Lone Star College System
Name:_____________________ Math 2412 Activity 3(Due by Apr. 4)
Graph the following exponential functions by modifying the graph of [pic]. Find the
range of each function.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
Find a formula for the exponential function whose graph is given
7. 8.
Write each equation in its equivalent exponential form.
9. [pic] 10. [pic] 11. [pic]
Write each equation in its equivalent logarithmic form.
12. [pic] 13. [pic] 14. [pic]
Simplify the following expressions.
15. [pic] 16. [pic] 17. [pic]
18. [pic] 19. [pic] 20. [pic]
21. [pic] 22. [pic] 23. [pic]
24. [pic] 25. [pic] 26. [pic]
27. [pic] 28. [pic] 29. [pic]
30. [pic]
Graph the following logarithmic functions by modifying the graph of [pic]. Find the domain of each function.
31. [pic] 32. [pic]
33. [pic] 34. [pic]
Expand each expression as much as possible, and simplify whenever possible.
35. [pic] 36. [pic] 37. [pic] 38. [pic]
39. [pic] 40. [pic] 41. [pic] 42. [pic]
43. [pic] 44. [pic]
Compress each expression as much as possible, and simplify whenever possible.
45. [pic] 46. [pic] 47. [pic]
48. [pic] 49. [pic]
Use a calculator to evaluate the following logarithms to four places.
50. [pic] 51. [pic] 52. [pic]
53. Simplify [pic], for [pic].
54. Find the value of x for [pic] so that [pic]
Solve the following exponential equations.
55. [pic] 56. [pic] 57. [pic] 58. [pic]
59. [pic] 60. [pic] 61. [pic]
Solve the following logarithmic equations.
62. [pic] 63. [pic] 64. [pic]
65. [pic] 66. [pic]
67. [pic] 68. [pic]
69. [pic] 70. [pic]
Solve the following mixed equations.
71. [pic] 72. [pic]
73. [pic] 74. [pic]
75. [pic] 76. [pic]
77. [pic] 78. [pic]
79. For what values of b is [pic] an increasing function? a decreasing function?
80. For [pic], [pic], find the value of [pic].
81. Find values of a and b so that the graph of [pic] contains the points [pic] and [pic].
82. Suppose that x is a number such that [pic]. Find the value of [pic].
83. Suppose that x is a number such that [pic]. Find the value of [pic].
84. Suppose that x is a number such that [pic]. Find the value of [pic].
85. Suppose that x is a number such that [pic]. Find the value of [pic].
86. For [pic] and [pic], show that [pic].
Find a simplified formula for [pic] for the given functions.
87. [pic]
88. [pic]
89. [pic]
90. [pic]
When solving inequalities, it is common to apply a function to all sides of the inequality. If the function is an increasing function, then the direction of the inequality is preserved, but if the function is a decreasing function, then the direction of the inequality is reversed. Here’s why:
If f is increasing, then for [pic], [pic], and you see that the inequality direction is preserved. If f is decreasing, then for [pic], [pic], and you see that the inequality direction is reversed.
Here are some examples:
a) To solve the inequality [pic], you apply the function [pic] to both sides of the inequality. Since it’s an increasing function, the direction of the inequality is preserved, and you get [pic].
b) To solve the inequality [pic], you apply the function [pic] to both sides of the inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and you get [pic].
c) To solve the inequality [pic], you apply the function [pic] to all sides of the inequality. Since it’s an increasing function, the direction of the inequality is preserved, and you get [pic].
d) To solve the inequality [pic], you apply the function [pic] to all sides of the inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and you get [pic].
e) To solve the inequality [pic], you apply the function [pic] to all sides of the inequality. Since it’s an increasing function, the direction of the inequality is preserved, and you get [pic].
f) To solve the inequality [pic], you apply the function [pic] to all sides of the inequality. Since it’s a decreasing function, the direction of the inequality is reversed, and you get [pic].
