Calculus AB Limits Flashcards

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Questions and Answers

What are limits?

  • The value a function approaches (correct)
  • The slope of a function
  • The minimum value of a function
  • The maximum value of a function

What does DNE stand for?

Does Not Exist

Unbounded limits result in DNE.

True (A)

What does Lim f(x) x --> c^+ represent?

<p>The limit as x approaches c from the right</p>
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What does Lim f(x) x --> c^- represent?

<p>The limit as x approaches c from the left</p>
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The limit of f(x) x --> c exists when Lim f(x) x --> c^+ and Lim f(x) x --> c^- are the same.

<p>True (A)</p>
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If a one-sided limit is unbounded, the limit exists.

<p>False (B)</p>
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How can you estimate limits from a table?

<p>Compare lower values from the left and higher values from the right.</p>
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A function is undefined where the denominator equals what?

<p>zero</p>
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What is the sum property of limits?

<p>Two limits added together equal the individual limits added together.</p>
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What is the product property of limits?

<p>Individual limits multiplied together equal the limit of the product.</p>
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What is the Constant Multiple Rule?

<p>d/dx[cf(x)] = cf'(x)</p>
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What is the exponent limit rule?

<p>The derivative raised to the exponent.</p>
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What should you do with piecewise function limits if DNE occurs?

<p>Break it up into left and right limits.</p>
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What is 'f o g'?

<p>f(g(x))</p>
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What should be done when the internal or external limit does not exist?

<p>Use the right and left limits to find the overall limit.</p>
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What is Direct Substitution?

<p>Evaluating a function by plugging a given value into the function.</p>
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When evaluating a limit, the denominator can equal zero.

<p>False (B)</p>
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How do you find limits of piecewise functions?

<p>Use x and align it with the chart to find the correct function.</p>
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What is the process for limits by factoring?

<p>Factor out any crossing values, which become undefined, to get a 'new' function.</p>
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What does (a+b)(a-b) equal?

<p>a²-b²</p>
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What does (√a+b)(√a-b) equal?

<p>a-b²</p>
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What is sin²x + cos²x equal to?

<p>1</p>
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What is cos 2x equal to?

<p>cos²x - sin²x</p>
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What is cos²x - sin²x equal to?

<p>1 - 2sin²x</p>
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What is 1 - 2sin²x equal to?

<p>2cos²x - 1</p>
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What is the Squeeze Theorem?

<p>If f(x) ≤ g(x) ≤ h(x) and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L.</p>
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What is lim x->0 sin(x)/x equal to?

<p>1</p>
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What is lim -> 0 (1 - cos(x) / x) equal to?

<p>0</p>
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What is point discontinuity?

<p>A point in the graph where the function does not exist, creating a removable hole.</p>
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What is jump discontinuity?

<p>A graph that has discontinuity where the function moves to a different y-value.</p>
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What is asymptotic discontinuity?

<p>A graph that approaches one or more asymptotes.</p>
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F is continuous at x=c if lim f(x) x -> c = f(c).

<p>True (A)</p>
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What is the average rate of change formula?

<p>f(b) - f(a) / b - a</p>
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What does the Mean Value Theorem state?

<p>The instantaneous rate of change equals the mean rate of change somewhere in the interval.</p>
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What does the Extreme Value Theorem state?

<p>If f is continuous on [a,b], then f has an absolute maximum and minimum on [a,b].</p>
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What are critical points?

<p>Points in the domain of a function where f'=0 or f' does not exist.</p>
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How do you find critical points?

<p>Find the derivative, set to zero, and test using a number line.</p>
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The function is decreasing when f'(x) < 0.

<p>True (A)</p>
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What is relative maximum?

<p>A point on the graph of a function where no other nearby points are higher.</p>
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Study Notes

Limits

  • Limits indicate the value that a function approaches as the variable approaches a certain point, denoted as ( \lim_{x \to c} f(x) ).
  • A limit can exist even if the function is undefined at that point.
  • DNE (Does Not Exist) occurs when a left-hand limit and right-hand limit yield different values.
  • Unbounded limits also result in DNE.

One-sided Limits

  • ( \lim_{x \to c^+} f(x) ): Represents the limit as ( x ) approaches ( c ) from the right.
  • ( \lim_{x \to c^-} f(x) ): Represents the limit as ( x ) approaches ( c ) from the left.
  • If both one-sided limits are the same, the two-sided limit exists.

Calculating Limits

  • Estimating limits using a table: Lower values indicate left limits, and higher values indicate right limits.
  • Direct substitution involves plugging the value directly into the function to evaluate limits.

Properties of Limits

  • The Sum Property: ( \lim (f(x) + g(x)) = \lim f(x) + \lim g(x) )
  • The Product Property: ( \lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x) )
  • Constant Multiple Rule: If ( c ) is a constant, then ( \frac{d}{dx}[c f(x)] = c f'(x) ).

Special Limit Cases

  • The limit of ( \frac{\sin x}{x} ) as ( x \to 0 ) is 1.
  • The limit of ( \frac{1 - \cos x}{x} ) as ( x \to 0 ) is 0.

Piecewise Functions

  • For limits involving piecewise functions, evaluate the limits separately for each side of the point of interest.
  • Factorization can help simplify limit evaluation; undefined values indicate crossings of the function.

Types of Discontinuities

  • Point Discontinuity: Exists where the function has a hole in the graph; it's removable.
  • Jump Discontinuity: Function jumps to a different y-value, non-removable.
  • Asymptotic Discontinuity: Approaches an asymptote, resulting in non-removable discontinuity.

Continuity and Critical Points

  • A function ( f ) is continuous at ( x = c ) if ( \lim_{x \to c} f(x) = f(c) ).
  • A critical point occurs where ( f' = 0 ) or ( f' ) is undefined, which may indicate local maxima or minima.

Theorems

  • Mean Value Theorem: There exists at least one point where the instantaneous rate of change equals the average rate of change on an interval.
  • Extreme Value Theorem: A continuous function on a closed interval has both an absolute maximum and minimum.

Additional Trigonometric Identities

  • ( \sin^2 x + \cos^2 x = 1 )
  • ( \cos(2x) = \cos^2 x - \sin^2 x )
  • Alternative forms include ( \cos^2 x - \sin^2 x = 1 - 2 \sin^2 x ) and ( 1 - 2 \sin^2 x = 2 \cos^2 x - 1 ).

Finding Critical Points

  • To find critical points, take the derivative, set it to zero, and test the intervals using a number line.

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