Teacher Alina Sebe - PDF document

1 Teacher Alina Sebe
1 Teacher Alina Sebe AP Calculus (AB) 2019-2020 Prerequisite Summer Assignment Welcome upcoming AP Calculus students! About the AP Calculus AB and AP Calculus BC courses Building enduring mathematical understanding requires students to understand the WHY and HOW of mathematics in addition to mastering the necessary procedures and skills. To foster this deeper level of learning, AP Calculus is designed to develop mathematical knowledge conceptually, guiding students to connect topics and representations throughout each course and to apply strategies and College Course Equivalents AP Calculus AB is equivalent to a first semester college calculus course devoted to topics in differential and integral calculus. AP Calculus BC is equivalent to both first and second semester college calculus courses. It extends the content learned in AB to different types of equations (parametric, polar) and introduces the topics of sequences and series. Both courses are intended to be challenging and demanding, each of them being taught over a full academic year. Prerequisites Before studying calculus, you must be familiar with elementary functions: linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise – defined functions. You must also understand the language of functions (domain, range, odd and even, periodic, symmetry, zeros, intercepts and descriptors such as increasing or decreasing). As a prospective calculus student you should also know how the sine and cosine functions are defined from the unit circle and know the values of the trigonometric functions at the numbers: 0, π/6, π/4, π/ This Summer Review Packet contains many of the prerequisite skills from Algebra and Pre- Calculus which we will use in calculus and with which you should be familiar. You have seen and mastered these topics in previous classes, but I understand some skills may be rusty. Take the time to complete the problems in this packet, r

eviewing where necessary and come prepar
eviewing where necessary and come prepared at the beginning of next school year. These prerequisite skills will make your time in AP Calculus much easier and greatly increase the likelihood of a successful semester. Make sure to check your answers. Do not despair if you have difficulty with some of the problems. Please bring your completed packet the first day of school and be ready to ask questions about anything you did not understand. Have a great summer! Sincerely, Ms Sebe 2 Part A (Easy problems and problems of moderate difficulties) 1) For each y = f(x) find domain, range, x – intercept, y – intercept and sketch the graph (do not use your graphic calculator) a) ݕ=2௫ ݂) ݕ=݈݋݃2ݔ b) ݕ=1+3−2௫ ݃) ݕ=ln1௫ c) ݕ=41−௫ ℎ) ݕ=݈݋݃12 (ݔ+2) d) ݕ=(15)௫−4 ݅) ݕ=݈݋݃3௫−19 e) ݕ=(12)−3௫+1 ݆) ݕ= ݈݋݃௢.5(8ݔ) 2) Solve the equation ܽ) ݈݋݃5(ݔ−3)=݈݋݃5√ݔ+3 ݁) 4௫+2௫−2=0 ܾ) ݁ln(1+௫)=(2ݔ+2)2 ݂) (12)2௫−(12)௫−6=0 ܿ) ݈݋݃2ݔ−݈݋݃2(√ݔ−1)=2 ݃) ݈݋݃32ݔ+݈݋݃3ݔ−2=0 ݀) ݁ln(1−௫)2=ݔ2−4ݔ+3 3) Solve each inequality (do not use your graphing calculator) ܽ) 4௫<8

݅) ln(1௫)>3 ܾ) 3௫<19 ݆) ݈݋݃2(݈݋݃13ݔ)<1 ܿ) 2∙2−௫<4 ݇) ݔ2−3ݔ+2<0 ݀) 9௫−2>13 ݈) ݔ2−3ݔ+2≥0 ݁) 2∙4௫+1<18 ݉) 4௫−3∙2௫+2<0 ݂) ݈݋݃2ݔ<1 ݊) (19)௫−3∙(13)௫+2>0 ݃) ݈݋݃12(ݔ+2)>4 ݋) ݈݋݃22ݔ−3݈݋݃2ݔ+2<0 ℎ) log(ݔ−3)<−2 4) Find the domain for each given function: ܽ) ݂(ݔ)=݈݋݃3(ݔ−1) ݂) ݂(ݔ)=44−9௫2+1௫3−27 ܾ) ݂(ݔ)=log(ݔ2+3) ݃) ݂(ݔ)=௫−1௫2−1−4√௫2−4 ܿ) ݂(ݔ)=݈݋݃2(4−8ݔ) ℎ) ݂(ݔ)=3௫√1−௫+7−௫√௫+6 ݀) ݂(ݔ)=݈݋݃௫(4−8ݔ) ݅) ݂(ݔ)=√௫2−௫௫+1 ݁) ݂(ݔ)=2௫+1௫2−1+7√௫−1−3௫௫2 ݆) ݂(ݔ)=ln (3−௫)√௫2+௫+1 3 5) ݃(ݔ)=ݔ3+4ݔ Find a) g(2a) b) g(-3ܽ2)

