Montgomery County Public Schools



AP Calculus AB Summer Review PacketWelcome to AP Calculus AB. This packet is a set of problems that you should be able to do before entering this course. You are not required to do EVERY problem in this packet, but you are responsible for all content. We will not collect these problems, but you will have an online quiz on these questions during the first week of school. We’ll give you a few days to ask us questions and then you’ll have a graded test on this background material. Make sure you circle any questions that you are unsure about so that you can get clarification during the Q & A period before the test is given. We would recommend reviewing instructional videos on any topics that you may find unclear. We’d recommend Khan Academy, but there are numerous, excellent web sites that can help you. You need to be an active learner in Calculus AB, this is good practice for you. At the end of the problems, We’ve included an answer key for you to do some self-check. Probably the biggest difficulty for students is the limitations placed on the use of a calculator. Approximately 70% of the AP exam is without a calculator, so that is the way the course is taught. Approximately 70% of your evaluations will be without a calculator. (That includes a four function calculator, no calculator is allowed at all). When tested on this material, it will be without a calculator. Finally, we cannot stress enough the importance of these background skills. Calculus is easy, it’s the algebra that is hard. Most of the time, you’ll understand the calculus concept being taught, but will struggle to get the correct answer because of your background skills. Please be diligent and figure out what you need help with so that we can clarify for you. A little extra work this summer will go a long way to help you succeed in the upcoming year.We look forward to having you in class and working together to accomplish your goals. We think you’ll find this course challenging but enjoyable at the same time. Should you have any questions throughout the summer, please feel free to contact us via email at james_m_kuhn@. Hope you have a great summer!Mr. Kuhn and Ms. ThatcherAP Calculus AB Summer Review PacketSimplifyx3-9xx2-7x+12x2-2x-8x3+x2-2x1x-151x2-1259-x-23-x-1Rationalize the denominator23+241-51-51+3Write each of the following expressions in the form of capbq where c, p, and q are numbers2a23b9ab3a2b3aab-ab2-ba-1b-1aa23b122b32a12Solve for x. Do not use a calculator5x+1=2513=32x+2log2x=3log3x2=2log34-4log35Simplifylog25+log2x2-1-log2x-12log49-log2332log35Simplifylog101012log10110x2log10x+3log10x13Solve the following equations for the indicated variablexa+yb+zc=1, for aV=2ab+bc+ca, for aA=2πr2+2πrh , for positive hA=P+πrP, for P2x-2yd=y+xd, for d2x4π+1-x2=0, for xFor each equation complete the square and reduce to one of the standard forms y-y1=Ax-x12 or x-x1=y-y12 y=x2+4x+33x2+3x+2y=09y2-6y-9-x=0Factor completelyx6-16x44x3-8x2-25x+508x3+27x4-1Find all real solutionsx6-16x4=04x3-8x2-25x+50=08x3+27 =0Solve for x3sin2x=cos2x;0≤x<2πcos2x-sin2x=sinx; -π<x≤πtanx+secx=2cosx; -∞<x<∞Without using a calculator, evaluate the following:cos210°sin5π4tan-1-1sin-1-1cos9π4sin-132tan7π6cos-1sin-π4Given the graph of y=sinx, sketch the graphs of: 1692910106045sinx-π4sinx22sinxcosx1sinxSolve the equations4x2+12x+3=02x+1=5x+2x+1x-xx+1=0Find the remainders on division ofx5-4x4+x3-7x+1 by x+2x5-x4+x3+2x2-x+4 by x3+1The equation 12x3-23x2-3x+2=0 has a solution x=2. Find all other solutions. Solve for x, the equation 12x3+8x2-x-1=0 (all solutions are rational and between ±1)Solve