Chapter Review 905 56. Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points. ≥ ≥ + ≤ + ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ x y x y x y 0 0 6 2 10 57. If ( ) = − f x x 6 2 and ( ) = + g x x 2, find ( ) ( ) g f x and state its domain. 58. If = + + y x x C 5 3 2 3 and = y 5 when = x 3, find the value of C. 59. Establish the identity θ θ θ θ + = sin sin tan tan . 2 2 2 2 60. Simplify: ( ) ( ) ( ) + ⋅ − + − x x x x x 1 1 3 3 1 3 2 3 1 3 2 3 2 61. Find the vertical asymptotes, if any, of the graph of ( ) ( )( ) = − + f x x x x 3 3 1 2 62. If ( ) = + + f x x x 1 2 5 , 2 find ( ) − f 2 . What is the corresponding point on the graph of f? Chapter Review Things to Know Sequence (p. 855) A function whose domain is the set of positive integers and whose range is a subset of the real numbers Factorials (p. 858) ( ) = = = − ⋅ ⋅ ⋅ ⋅ n n n 0! 1,1! 1, ! 1 3 2 1 if ≥ n 2 is an integer Arithmetic sequence (pp. 869 and 870) = = + − a a a a d , , n n 1 1 where = = = a a d first term, common difference 1 ( ) = + − a a n d 1 n 1 Sum of the first n terms of an arithmetic sequence (p. 878) ( ) [ ] ( ) = + − = + S n a n d n a a 2 2 1 2 n n 1 1 Geometric sequence (pp. 876 and 877) = = − a a a ra , , n n 1 1 where = = = a a r first term, common ratio 1 = ≠ − a a r r 0 n n 1 1 Sum of the first n terms of a geometric sequence (p. 878) = − − ≠ S a r r r 1 1 0, 1 n n 1 Infinite geometric series (pp. 879 and 892) ∑ + + + + = − = ∞ − a a r a r a r n k k 1 1 1 1 1 1 1 Sum of a convergent infinite geometric series (pp. 879 and 893) If ∑ < = − = ∞ − r a r a r 1, 1 k k 1 1 1 1 Amount of an annuity (p. 882) ( ) = + − A P i i 1 1 , n where = P the deposit (in dollars) made at the end of each payment period, = i interest rate per payment period (as a decimal), and = A the amount of the annuity after n deposits. Principle of Mathematical Induction (p. 894) If the following two conditions are satisfied, Condition I: The statement is true for the natural number 1. Condition II: If the statement is true for some natural number k, then it can be shown to be true for +k 1. then the statement is true for all natural numbers. Binomial coefficient (p. 899) ( ) ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = − n j n j n j ! ! ! The Pascal triangle (p. 900) See Figure 22. Binomial Theorem (p. 900) ∑ ( ) + = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ + + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − − = − x a n x n ax n j a x n n a n j a x 0 1 n n n j n j n j n j n j 1 0
RkJQdWJsaXNoZXIy NjM5ODQ=