Mathematics
Sequences
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CALCULUS BC NOTES: SEQ 11.1 (revised 2010) SEQUENCE: a list of numbers OR a function whose domain is the set of positive integers FIND FIRST 5 TERMS OF EACH SEQUENCE: 1) 2) 3) LIMIT of a SEQUENCE: (p 695) If where L is a finite number, the sequence CONVERGES. If no limit, the sequence DIVERGES. THEOREM: If (p 696) [Sec 11.1: p 2] FIND THE LIMIT OF EACH SEQUENCE. TELL WHETHER SEQUENCE CONVERGES OR DIVERGES. 4) 5) INCREASING/DECREASING SEQUENCES: increases if for all n decreases if for all n BOUNDED SEQUENCE : (p 700) Above: for all Below: for all [Sec 11.1: p 3] MONOTONIC SEQUENCE: always increasing or always decreasing
Lesson 3 Precalculus Online
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Lesson 3: Trigonometric Functions Topic 3: Reference Angles Examples : Find the reference angle for each angle. 1. Find the reference angle for 218?. Find the positive acute angle made by the terminal side of the angle and the x-axis: The reference angle for 218? is 218? - 180? = 38? 2. Find the reference angle for 1387 ? First find a coterminal angle between 0? and 360?. Divide 1387 by 360 to get a quotient of about 3.9. So subtract 360 three times. 1387? ? 3(360? ) = 307?. The reference angle for 307 ? is 360? ? 307? = 53? 360? ? 307? = 53? 3. Find the reference angle for -237? Find a coterminal positive angle by adding 360?: -237? + 360? = 123? The reference angle for 123? is 180? - 123? = 57?. 180 ? ? 123 ? = 57 ? 180? ? 123? = 57? Practice
Trig functions
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Review : Trig Functions The intent of this section is to remind you of some of the more important (from a Calculus standpoint?) topics from a trig class. One of the most important (but not the first) of these topics will be how to use the unit circle. We will actually leave the most important topic to the next section. First let?s start with the six trig functions and how they relate to each other. Recall as well that all the trig functions can be defined in terms of a right triangle. From this right triangle we get the following definitions of the six trig functions. Remembering both the relationship between all six of the trig functions and their right triangle definitions will be useful in this course on occasion.
AP inversfunction
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Review : Inverse Functions In the last example from the previous section we looked at the two functions and and saw that and as noted in that section this means that there is a nice relationship between these two functions. Let?s see just what that relationship is. Consider the following evaluations. In the first case we plugged into and got a value of -5. We then turned around and plugged into and got a value of -1, the number that we started off with. In the second case we did something similar. Here we plugged into and got a value of , we turned around and plugged this into and got a value of 2, which is again the number that we started with. Note that we really are doing some function composition here. The first case is really, and the second case is really,
Human Population Outline
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Human population Out line .4.1 How population change over time A. B. C. D. II. Age structure A. B. C. D. E. F. G. H. I. J. K. 4.2 Kinds of population Growth III Exponential growth A B C IV A brief history of human population Growth 1. 2. 3. 4. 4.3 Present Human Population Rates of Growth A. B. C. D. E. F. 4.4 Project Future Population Growth A. V. Exponential Growth and daubing Time A. B. C. D. E. The logistic Growth Curve A. B C D E. F. G. VII. Forecasting Human Population Growth Using the logistic Curve A. B. C. D. E.
Continuity
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Function continuity at x=a (bridges) F(x) is continuous at x=a if must exist f(a) must be defined f(a)= Ex. Continuous: f(a) is defined f(a)= Ex. Discontinuous: f(a) is defined f(a)
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INT SCI physics review/basic unit study guide
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Science Study Guide Metric Prefixes: Deka da ten Deci d tenth Hecto h hundred Centi c hundredth Kilo k thousand Milli m thousandth Mega M million Micro ? millionth Giga G billion Nano n billionth Tera T trillion Pico p trillionth Difference between theory and hypothesis: A scientific theory is a hypothesis that has been tested and proven Scientific method: a method used in gaining, organizing, and applying new knowledge Recognize a problem Make an educated guess (hypothesis) about the answer Predict the consequences of the hypothesis Perform experiments to test it Make conclusions Experimental Components: Independent variable: part of the experiment that is purposely manipulated
AP CALC AB MIDTERM REVIEW
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AB CALC AP MIDTERM REVIEW SHEET Limits Limit of a constant is a constant 0/? -> 0 Value/0 -> ?? or DNE 0/0 or ?/? Factor Rationalize L?Hospital?s Rule (derivative of numerator? derivative of denominator) End behavior (x->?) look at highest power of numerator and denominator Special Limits: (used in trig limits) Lim(x->0) [sinx/x] =1 Lim(x->0) [(cosx -1)/x] =0 Lim(x->0) [tanx/x] =1 Local Linear Approximation Approximation of a value on a function using a linear function f(x) ? f(xo) + f?(xo) (x - xo) Continuity A function f is continuous at point c if: F(c) is defined Lim(x->c) f(x) exists F(c) = limit Intermediate Vale Theorem
AP CALCULUS BC REVIEW GUIDE
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2009?2010 AP BC Calculus First Semester Exam Review Guide I. ?BIG 7? THEOREMS (be able to state and use theorems especially in justifications) Intermediate Value Theorem Extreme Value Theorem Rolle?s Theorem Mean Value Theorem for Derivatives & Definite Integrals FUNDAMENTAL THEOREM OF CALCULUS ? Be sure you remember the 2nd part is called the Total Change Theorem Also, in your justifications, if a problem says it is differentiable at x = a, then the function is continuous at x = a. (Differentiability implies continuity, but not vice-versa.) II. VOCABULARY/KEY CONCEPTS CHAPTER 1: 1. Distinguish the graphs of parent functions without the use of a calculator. (handout) 2. Define an absolute value function as a piecewise function. (pg. 17)
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