calculus definitions Flashcards
Define the density of rationals within the real numbers using two equivalent definitions.
The density of rationals states that for any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Equivalently, there exists a sequence of rationals that converges to any real number.
This property illustrates how densely packed the rational numbers are within the real numbers.
Define the density of irrationals within the real numbers using two equivalent definitions.
The density of irrationals states that for any two real numbers a and b with a < b, there exists an irrational number q such that a < q < b. Equivalently, there exists a sequence of irrationals that converges to any real number.
This demonstrates that irrational numbers are also densely distributed among real numbers.
State the Triangle Inequality (all three versions).
The three versions of the Triangle Inequality are:
* |x - z| ≤ |x - y| + |y - z|
* |x + y| ≤ |x| + |y|
* |x - y| ≥ ||x| - |y| |.
What is the Archimedean Property?
For any two positive real numbers x and y, there exists a natural number n such that nx > y.
State Bernoulli’s Inequality.
For any real number x ≥ -1 and any natural number n, (1 + x)^n ≥ 1 + nx.
State the Inequality of Means.
For non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. This can be extended to include the root mean square, which is always greater than or equal to the arithmetic mean.
State Newton’s Binomial Theorem.
For any non-negative integer n, and any real numbers a and b, (a + b)^n = ∑_(k=0)^n (n choose k) a^(n-k) b^k.
Define an upper bound of a set.
An upper bound of a set A is a real number α such that α ≥ a for all a ∈ A.
Define a supremum of a set.
The supremum of a set is the least upper bound of that set, if it exists.
Define a lower bound of a set.
A lower bound of a set A is a real number α such that α ≤ a for all a ∈ A.
Define an infimum of a set.
The infimum of a set is the greatest lower bound of that set, if it exists.
Define the minimum of a set.
The minimum of a set is the smallest element within that set, and it must belong to the set.
Define the maximum of a set.
The maximum of a set is the largest element within that set, and it must belong to the set.
State the Completeness Axiom of the real numbers.
Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the set of real numbers.
Define a sequence.
A sequence is an ordered list of numbers.
Define the limit of a sequence.
The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity.
Define a convergent sequence.
A sequence that approaches a finite limit is a convergent sequence.
Define a divergent sequence.
A sequence that does not approach a finite limit is a divergent sequence.
Define a monotone sequence.
A monotone sequence is a sequence that is either non-decreasing or non-increasing.
Define a subsequence.
A subsequence is a sequence formed by selecting elements from another sequence.
Define a Cauchy sequence.
A sequence where the terms become arbitrarily close to each other as the index increases.
State the Monotone Convergence Theorem.
A monotonic and bounded sequence converges.
Define a series.
A series is the sum of the terms of a sequence.
Define a partial sum.
A partial sum is the sum of a finite number of terms in a series.
State the Comparison Test for series.
If 0 ≤ an ≤ bn for all n, and Σbn converges, then Σan converges. If Σan diverges, then Σbn diverges.
State the Ratio Test for series, including its limitations.
Given a series Σan, compute L = lim |an+1 / an|. If L < 1, the series converges absolutely. If L > 1 or L = ∞, the series diverges. If L = 1, the test is inconclusive.
State the Root Test for series, including its limitations.
Given a series Σan, compute L = lim (|an|)^(1/n). If L < 1, the series converges absolutely. If L > 1 or L = ∞, the series diverges. If L = 1, the test is inconclusive.
Define absolute convergence.
A series Σan converges absolutely if the series of absolute values Σ|an| converges.
Define conditional convergence.
A series Σan converges conditionally if Σan converges but Σ|an| diverges.
State the Alternating Series Test.
An alternating series Σ(-1)^n * bn converges if the sequence bn is monotonically decreasing and lim bn = 0.
Define a one-to-one (injective) function.
A function where each output corresponds to only one input.
Define an onto (surjective) function.
A function where every output value can be achieved from an input value.
Define a bounded function.
A function for which the output values stay within a given range.
Define the limit of a function at a point using the Cauchy definition.
For every ε > 0, there exists δ > 0 such that if 0 < |x - x0| < δ, then |f(x) - L| < ε.
Define the limit of a function at a point using the Heine definition.
For every sequence xn approaching x0, the sequence f(xn) approaches L.
Define continuity of a function at a point.
A function f is continuous at a point x0 if lim_(x→x0) f(x) = f(x0).
Define continuity of a function on an interval.
A function is continuous on an interval if it is continuous at every point in that interval.
Describe a removable discontinuity.
A discontinuity where the limit exists but is not equal to the function’s value at that point.
Describe a jump discontinuity.
A discontinuity where the left-hand and right-hand limits exist but are not equal.
Describe an infinite discontinuity.
A discontinuity where the limit does not exist because it approaches infinity.
State the Squeeze Theorem for functions.
If g(x) ≤ f(x) ≤ h(x) for all x near a, and lim_(x→a) g(x) = lim_(x→a) h(x) = L, then lim_(x→a) f(x) = L.