c) g(3-4a) 6) Simplify: ܽ)
c) g(3-4a) 6) Simplify: ܽ) ௫−2√௫+2−2 ܾ) ௫2−1௫÷௫+1௫3 Ü¿) 1�+4�23−1� ݀) 11−2௔−21+2௔+6௔+24௔2−1 ݁) �2�+ℎ−�2�ℎ ݂) 3(௫+ℎ)2−3௫2ℎ ݃) √ݔ∙√ݔ3∙ݔ16 7) True / False problems: ܽ) ௫+௬2=௫2+௬2 (�/� ) ܾ) 1௣+�=1௣+1� (�/� ) Ü¿) 3(௔+௕)௖=3௔+௕௖ (�/�) ݀) √ܽ2+ܾ2=ܽ+ܾ (�/�) ݁) 2�2௫+ℎ=�௫+ℎ (�/�) 8) xy + y + x = y a) Solve for x b) Solve for y 9) Factor: ܽ) ݔ2(ݔ−1)−4(ݔ−1) ܾ) ݔ4(2−ݔ)−16(2−ݔ) 10) Evaluate each of the following – leave answer in radical form (do not use calculator) ܽ) sin�6−cos2�3= c) cos3�4−cos−1(12)= ܾ) sin(−�4)+tan�= ݀) sin−1�6+ݐܽ݊−1(1)= 11) Solve:

ܽ) cos௫2=√2
ܽ) cos௫2=√22 0≤ݔ≤2� ܾ) 2ݏ݅݊2ݔ+sinݔ−1=0 0≤ݔ≤2� 12) Show work to determine if the relation is even, odd or neither: ܽ) ݂(ݔ)=3ݔ2−6 ܾ) ݂(ݔ)=−2ݔ3−7ݔ Ü¿) ݂(ݔ)=3ݔ2−3ݔ+3 13) Find the equation of a line: a) Passing through (-2,4) and (3,-5) b) Passing through (3,7) and parallel to the line 3x+2y-8=0 c) That is perpendicular to the line 3x+2y-8=0 at the point (3,−12) 4 14) The line with slope 4 that passes through (-3,7) interests the x – axis at point A and y – axis at point B. Find the coordinates of each point. 15) ݂(ݔ)=1௫−1 ܽ݊݀ ݃(ݔ)=ݔ2−2 Find: a) f(g(x)), b) g(f(x)), c) f(f(x)), d) g(g(x)) 16) Find the surface area of a box of height h whose base dimensions are l and w that satisfies the following condition: a) The box is closed, the base is rectangle. b) The box has an open top, the base is a right triangle with legs l and w c) The box has an open top and the base is square with side x 17) For each function: find domain, find range, sketch the graph. (do not use your calculator) ܽ) ݕ=√ݔ−2+1 ݁) ݕ=√4+ݔ2 ܾ) ݕ=௫−2௫2−4 ݂) ݕ=ݔ3−ݔ Ü¿) ݕ=|ݔ+3|−2 ݃) ݕ= ݔ3+ݔ ݀) ݕ= ݕ=√4−ݔ2 18) Three sides of a fence and an existing wall form a rectangular enclosure. The total length ef a fence

