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Functions and mappings

AP Calculus - Limits, Derivatives, and Integrals.

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transformation and compositions notes

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I?. Ei- Pre-AP Algebra 2 Day 3 Notes: Transformations and Compositions Date Transformations y=?af[?b(x?c)](+c? c H &1M s C) t ( c) o &A \ \ on j. vri (G vv I I Graph and state the transformation y = f(x) to y = f(x - 3) + 2 y = 2f(x) y = f(2x) (, ::; 91 dt L j-vcLLNc I - VI aizA. pxvawc' 10 ticw ti a 0 10 y = f(x)to y = f(4x) 9 OY7.A4'iit cWrcc1 It C, 10 y=f(x) to y= ?f(x+2) y=f(?x) y=f(x) onicrt Cftc4ioj k horl SkICkI 10~ of O-i CtWi&'\ [r1 C,-& Lmi1z H- y = f(x) - 2 _& y = 2f(3x) + 3 -7-- - V ho?i\ cecA s\'\E4' b 3 o'iziA 1;S aN C_ Yify Compositon of functions: Putting one function into another. Ex. Given f(x) and gf4piS would be f(g(x))?"f of g of x'. **When

hw for Algebra 2

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2.1.4 Even More Graphing from Vertex Form Name: ____________________ Date: _________ Per: _____ For each of the following FUNctions: sketch a graph, state the domain and range, describe any transformations that the FUNction has undergon compared to its parent, state the y-intercept, the location of the vertex, and describe the intervals of the domain over which the FUNction is increasing or decreasing.
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Ap Calc Test - First Semester

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AP Calculus AB ? Semester Exam Review No Calculator Portion A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: DO NOT WRITE ON THIS TEST. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No partial credit will be given. Do not spend too much time on any one problem. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. 1. Let f(x) be the function whose graph is shown to the right: A. B. C. D. E. none of the above 2. The graph of f(x) is shown in the figure to the right. What is ? A. ?1 B. 1 C. 2 D. it varies E. does not exist

LIMITS AND CONTINUITY

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Substitution
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Calculus

Limits and Continuity Brief Review Limit ? intended height (y-value) of the function. Properties: add, subtract, divide, multiply, multiply constant and raise to any power. Techniques to Evaluation: Direct Substitution ? plug the x-value in?if you get a number you are done?if you get an indeterminate form?. 1.) Try to factor the expression. Cancel common factors and try direct substitution again. 2.) Try tables or graphs?.try plugging in a number close to the x-value to the right and the left. 2. 3. 4. 5. 6 7. 8.

Ch9 SG

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Ordinary differential equations
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Natural logarithm
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law

372 CHAPTER 9 Mathematical Modeling with Differential Equations EXERCISE SET 9.1 1. y? = 2x2ex 3/3 = x2y and y(0) = 2 by inspection. 2. y? = x3 ? 2 sinx, y(0) = 3 by inspection. 3. (a) ?rst order; dy dx = c; (1 + x) dy dx = (1 + x)c = y (b) second order; y? = c1 cos t? c2 sin t, y?? + y = ?c1 sin t? c2 cos t+ (c1 sin t+ c2 cos t) = 0 4. (a) ?rst order; 2 dy dx + y = 2 ( ? c 2 e?x/2 + 1 ) + ce?x/2 + x? 3 = x? 1 (b) second order; y? = c1et ? c2e?t, y?? ? y = c1et + c2e?t ? ( c1et + c2e?t ) = 0 5. 1 y dy dx = x dy dx + y, dy dx (1? xy) = y2, dy dx = y2 1? xy 6. 2x+ y2 + 2xy dy dx = 0, by inspection. 7. (a) IF: ? = e3 ? dx = e3x, d dx [ ye3x ] = 0, ye3x = C, y = Ce?3x separation of variables: dy y = ?3dx, ln |y| = ?3x+ C1, y = ?e?3xeC1 = Ce?3x

Limits

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Limits Number value that f(x) approaches as the values of x approach a specific numberIf f(x) is a polynomial or a rational function and a is in the domain of f(x), then Limits that fail to exist 1. If f(x) approaches a different number from the right than from the left: 2. If f(x) increases or decreases without bound as it approaches a number: 3. If f(x) oscillates between 2 fixed value One-sided limits If only approaching from the right If only approaching from the left Limit Existence Theorem = L iff = L = Vertical Asymptote If = or = Then x=a is a vertical asymptote Horizontal Asymptote If or Then y=b is a horizontal asymptote

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

First derivatives

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Calculus
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World Wide Web
The first derivative of a function can tell a lot about how the original function looks. The first derivative tells us the critical numbers, and when the original function is increasing or decreasing. Finding the critical numbers of a function is easy. You have a critical number anytime the derivative function equals 0, or if it does not exist. Critical numbers in the derivative usually tell us that there is a minimum, maximum, or a vertical asymptote. However, it does not guarantee a min/max.

Definition and Domain of Rational Functions

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Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. f(x) = P(x) / Q(x) Here are some examples of rational functions: g(x) = (x2 + 1) / (x - 1) h(x) = (2x + 1) / (x + 3) The rational functions to explored in this tutorial are of the form f(x) = (ax+b)/(cx + d) where a, b, c and d are parameters that may be changed, using sliders, to understand their effects on the properties of the graphs of rational functions defined above. Example: Find the domain of each function given below. g(x) = (x - 1) / (x - 2) h(x) = (x + 2) / x Solution

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