State the Intermediate Value Theorem.
If a function is continuous on a closed interval, it takes on every value between the values at the interval endpoints.
State the Extreme Value Theorem (Weierstrass Theorem).
A continuous function on a closed interval has both a maximum and a minimum value on that interval.
Define the derivative of a function.
The derivative measures the instantaneous rate of change of a function with respect to its input and is defined as f’(x) = lim_(h→0) [f(x+h) - f(x)]/h.
Describe the geometric interpretation of the derivative.
The derivative represents the slope of the tangent line to the function’s graph at a given point.
How do derivatives relate to increasing/decreasing functions?
If f’(x) > 0 on an interval, then f is increasing; if f’(x) < 0, then f is decreasing.
How do derivatives help find local extrema?
Local extrema occur at critical points where the derivative is zero or undefined.
State Rolle’s Theorem.
If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists a c in (a,b) such that f’(c) = 0.
State the Mean Value Theorem.
If f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f’(c) = [f(b) - f(a)] / (b - a).
Define a Taylor Polynomial.
A Taylor polynomial is an approximation of a function using a polynomial that uses the function’s derivatives at a single point.
When can you use the Squeeze Theorem (Sandwich Theorem) for functions?
You can use the Squeeze Theorem if you have a function f(x) that is bounded between two other functions, g(x) and h(x), such that g(x) ≤ f(x) ≤ h(x) for all x near a point a. Additionally, the limits of g(x) and h(x) as x approaches a must exist and be equal.
What are the requirements to use the Intermediate Value Theorem?
The Intermediate Value Theorem applies if a function f(x) is continuous on a closed interval [a, b]. It states that the function will take on every value between f(a) and f(b) within that interval.
What are the requirements to use the Extreme Value Theorem (Weierstrass Theorem)?
The Extreme Value Theorem applies if a function f(x) is continuous on a closed interval [a, b]. The theorem guarantees the existence of both a maximum and a minimum value for the function within that interval.
State the requirements to use Rolle’s Theorem.
To apply Rolle’s Theorem, f(x) must be continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) must equal f(b).
State the requirements to use the Mean Value Theorem.
To use the Mean Value Theorem, f(x) must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
Define ‘lim sup’ ( גבול עליון) of a sequence.
The lim sup of a sequence is the largest limit point of the sequence.
Define ‘lim inf’ ( גבול תחתון) of a sequence.
The lim inf of a sequence is the smallest limit point of the sequence.
When can you use the Comparison Test for series convergence?
You can use the Comparison Test if you have two series, Σa_n and Σb_n, where 0 ≤ a_n ≤ b_n for all n. If Σb_n converges, then Σa_n also converges. If Σa_n diverges, then Σb_n also diverges.
When is the Ratio Test ( מבחן המנה) inconclusive?
The Ratio Test is inconclusive when lim |a_n+1 / a_n| = 1. In this case, another convergence test is needed.
When is the Root Test ( מבחן השורש) inconclusive?
The Root Test is inconclusive when lim (|a_n|)^(1/n) = 1. In this case, another convergence test is needed.
What does it mean for a function to be uniformly continuous?
A function f(x) is uniformly continuous on an interval if for every ε > 0, there exists a δ > 0 such that for all x, y in the interval, if |x - y| < δ, then |f(x) - f(y)| < ε. The key is that δ does not depend on the specific location in the interval.
What is the Completeness Axiom (אקסיומת השלמות) and why is it important?
The Completeness Axiom states that every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers. This distinguishes the real numbers from rational numbers, and is crucial for the convergence theorems.
What is the Monotone Convergence Theorem?
If a sequence is both monotonic (either non-decreasing or non-increasing) and bounded, then the sequence must converge to a finite limit.
What are the conditions for using the Alternating Series Test (מבחן לייבניץ)?
The Alternating Series Test applies if the terms alternate in sign and the absolute values of the terms are monotonically decreasing and approach zero.
What is a Cauchy Sequence (סדרת קושי) and why is it important?
A Cauchy Sequence is a sequence where the terms become arbitrarily close to each other as the index increases. Every convergent sequence is a Cauchy sequence, and in the real numbers, every Cauchy sequence is convergent.
Define a partial sum of a series.
A partial sum is the sum of a finite number of terms in a series. For a series ∑ a_n, the partial sum S_N is given by S_N = a_1 + a_2 + … + a_N.
How do you determine if a series converges absolutely?
A series ∑a_n converges absolutely if the series of the absolute values of its terms, ∑|a_n|, converges.
What is the condition for a series to be conditionally convergent?
A series is conditionally convergent if it converges, but it does not converge absolutely. That is, ∑a_n converges, but ∑|a_n| diverges.
What are the two definitions of a limit of a function at a point?
The two equivalent definitions are: the Cauchy definition (using ε-δ) and the Heine definition (using sequences).
How can you classify a discontinuity of a function?
Discontinuities can be classified as removable, jump, or infinite based on the behavior of the limit at the point of discontinuity.