the inequalities. Give the solution in interval notationx2+2x-3≤02x-13x-2≤122x+3>2x-5Solve for x. Give the solution in interval notation–x+4≤15x-2=82x+1>3Determine the equation of the following linesThe line through -1, 3 and 2, -4The line through -1, 2 and perpendicular to the line 2x-3y+5=0The line through 2, 3 and the midpoint of the line segment from -1, 4 to 3, 2Find the point of intersection of the lines: 3x-y-7=0 and x+5y+3=0Shade the region in the xy-plane that is described by the inequalities 3x-y-7<0x+5y+3≥0Find the equations of the following circles:The circle with center at 1, 2 that passes through the point -2, -1The circle that passes through the origin and has intercepts equal to 1 and 2 on the x and y axes respectively. For the circle x2+y2+6x-4y+3=0 find the center and the radiusFind the domain of 3x+1x2+x-2Find the domain and range of:fx=7gx=5x-32x+1fx=xxSimplify fx+h-fxh whenfx=2x+3fx=1x+1fx=3x2-x+51582180274371The graph of the functions y=fx is given as follows: Determine the graphs of the functions:fx+1f-xfxSketch the graphs of the functionsgx=3x+2hx=xx-1The graph of a quadratic function has x-intercepts -1 and 3 and a range consisting of all numbers less than or equal to 4. Determine an expression for the function. Sketch the graph of the quadratic function y=2x2-4x+3Find the inverse of the functionsfx=2x+3fx=x+25x-1fx=x2-2x-1, x>0544987200059A function fx has the graph below. Sketch the graph of the inverse function f-1x.476250263525For problems 96 and 97, express x in terms of the other variables in the picture:476251-1030Find the ratio of the area inside the square but outside the circle to the area of the square in the picture below27876540640Find the formula for the perimeter of the window of the shape in the picture below537845-1270A water tank has the shape of a cone (like an ice cream cone without the ice cream). The tank is 10m high and has a radius of 3 m as the top. If the water is 5 m deep (in the middle) what is the surface area of the top of the water?Two cars start moving from the same point. One travels south at 100 kmhr, the other west at 50 kmhr. How far apart are they two hours later?A kite is 100 m above the ground. If there are 200 m of string out, what is the angle between the sting and the horizontal. (Assume that the string is perfectly straight.)If fx=2x-3 and gx=3x-1, Find:fgxgfxIf fx=3x and gx=x2x-1, Find fgx and state its domain. Decompose each composition function into individual function. (If y=fu, identify uand rewrite y in terms of u)y=sin3xy=52x+1y=x2-2x+55y=cos2xAnswersx2+3xx-4x-4x2-x5xx+53x+1x23-2-1-51-3-5+15-28a6b-13a12b3223a2b-1ab-1a-32ba56b121-328±425log25x+1log232512–x2log10xbcxbc-cy-bzV-2bc2b+cA-2πr22πrA1+πr2x-yx+2yππ-1y+1=x+22y-38=-32x+122x+10=9y-132x4x-4x+4x-22x-52x+52x+34x2-6x+9x-1x+1x2+10, ±42, ±52-32π6, 5π6, 7π6, 11π6-π2, π6, 5π6π6+2kπ and 5π6+2kπ where k ? I-32-22-π4-π222π3333π445720054187 5842001739906934202222558547013017551625594827-3±6212 or -3-12-89x2+3-13 or 14-12, , 13, -12-3, 1-∞, 23∪[1, ∞)-∞, -8∪-32, 53, 52 and -65-∞, -2∪1, ∞7x+3y=23x+2y=1y=32, -146566764559x-12+y-22=18x-122+y-12=54Center = -3, 2, radius = 10-∞, -2∪1, ∞Domain -∞, ∞ Range 7Domain -∞, -12∪-12, ∞ Range -∞, 52∪52, ∞Domain -∞, 0∪0, ∞ Range -1, 12-1x+1x+h+16x+3h-146566748683676275-88906089656096060896520955061023510795 y=-x2+2x+3609600-6350f-1=x-32f-1=x+25x-1f-1=1+x+2 for x>-156726753340x=tr-hhx=rtr2-h21-π44r+πr9π41005 KMπ623x-1-36x-106x-3x Domain -∞, 0∪0, 12∪12, ∞Let u=3x, then y=sinuLet u=2x+1, then y=5uLet u=x2-2x+5, then y=u5Let u=cosx, then y=u2 ................
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