AP Calc AB - Chapter 1

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Limits and Their Properties

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An even function is…

symmetric w/ respect to y-axis like:

  • y=x²

  • y=cos x

  • y=|x|

f(-x)=f(x)

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An odd function is…

symmetric w/ respect to origin like”

  • y=x³

  • y=sin x

  • y=tan x

f(-x)=-f(x)

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Unit Circle

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Point-slope form of linear equation

y - y1 = m(x - x1)

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Ways to solve limits algebraically

  1. Substitute directly

  2. Factoring

  3. Rationalize w/ conjugate

  4. LCM

  5. Apply trig identities for trig limits

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Two Special Trig Limits

  • lim of sin x/x = 1 as x→0

  • lim of (1-cos x)/x = 0 as x→0

  • *lim of tan x/x = 1 as x→0

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The Sanwich/Squeeze Theorem

If h(x) is less than or equal to f(x) which is less than or equal to g(x), all x in open interval containing c:

lim of h(x) = L = lim as x→x g(x) as x→c, then

lim of f(x) as x→c exists & = L

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Continuity Test

Function f(x) is continuous at x = c if & only if it meets three conditions:

  1. f(c) exists (c lies in domain f)

  2. lim of f(x) as x→c exists (f has limit x→c)

  3. lim of f(x) as x→c = f(c) (limit = function value)

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Limits that DNE (do not exist)

  • lim of f(x) as x→c from the left is not equal to lim of f(x) as x→c from the right

  • f(x) goes up or down w/o bound

  • f(x) oscillates between two fixed values as x→c (shown in image)

<ul><li><p>lim of f(x) as x→c from the left is not equal to lim of f(x) as x→c from the right</p></li><li><p>f(x) goes up or down w/o bound</p></li><li><p>f(x) oscillates between two fixed values as x→c (shown in image)</p></li></ul>
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Types of Discontinuity

3 types:

  1. Jump (nonremovable)

  2. Infinite (nonremovable)

  3. Point (removable)

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Intermediate Value Theorem

Suppose f continuous on closed interval [a,b]. Let w be any number strictly between f(a) and f(b). Then there exists #c in (a,b) such that f(c)=w.

  1. Identify if function’s continuous

  2. Plug intervals into equation

  3. See if f(c) is between f(a) and f(b)

<p>Suppose f continuous on closed interval [a,b]. Let <em>w</em> be any number strictly between f(a) and f(b). Then there exists #c in (a,b) such that f(c)=<em>w. </em></p><ol><li><p>Identify if function’s continuous</p></li><li><p>Plug intervals into equation</p></li><li><p>See if f(c) is between f(a) and f(b)</p></li></ol>
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Limits Involving Infinity

2 types:

  1. lim of f(x) as x→±infinity = #, infinity

  2. lim of f(x) as x→c = infinity

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Limits w/ Infinity Rules

y = ax^m/bx^n

  • Degree of numerator larger (m>n), no horizontal asymptote, limit of f(x) as x→±infinity = DNE (does not exist)

  • Denominator’s larger (m<n), horizontal asymptote: y=0, limit of f(x) as x→±infinity = 0

  • Degrees are the same (m=n), ratio between leading coefficients, lim of f(x) as x→±infinity = a/b

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Graph of y=square root of x

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Graph of y=square root of 1-x²

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Graph of y=|x|

*Absolute Value Function

<p>*Absolute Value Function</p>
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Graph of y=ln x

*Natural Log Function

<p>*Natural Log Function</p>
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Graph of y=e^x

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Graph of y=1/x

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Graph of y=sec x

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Graph of y=tan x

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Graph of y=sin x

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Graph of y=cos x

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Graph of y=csc x

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Graph of y=cot x

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Graph of y=x³

*Cubic Function

<p>*Cubic Function</p>
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Greatest Integer Function

f(x) = [[x]]

<p>f(x) = [[x]]</p>