calculus definitions

Question 1

Define the density of rationals within the real numbers using two equivalent definitions.

Answer 1

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.

Question 2

Define the density of irrationals within the real numbers using two equivalent definitions.

Answer 2

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.

Question 3

State the Triangle Inequality (all three versions).

Answer 3

The three versions of the Triangle Inequality are:
* |x - z| ≤ |x - y| + |y - z|
* |x + y| ≤ |x| + |y|
* |x - y| ≥ ||x| - |y| |.

Question 4

What is the Archimedean Property?

Answer 4

For any two positive real numbers x and y, there exists a natural number n such that nx > y.

Question 5

State Bernoulli’s Inequality.

Answer 5

For any real number x ≥ -1 and any natural number n, (1 + x)^n ≥ 1 + nx.

Question 6

State the Inequality of Means.

Answer 6

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.

Question 7

State Newton’s Binomial Theorem.

Answer 7

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.

Question 8

Define an upper bound of a set.

Answer 8

An upper bound of a set A is a real number α such that α ≥ a for all a ∈ A.

Question 9

Define a supremum of a set.

Answer 9

The supremum of a set is the least upper bound of that set, if it exists.

Question 10

Define a lower bound of a set.

Answer 10

A lower bound of a set A is a real number α such that α ≤ a for all a ∈ A.

Question 11

Define an infimum of a set.

Answer 11

The infimum of a set is the greatest lower bound of that set, if it exists.

Question 12

Define the minimum of a set.

Answer 12

The minimum of a set is the smallest element within that set, and it must belong to the set.

Question 13

Define the maximum of a set.

Answer 13

The maximum of a set is the largest element within that set, and it must belong to the set.

Question 14

State the Completeness Axiom of the real numbers.

Answer 14

Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the set of real numbers.

Question 15

Define a sequence.

Answer 15

A sequence is an ordered list of numbers.

Question 16

Define the limit of a sequence.

Answer 16

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity.

Question 17

Define a convergent sequence.

Answer 17

A sequence that approaches a finite limit is a convergent sequence.

Question 18

Define a divergent sequence.

Answer 18

A sequence that does not approach a finite limit is a divergent sequence.

Question 19

Define a monotone sequence.

Answer 19

A monotone sequence is a sequence that is either non-decreasing or non-increasing.

Question 20

Define a subsequence.

Answer 20

A subsequence is a sequence formed by selecting elements from another sequence.

Question 21

Define a Cauchy sequence.

Answer 21

A sequence where the terms become arbitrarily close to each other as the index increases.

Question 22

State the Monotone Convergence Theorem.

Answer 22

A monotonic and bounded sequence converges.

Question 23

Define a series.

Answer 23

A series is the sum of the terms of a sequence.

Question 24

Define a partial sum.

Answer 24

A partial sum is the sum of a finite number of terms in a series.

Question 25

State the Comparison Test for series.

Answer 25

If 0 ≤ an ≤ bn for all n, and Σbn converges, then Σan converges. If Σan diverges, then Σbn diverges.

Question 26

State the Ratio Test for series, including its limitations.

Answer 26

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.

Question 27

State the Root Test for series, including its limitations.

Answer 27

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.

Question 28

Define absolute convergence.

Answer 28

A series Σan converges absolutely if the series of absolute values Σ|an| converges.

Question 29

Define conditional convergence.

Answer 29

A series Σan converges conditionally if Σan converges but Σ|an| diverges.

Question 30

State the Alternating Series Test.

Answer 30

An alternating series Σ(-1)^n * bn converges if the sequence bn is monotonically decreasing and lim bn = 0.

Question 31

Define a one-to-one (injective) function.

Answer 31

A function where each output corresponds to only one input.

Question 32

Define an onto (surjective) function.

Answer 32

A function where every output value can be achieved from an input value.

Question 33

Define a bounded function.

Answer 33

A function for which the output values stay within a given range.

Question 34

Define the limit of a function at a point using the Cauchy definition.

Answer 34

For every ε > 0, there exists δ > 0 such that if 0 < |x - x0| < δ, then |f(x) - L| < ε.

Question 35

Define the limit of a function at a point using the Heine definition.

Answer 35

For every sequence xn approaching x0, the sequence f(xn) approaches L.

Question 36

Define continuity of a function at a point.

Answer 36

A function f is continuous at a point x0 if **lim_(x→x0) f(x) = f(x0)**.

Question 37

Define continuity of a function on an interval.

Answer 37

A function is continuous on an interval if it is continuous at every point in that interval.

Question 38

Describe a removable discontinuity.

Answer 38

A discontinuity where the limit exists but is not equal to the function's value at that point.