used for the three sides is 240 ft. Let
used for the three sides is 240 ft. Let x be the length of two sides perpendicular to the wall as shown below. ? x x Existing wall a) Write the equation of area A of the enclosure as a function of the length x of the rectangular area as shown in the above figure. b) Find value(s) of x for which the area is 5500 ݂ݐ2 c) What is the maximum area? 5 Part B (problems of moderate difficulties and challenge problems) 1) Express |2ݔ−4| as a piece-wise function 2) Find the equation of the line tangent to the graph of ݔ2−4 at point (1, -3) 3) Simplify : a) (௫2+2௫)−(௔2+2௔)௫−௔ b) − 2�−3 − ௫௫2−1 4) Simplify : a) (௫−1)2(௫−2)(௫6−64)(4௫+8)2(௫2−1)(௫3+8) b) ௫6+64௫2+4 5) Solve the equation: 4௫−2௫−12=0 6) Solve the inequality: 4௫−2௫−12≥0 7) Solve the equation: ݈݋݃3ݔ2−݈݋݃32ݔ2=0 8) ݂(ݔ)=ݔ2+3ݔ Does y = f(x) have an inverse? Explain why. 9) ݂(ݔ)=−ݔ2+4ݔ Does y = f(x) have an inverse? Explain why. a) When x b) When x � 0 10) ݂(ݔ)=23−௫ a) Sketch the graph of y = f(x) b) Find ݕ=݂−1(ݔ) and sketch the graph. c) Check (both algebraic and geometrical method) that ݕ=݂−1(ݔ) is the inverse of y = f(x) 11) y = f(x) is a periodic function with period of length 5 and graph given below: (-3, 2) (-2, 2) (2, 2) (2, 2) (-5, -2) (0, -2) (5, -2) Find f(108) 12) ݂(ݔ)={

ݔ2−4 −2≤
ݔ2−4 −2≤ݔ≤2ݔ−2 2<ݔ≤6 Sketch the graph, find domain, find range. 6 13) (5,4) f 2 (3,1) g Solve a) f�(x) g(x) b) f(x) g(x) 14) Find m and b such that for any real value for x 2x – (mx+b) = m 15) - ln (3 – y) = sin x – ln 2 Find y = f(x) 16) ݂(ݔ)=2√5−4௫2 Find domain. 17) (-4,2) (0,2) Find the area of the shaded region. (-2,0) (4,-2) 18) Solve sin�ݔ=ݔ3−4ݔ 19) The graph of y = v(t) is given left a) Solve v(t) 0 0 3 5 t b) Solve v(t�) 0 20) Solve: cos(�6ݐ)<0 ݂݋ݎ 0≤ݐ≤12 21) Sketch the graph: ܽ) ݔ2+ݕ2=4 d) ݔ=−√4−ݕ2 ܾ) ݕ=+√4−ݔ2 ݁) ݔ=+√4−ݕ2 Ü¿) ݕ=−√4−ݔ2 7 22) ݂(ݔ)=√25−ݔ2 Sketch the graph and find domain and range ݃(ݔ)={݂(ݔ) −5≤ݔ≤−3ݔ+7 −3<ݔ≤5 a) for y = f(x) b) for y = g(x) 23) Express ݕ=|ݔ2−16| as a piece – wise function 24) 8ݔ3=sin�ݔ Solve for x. 25) 3

y = f(x)
y = f(x) The graph of y = f(x) is given. Find f(x) using a piece-wise function -3 (3,-3) 26) ݂(ݔ)={−2cosݔ ݔ<0 �݅݊݀ ݐℎ݁ ݔ ݒ݈ܽݑ݁(ݏ)݂݋ݎ ݓℎ݅ܿℎ ݂(ݔ)=−3−4݁−4௫ ݔ>0 27) The graph of y = g(x) is given below 2π ݂(ݔ)=݃(ݔ)−cos௫2 Find f(π), f(2π), f(3π) -2 π 0 4 π 28) (3,3) The graphs for y = g(x) and y = x are given. 1 a) Solve g(x) � x b) Solve g(x) g -1 +1 (5,-1) 29) ݒ(ݐ)=sinݐ2 a) Solve v(t) = 0 (general solution) b) Solve v(t) = 0 (when −2�≤ݐ<�) 30) ݒ(ݐ)=60√ݐ2−16 ݏ݅݊2(�3) Solve v(t) = 0 when 0 t 18 31) f(x) = periodic function with f(7)=4 and period of length 6. Find f(127) 8 32) ݒ(ݐ)=ln(ݐ2−3ݐ+3) with 0 5 a) Solve v(t耀)0 b) Solve v(t) 0 33) ݂(ݔ)=݁−�4∙sin2ݔ Solve f(x) = 0 34) Solve : ܽ) 3௫−2∙(ݔ2−9)≤0 ܾ) 3�−2∙(௫2−9)−௫2−௫−1â