Use the previous discussion to solve the following inequalities:
91. [pic] 92. [pic] 93. [pic] 94. [pic]
95. [pic] 96. [pic] 97. [pic] 98. [pic]
Simplify the following as much as possible:
99. [pic]
100. [pic]
101. [pic]
102. [pic]
103. [pic]
104. Find the exact value of the sum [pic].
105. Find the exact value of the sum [pic].
If all the terms of a sequence [pic] are positive, it is sometimes convenient to analyze the related sequence [pic]. If [pic], for [pic], then the sequence [pic] is eventually nondecreasing. If [pic], for [pic], then the sequence [pic] is eventually increasing. If [pic], for [pic], then the sequence [pic] is eventually nonincreasing. If [pic], for [pic], then the sequence [pic] is eventually decreasing. Test the following sequences by analyzing [pic].
106. [pic] 107. [pic] 108. [pic]
109. [pic] 110. [pic] 111. [pic]
112. [pic] 113. [pic] 114. [pic]
115. Find an explicit formula for [pic] in the following recursively defined sequence:
[pic].
{Hint: [pic], so [pic], [pic],
[pic], [pic],…, see if you can formulate the pattern.}
116. A sequence [pic] is given recursively by [pic]. If [pic], then what’s [pic]?
117. Find x so that [pic] are consecutive terms of an arithmetic sequence.
118. Find x so that [pic] are consecutive terms of an arithmetic sequence.
119. How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to get the sum 1092?
120. How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is -4 to get the sum 702?
121. a) Determine the common difference for the following arithmetic sequence:
[pic]
b) Find a formula for [pic] that generates the number in this sequence.
c) Is 71 a number in this arithmetic sequence?
122. a) Determine the common difference for the following arithmetic sequence:
[pic]
b) Find a formula for [pic] that generates the numbers in this arithmetic sequence.
c) What is the 99th number in this arithmetic sequence?
123. Find x so that [pic] are consecutive terms of a geometric sequence.
124. Find x so that [pic] are consecutive terms of a geometric sequence.
125. a) Determine the common ratio for the following geometric sequence:
[pic]
b) Find a formula for [pic] that generates the numbers in this geometric sequence.
c) Is [pic] a number in this geometric sequence?
126. a) Determine the common ratio for the following geometric sequence:
[pic]
b) Find a formula for [pic] that generates the numbers in this geometric sequence.
c) Is [pic] a number in this geometric sequence?
Determine if the following geometric series converge or diverge. If it converges, find its sum.
127. [pic] 128. [pic] 129. [pic]
130. [pic] 131. [pic] 132. [pic]
133. [pic] 134. [pic]
Prove the following using the Principle of Mathematical Induction:
135. [pic]
136. [pic]
137. [pic]
138. [pic]
139. [pic]
140. [pic]
141. [pic]
142. [pic]
143. [pic] is divisible by 3
144. [pic] is divisible by 6
145. [pic]
146. [pic]
147. [pic]
148. [pic] is divisible by 5
Find a formula for the nth term of the following sequences:
149. [pic] 150. [pic] 151. [pic]
152. [pic] 153. [pic] 154. [pic]
155. [pic] 156. [pic]
157. In a geometric sequence of real number terms, the first term is 3 and the fourth term is 24. Find the common ratio.
158. Find the seventh term of a geometric sequence whose third term is [pic] and whose fifth term is [pic].
159. For what value(s) of k will [pic] form a geometric sequence?
Find the sum of the following series:
160. [pic] 161. [pic] 162. [pic]
163. [pic] 164. [pic] 165. [pic]
166. [pic] 167.[pic] 168. [pic]
169. [pic] 170. [pic] 171. [pic]
172. Solve the equation [pic] for n.
173. Find the second term of an arithmetic sequence whose first term is 2 and whose first, third, and seventh terms form a geometric sequence.
174. The figure shows the first four of an infinite sequence of squares. The outermost square has an area of 4, and each of the other squares is obtained by joining the midpoints of the sides of the square before it. Find the sum of the areas of all the squares.
175. The equation [pic] has 0 as its only solution. If the Method of Successive Approximations is applied to approximate this solution, graphically indicate the result if the starting guess is
a) a positive number
b) a negative number
176. The equation [pic] has 0 and 2 as its solutions. If the Method of Successive Approximations is applied to approximate these solutions, graphically indicate the result if the starting guess is
a) greater than 2
b) between 0 and 2
c) less than 0
177. Consider the statement [pic].
a) Show that if the statement is true for [pic], then it must be true for [pic].
b) For which natural numbers is the statement true?
178. If [pic], then what’s the value of b?
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[pic]
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