Question 39

Describe a jump discontinuity.

Answer 39

A discontinuity where the left-hand and right-hand limits exist but are not equal.

Question 40

Describe an infinite discontinuity.

Answer 40

A discontinuity where the limit does not exist because it approaches infinity.

Question 41

State the Squeeze Theorem for functions.

Answer 41

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.

Question 42

State the Intermediate Value Theorem.

Answer 42

If a function is continuous on a closed interval, it takes on every value between the values at the interval endpoints.

Question 43

State the Extreme Value Theorem (Weierstrass Theorem).

Answer 43

A continuous function on a closed interval has both a maximum and a minimum value on that interval.

Question 44

Define the derivative of a function.

Answer 44

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**.

Question 45

Describe the geometric interpretation of the derivative.

Answer 45

The derivative represents the slope of the tangent line to the function's graph at a given point.

Question 46

How do derivatives relate to increasing/decreasing functions?

Answer 46

If f'(x) > 0 on an interval, then f is increasing; if f'(x) < 0, then f is decreasing.

Question 47

How do derivatives help find local extrema?

Answer 47

Local extrema occur at critical points where the derivative is zero or undefined.

Question 48

State Rolle's Theorem.

Answer 48

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.

Question 49

State the Mean Value Theorem.

Answer 49

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)**.

Question 50

Define a Taylor Polynomial.

Answer 50

A Taylor polynomial is an approximation of a function using a polynomial that uses the function's derivatives at a single point.

Question 51

When can you use the **Squeeze Theorem (Sandwich Theorem)** for functions?

Answer 51

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.

Question 52

What are the requirements to use the **Intermediate Value Theorem**?

Answer 52

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.

Question 53

What are the requirements to use the **Extreme Value Theorem (Weierstrass Theorem)**?

Answer 53

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.

Question 54

State the requirements to use **Rolle's Theorem**.

Answer 54

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)**.

Question 55

State the requirements to use the **Mean Value Theorem**.

Answer 55

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*).

Question 56

Define **'lim sup' ( גבול עליון)** of a sequence.

Answer 56

The *lim sup* of a sequence is the **largest limit point** of the sequence.

Question 57

Define **'lim inf' ( גבול תחתון)** of a sequence.

Answer 57

The *lim inf* of a sequence is the **smallest limit point** of the sequence.

Question 58

When can you use the **Comparison Test** for series convergence?

Answer 58

You can use the Comparison Test if you have two series, Σ*a_n* and Σ*b_n*, where 0 ≤ *a_n* ≤ *b_n* for all *n*. If Σ*b_n* converges, then Σ*a_n* also converges. If Σ*a_n* diverges, then Σ*b_n* also diverges.

Question 59

When is the **Ratio Test ( מבחן המנה)** inconclusive?

Answer 59

The Ratio Test is inconclusive when **lim |a_n+1 / a_n| = 1**. In this case, another convergence test is needed.

Question 60

When is the **Root Test ( מבחן השורש)** inconclusive?

Answer 60

The Root Test is inconclusive when **lim (|a_n|)^(1/n) = 1**. In this case, another convergence test is needed.

Question 61

What does it mean for a function to be **uniformly continuous**?

Answer 61

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**.

Question 62

What is the **Completeness Axiom (אקסיומת השלמות)** and why is it important?

Answer 62

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.

Question 63

What is the **Monotone Convergence Theorem**?

Answer 63

If a sequence is both **monotonic** (either non-decreasing or non-increasing) and **bounded**, then the sequence must **converge** to a finite limit.

Question 64

What are the conditions for using the **Alternating Series Test (מבחן לייבניץ)**?

Answer 64

The Alternating Series Test applies if the terms alternate in sign and the absolute values of the terms are monotonically decreasing and approach zero.

Question 65

What is a **Cauchy Sequence (סדרת קושי)** and why is it important?

Answer 65

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**.

Question 66

Define a **partial sum** of a series.

Answer 66

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*.

Question 67

How do you determine if a series converges **absolutely**?

Answer 67

A series ∑*a_n* converges absolutely if the series of the absolute values of its terms, ∑|*a_n*|, converges.

Question 68

What is the condition for a series to be **conditionally convergent**?

Answer 68

A series is conditionally convergent if it converges, but it does not converge absolutely. That is, ∑*a_n* converges, but ∑|*a_n*| diverges.

Question 69

What are the two definitions of a limit of a function at a point?

Answer 69

The two equivalent definitions are: the Cauchy definition (using ε-δ) and the Heine definition (using sequences).

Question 70

How can you classify a discontinuity of a function?

Answer 70

Discontinuities can be classified as removable, jump, or infinite based on the behavior of the limit at the point of discontinuity.