, Example: Find the limit of the following functions. lim(3x - 1),..., Solution: We are looking for the limit as x approaches 3, so plug~ illfo t)-s expression 3x - L You get 8 as an answer. Therefore,!l(3x-l - Your turn: Find the following limits. 1, lim(xl - 1). lim(~ -6x+l) z-+l z-t-4 3, 1im(c05x) s-+tr ~13;Z.57;3,-1

12 A nswers to Summer Assignment (x+l)(x+3) 1. 3x -4x-5 b-b x(4x+3) 15x +15xh+5h (x-t b +a xy Ba. b>a 5. X=-- 6. X= ' ' Y = xy +x-5 5 8b. a>b sx --4x u 6 +u u hor shrink of _!_ 16. hor 17. reflect over y- 18. same graph 19. g(x) = f(x)-3 stretch of axis 0. f(x-4)-3 1. g(x),h(x). lf(x)j+ 3. [-3,1] 4. x= f(x)=-jxj+5 6. min is X= -1,8.5 8a. 8b. g(0) = 30 gal { ::::; t::::; 0 9b. D: all real; 8c.f(t)= g(t) = 360-t left in tank. 30+4! t>0 R:[ -<X),] 9c. -8 9d. -, -4 9 t+l x5-3 30b. 9~ 1, undefined e. f(x) = -x-5 x>-3 30c. not 1 31a. - 31c. undefined 3a. (0, A) continuous 3 3b. (F, ) 3c. (A, F) 3d. (D, E) 33a. x=- 33b. x=-1 33c. y= (x-1) 35.f(x) (x+3)(x+) y =x+ 39. (3, 5) a= 6;b = 3;c,d = 9, ±J , 43. y=3x check on 45a.4 45b.O 45c.1 calc 45d. O 45e.1 45f f- 1 (x) 47., 48. 7(3b) 49. check on calc 50. (-.5, -.368) 51. (3, ) (-3,- 5. ( -37 _!) fi -1 54a. ) 3 ' 3 Jr.J3 lnx+ 54b. _!_ 54c. 0 Jr 54d. Jr 54e Jr 54f check on calc 56. even: cosine, secant 57. [-3, -1] 3 Odd: sine, cosecant, tangent 58. 7Jr lljr 59. (3.544, 9.0) 60. Jr 61. check on calc 6 ' (;,J 64. Jr 5Jr 3Jr 65. work may vary 6'6'