## AP Calculus - Limits, Derivatives, and Integrals.

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AP Notes, Outlines, Study Guides, Vocabulary, Practice Exams and more!

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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

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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

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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

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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

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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.

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The derivatives of the basic trig. functions are as follows:
f(x)=sin(x) / f`(x)=cos(x)
f(x)=cos(x) / f`(x)=-sin(x)
f(x)=tan(x) / f`(x)=sec^2(x)

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y + xy = 7x
Implicit differentiation?
(dy/dx) + (x(dy/dx)+ y(1)) = 7(1)
(dy/dx)(x + 1) = 7 - y
(dy/dx) = (7 - y) / (x + 1)
Easy?

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Instead of doing the limit process of finding a derivative, there is a much more effective method of finding a derivative.
X^N -> the derivative would be -> N(X)^N-1
Say for example, you have f(x) = x^2, it would be 2(x)^(2-1), which would be 2x^1, or just 2x.
if you have 3x^3, it would be 3(3)x^(3-1), which would be 9x^2

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