Summer Review for Students Taking Calculus in 2016-2017 No calculators allowed. To earn credit: Be sure to show all work in the area provided. 1 Graph each equation on the axes provided. Include any relevant information, such as domain, intercepts, end behavior, period, amplitude, and transformations. Graphing polynomial functions: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm Graphing using transformations: http://www.purplemath.com/modules/fcntrans.htm Graphing exponential and logarithmic functions: http://www.sosmath.com/algebra/logs/log4/log42/log42.html Graphing trigonometric functions: https://www.youtube.com/watch?v=ijtir-aykuk http://www.germanna.edu/documents/howtographtrigfunctions.pdf http://www.mathguide.com/lessons2/graphingtrig.html 1. y = 2x 2 6x 1 2. f ( x) = 2x 3 +15x 2 31x +12
2 3. h( x) = 2sin( x π ) 4. M(x) = 5+ log 3 (4 x) 5. k(x) = 3 e 6+x 6. r(x) = x 3 x 2 6x x 4 + 3x 3 9x 2 + 5x
3 Factor each expression completely. Factoring polynomials: http://www.purplemath.com/modules/simpfact.htm https://www.khanacademy.org/math/algebra2/polynomial-functions/factoring-polynomials-quadraticforms-alg2/e/factoring_polynomials_by_grouping_1 http://www.montereyinstitute.org/courses/algebra1/course_text_resource/u09_l1_t3_tex t_container.html 7. 3x 4 48 8. 2x 3 +8x 2 5x 20 9. e 2 x + 5e x 6 10. 2m 3 + 54 7 5 11. 15x 2 +19x 10 12. 2x 2 + 6x 2 2x 6 +10x 11 3 2 13. 3sin 2 x cos x + 2sin 2 x 3sin x cos x 2sin x
4 Simplify each expression completely. Operations on polynomials: https://www.youtube.com/watch?v=trpxmv29rx0 Division of polynomials: https://www.youtube.com/watch?v=smskmwf8zcs Simplifying complex fractions: http://www.regentsprep.org/regents/math/algtrig/atv2/simpcomplex.htm Rationalizing denominators: http://www.regentsprep.org/regents/math/algtrig/ato3/rdlesson.htm Evaluating trigonometric functions: http://www.mathsisfun.com/geometry/unit-circle.html Fractional exponents: http://www.purplemath.com/modules/exponent5.htm 2 1 14. x 5 15. x x 3 x 2 5x 2 45 2x 3 x 2 x 12 3x 4 +8x +1 15. Distribute over the denominator to simplify. 2x 12 17.. 18. x 2 + 5x 1 4 3i ( ) 2 19. 64 5 2 20. Divide: x 4 16 x 8
5 21. cos π 4 22. tan 2π 3 23. secπ 24. sin 7π 6 25. 150 26. 27 4 3 3 27. 216 28. arctan ( 1) 1 3! 29. sin 30. arcsin# cos 2π 2 " 3 $ & % Find the domain and range of each function. Express your answers in interval notation. Finding domain of a function: http://www.coolmath.com/algebra/15-functions/06-finding-thedomain-01.htm Finding the range of a function: http://www.purplemath.com/modules/fcns2.htm 31. y = 2x x + 5 32. f ( x) = x 2 4x + 2 33. g(x) = (2x 1)(x + 3)(3 x) x(x 3)(5x 4) 34. h(x) = 3 2 (4+x) 35. p(x) = ln(2x 3) 7 36. r(x) = 4 + csc( x π ) over the interval [ 2π,π ]
6 Solve each equation. Solving rational equations: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut15_rateq.htm Solving trigonometric equations: https://www.youtube.com/watch?v=fcrxdkwpspu Solving quadratic equations using completing the square: http://www.purplemath.com/modules/sqrquad.htm Solving exponential and logarithmic equations: http://tutorial.math.lamar.edu/classes/calci/explogeqns.aspx https://www.khanacademy.org/math/algebra2/logarithms-tutorial/logarithm_basics/v/exponentialequation http://cims.nyu.edu/~kiryl/precalculus/section_4.5- Exponential%20and%20Logarithmic%20Equations/Exponential%20and%20Logarithmic%20Equatio ns.pdf 37. x 8 + 5 x 6 = 4 38. cos x sin x = 0 on [ 2π, 2π ] 39. Find all solutions to tan2x = 1. 40. Solve 2x( x 5) =12 using the Quadratic Formula.
7 41. 2x 3 + 7x 2 19x 60 = 0 42. Solve xy + 2x 5 = 5y + 7x for y. 43. 2 e 5x = 13 44. 5+ ln(x + 3) = 2 Find all horizontal and vertical asymptotes for each function. Finding horizontal and vertical asymptotes: http://www.math-magic.com/algebra/asymptote.htm 45. y = x2 + 3 4x 2 1 46. p( x) = 2 x 2 9x 47. The page of a book has one-inch margins on all four sides. Express the area of the printed section of the page in terms of x, the width of the page. x in. (x + 4) in.
8 48. The bases on a baseball diamond form a square 90 feet to a side. A runner is x feet away from second base. Express his distance from first base in terms of x. x 49. Find the maximum value of the following expression. ( ) 2 21 2 x + 4 ( ) 2 50. Given the equation 24 +10x x 2 = c x 5, find the value of c.
9 Use the graph of the function to fill in the blanks. Analyzing graphs of functions: http://academics.utep.edu/portals/1788/calculus%20material/1_5%20analyzing%20gr APHS%20OF%20EQNS.pdf 51. Domain Range Interval(s) of Increase Interval(s) of Decrease Constant Interval(s) One-to-One? x-intercept(s) y-intercept Interval(s) on Which f(x) is Positive f(3) = Absolute Minimum Absolute Maximum Relative Minima Relative Maxima
10 52. You may use a calculator for this example. (from Illustrative Mathematics) The population of a country is initially 2 million people and is increasing at 4% per year. The country s annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year. a. Based on these assumptions, in approximately what year will this country first experience shortages of food? b. If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? In approximately which year? c. If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur?