Master Real Analysis I: Foundations of Calculus and Beyond

Question 1

Question: Define continuity for a function f(x) at a point x = a using the epsilon-delta definition. Then, demonstrate that the function f(x) = |x| is continuous at x = 0 using this definition.

Accepted Answer

Option 1: A function f(x) is continuous at a point x = a if for every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - f(a)| < ε.

Answer

A function f(x) is continuous at a point x = a if its graph has no breaks or jumps at x = a.

Answer

A function f(x) is continuous at a point x = a if its graph has no breaks or jumps at x = a.

Answer

A function f(x) is continuous at a point x = a if its graph has no breaks or jumps at x = a.

Question 2

Evaluate the integral ∫[0, 1] sin(x) dx using the Fundamental Theorem of Calculus, Part II.

Accepted Answer

cos(1) - cos(0)

Answer

sin(1) - sin(0)

Answer

-cos(1) + cos(0)

Answer

sin(1) + cos(0)

Question 3

Prove that between any two real numbers, there exists a rational number.

Accepted Answer

B. True, because the set of rational numbers is dense in the set of real numbers.

Answer

A. True, because every rational number can be written as a fraction of two integers.

Answer

C. False, because there are irrational numbers that cannot be expressed as a fraction of two integers.

Answer

D. False, because the set of rational numbers is countable while the set of real numbers is uncountable.

Question 4

Find the directional derivative of the function f(x, y) = e^(x^2 + y^2) at the point (1, 1) in the direction of the unit vector <1, 2>.

Accepted Answer

2e

Answer

e

Answer

1

Answer

0

Question 5

Which of the following is **not** a property of a continuous function on a closed interval?

Accepted Answer

It has breaks or jumps in its graph.

Answer

It assumes a minimum value.

Answer

It assumes a maximum value.

Answer

Its range is bounded.

Question 6

Evaluate the limit of the function (x^2 - 1)/(x - 1) as x approaches 1.

Accepted Answer

1

Answer

2

Answer

The limit does not exist.

Answer

0

Question 7

Let f(x) be a function such that f'(x) > 0 for all x. Which of the following statements is true?

Accepted Answer

f(x) is increasing.

Answer

f(x) is decreasing.

Answer

f(x) has a maximum at x = 0.

Answer

f(x) has a minimum at x = 0.

Question 8

Which of the following is the Taylor series expansion of the function f(x) = e^x around x = 0?

Accepted Answer

1 + x + x^2/2! + x^3/3! + ...

Answer

1 - x + x^2/2! - x^3/3! + ...

Answer

e^x + e^(x^2)/2! + e^(x^3)/3! + ...

Answer

x + x^2/2! + x^3/3! + x^4/4! + ...

Question 9

Find the derivative of the function f(x) = e^x + sin(x). (Apply the sum rule and derivative formulas for exponential and trigonometric functions.)

Accepted Answer

e^x + cos(x)

Answer

e^x - cos(x)

Answer

e^x * cos(x)

Answer

sin(x) / e^x

Question 10

Determine the critical points of the function f(x) = x^3 - 3x^2. (Find the derivative and set it equal to zero.)

Accepted Answer

x = 0, x = 1

Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

No critical points

Question 11

The limit of a function f(x) as x approaches c, denoted as lim_(x->c) f(x), is a value that:

Accepted Answer

The function approaches as x gets closer and closer to c

Answer

The function always equals at c

Answer

Is the slope of the function at c

Answer

Is the area under the curve of the function from c to infinity

Question 12

A continuous function is one that:

Accepted Answer

Has no sudden breaks or jumps in its graph

Answer

Has no local maxima or minima

Answer

Is always differentiable

Answer

Has a derivative that is continuous everywhere

Question 13

The derivative of the function f(x) = x^2 is:

Accepted Answer

2x

Answer

x^3

Answer

x

Answer

1/x^2

Question 14

The Fundamental Theorem of Calculus, Part 1 states that:

Accepted Answer

The integral of a derivative is the original function

Answer

The derivative of an integral is the original function

Answer

The area under the curve of a function from a to b is equal to the integral of the function from a to b

Answer

The volume of a solid of revolution is equal to the integral of the cross-sectional area of the solid from 0 to the height of the solid

Question 15

An improper integral of the form ∫[a,∞] f(x) dx is said to converge if:

Accepted Answer

The limit of the integral as b approaches infinity exists

Answer

The integral is always positive

Answer

The function f(x) is continuous on [a,∞)

Answer

The function f(x) is decreasing on [a,∞)

Question 16

A complete metric space is a metric space in which:

Accepted Answer

Every Cauchy sequence converges

Answer

Every bounded sequence converges

Answer

Every convergent sequence is Cauchy

Answer

Every open set is closed

Question 17

A Banach space is a normed vector space that is:

Accepted Answer

Complete

Answer

Hilbert

Answer

Finite-dimensional

Answer

Reflexive

Question 18

A reflexive space is a Banach space that:

Accepted Answer

Is isomorphic to its dual

Answer

Is Hilbert

Answer

Is finite-dimensional

Answer

Is complete

Question 19

Which of the following accurately defines the limit of a function at a point?

Accepted Answer

The limit of a function at a point is the value the function approaches as the input approaches the point from both the left and the right.

Answer

The limit of a function at a point is the value the function takes at that point.

Answer

The limit of a function at a point must be unique.

Answer

The limit of a function at a point exists only if the function is continuous at that point.

Question 20

What is the integral of ∫(x^2 + 1) dx?

Accepted Answer

(x^3 / 3) + x + C

Answer

(x^2 / 2) + C

Answer

x^3 + x + C

Answer

x^2 + x + C

Question 21

Which of the following sequences is convergent?

Accepted Answer

{n/(n+1)}

Answer

{(-1)^n}

Answer

{n^2}

Answer

{e^n}

Question 22

Which of the following is a Banach space?

Accepted Answer

The set of all continuous functions on [0, 1] with the supremum norm.

Answer

The set of all sequences of real numbers with the usual norm.

Answer

The set of all polynomials with the usual norm.

Answer

The set of all matrices with the Frobenius norm.

Question 23

If a function f(x) is differentiable at x = a, which of the following must hold true?

Accepted Answer

Both f(x) is continuous at x = a and f'(a) exists.

Answer

f(x) is continuous at x = a.

Answer

f'(a) exists.

Answer

f''(a) exists.

Question 24

Provide the epsilon-delta definition of a limit.

Accepted Answer

For any positive number epsilon, there exists a number delta such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon.

Answer

For any positive number delta, there exists a number epsilon such that if 0 < |f(x) - L| < epsilon, then |x - a| < delta.

Answer

For any positive number L, there exists a number delta such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon.

Answer

For any positive number L, there exists a number epsilon such that if 0 < |f(x) - L| < epsilon, then |x - a| < delta.

Question 25

Find the derivative of f(x) = x^3.

Accepted Answer

3x^2

Answer

2x

Answer

1/x

Answer

x^4

Question 26

Which condition is NOT necessary for a function to be differentiable at a point?

Accepted Answer

Function must be increasing at the point.

Answer

Function must be continuous at the point.

Answer

Function must have a derivative at the point.

Answer

Function must be defined at the point.

Question 27

Find the limit of f(x) = x^2 - 1 as x approaches infinity.

Accepted Answer

Infinity

Answer

-1

Answer

0

Answer

1

Question 28

Evaluate the definite integral of 1 / x from 1 to 2.

Accepted Answer

ln(2)

Answer

0

Answer

1

Answer

Infinity

Question 29

Question:

State the Stone-Weierstrass Theorem in the context of real analysis.

Accepted Answer

Any continuous function on a compact interval can be uniformly approximated by a polynomial.

Answer

Any continuous function on a closed interval can be uniformly approximated by a rational function.

Answer

Any differentiable function on a compact interval can be uniformly approximated by a trigonometric polynomial.

Answer

Any continuous function on a closed interval can be uniformly approximated by a sequence of continuous functions.

Question 30

The limit of a function f(x) at a point c is defined as the value that f(x) approaches as x approaches c. Which of the following statements about the limit of a function is true?

Accepted Answer

The limit exists if and only if the left-hand limit and the right-hand limit exist and are equal.

Answer

The limit exists if and only if the function is continuous at that point.

Answer

The limit exists if and only if the derivative of the function exists at that point.

Answer

The limit exists if and only if the function is differentiable at that point.

Question 31

The derivative of a function f(x) is defined as the limit of the difference quotient as the increment approaches zero. What is the derivative of f(x) = e^x?

Accepted Answer

e^x

Answer

x * e^x

Answer

e^x / x

Answer

0

Question 32

The Heine-Borel theorem states that every closed and bounded subset of R^n is compact. Which of the following statements about the Heine-Borel theorem is true?

Accepted Answer

Every closed and bounded subset of R^n is compact.

Answer

Every compact subset of R^n is closed and bounded.

Answer

Every open and bounded subset of R^n is compact.

Answer

Every closed and unbounded subset of R^n is compact.

Question 33

Calculate the derivative of the function f(x) = sin(x^3)

Accepted Answer

3x^2 * cos(x^3)

Answer

cos(x^3)

Answer

sin(x^3)

Answer

x^2 * sin(x^3)

Question 34

Determine the convergence of the series: Σ[n=1 to infinity] (n^2 / (n^2 + 1))

Accepted Answer

Converges

Answer

Conditionally convergent

Answer

Diverges

Answer

Cannot be determined

Question 35

Determine the uniform continuity of the function f(x) = |x| on the interval [-1, 1]

Accepted Answer

Uniformly continuous

Answer

Cannot be determined

Answer

Not continuous

Answer

Continuous but not uniformly continuous

Question 36

Find the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2. Use factoring to simplify the expression and avoid the indeterminate form.

Accepted Answer

4

Answer

0

Answer

Diverges

Answer

2

Question 37

Find the derivative of the function f(x) = x^3 - 2x^2 + x - 1 using the power rule and simplify the result.

Accepted Answer

3x^2 - 4x + 1

Answer

x^3 - 2x

Answer

x^2 - 4x + 1

Answer

3x^3 - 2x^2

Question 38

Evaluate the definite integral ∫[0, 1] x^2 dx using the fundamental theorem of calculus.

Accepted Answer

1/3

Answer

1

Answer

1/2

Answer

2

Question 39

Which of the following statements is true about the function f(x) = e^x? Provide a brief explanation.

Accepted Answer

It is always increasing.

Answer

It is always decreasing.

Answer

It has a minimum at x = 0.

Answer

It has an inflection point at x = 0.

Question 40

Determine the convergence or divergence of the series ∑[n=1 to ∞] 1/n^2 using the general term test.

Accepted Answer

Converges

Answer

Use the ratio test

Answer

Use the alternating series test

Answer

Diverges

Question 41

Determine the continuity of the function f(x) = |x - 2| at x = 2.

Accepted Answer

Yes

Answer

Only if f(2) = 0

Answer

No

Answer

Cannot be determined

Question 42

Determine the convergence of the series: ∑(n=1 to ∞) (n / (n² + 1)).

Accepted Answer

Converges

Answer

Cannot be determined

Answer

Diverges

Answer

Conditional convergence

Question 43

Define the limit of a function.

Accepted Answer

The value that the function approaches as the input approaches a specified value.

Answer

The value of the function at a specified input

Answer

The slope of the function at a specified input

Question 44

Which condition must be met for a function to be continuous at a point?

Accepted Answer

The function must be defined at that point.

Answer

The function must be differentiable at that point.

Answer

The function must have a zero derivative at that point.

Answer

The function must be integrable over a closed interval containing that point.

Question 45

Find the derivative of the function f(x) = x^2.

Accepted Answer

f'(x) = 2x

Answer

f'(x) = x^3

Answer

f'(x) = x

Answer

f'(x) = 2

Question 46

State the epsilon-delta definition of a limit.

Accepted Answer

For any given positive number epsilon, there exists a positive number delta such that whenever 0 < |x - a| < delta, then |f(x) - L| < epsilon.

Answer

For any given positive number delta, there exists a positive number epsilon such that whenever |f(x) - L| < epsilon, then 0 < |x - a| < delta.

Question 47

Enunciate the Mean Value Theorem.

Accepted Answer

If f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Answer

If f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in [a, b] such that f'(c) = (f(b) - f(a))/(b - a).

Question 48

A function f(x) is continuous at x = c. Which of the following statements is **always** true?

Accepted Answer

The limit of f(x) as x approaches c exists and is equal to f(c).

Answer

The function f(x) is differentiable at x = c.

Answer

The value of f(c) is equal to 0.

Answer

The limit of f(x) as x approaches c is greater than f(c).

Question 49

For a function to be differentiable at a point, which of the following is a **necessary** condition?

Accepted Answer

The function must be continuous at that point.

Answer

The function must be increasing at that point.

Answer

The derivative of the function must exist at that point.

Answer

The tangent line to the function at that point must have a slope of zero.

Question 50

Which of the following is a direct application of integration?

Accepted Answer

Calculating the area under a curve.

Answer

Solving systems of linear equations.

Answer

Solving trigonometric equations.

Answer

Finding the derivative of a function.

Question 51

Which of the following statements is **NOT** true about the limit of a function f(x) as x approaches a?

Accepted Answer

If the limit of f(x) as x approaches a exists, then f(a) must also exist.

Answer

If the limit of f(x) as x approaches a exists, then f(x) approaches a finite value as x approaches a.

Answer

If f(a) exists, the limit of f(x) as x approaches a may or may not exist.

Question 52

Which of the following conditions is **sufficient** to guarantee that a function f(x) is continuous at x = a?

Accepted Answer

The limit of f(x) as x approaches a exists, and f(a) is equal to that limit.

Answer

The limit of f(x) as x approaches a exists.

Answer

f(a) exists and is finite.

Question 53

Which of the following correctly states the Mean Value Theorem?

Accepted Answer

If f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Answer

If f(x) is differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Question 54

A sequence (a_n) is convergent if and only if:

Accepted Answer

The limit of (a_n) as n approaches infinity exists.

Answer

The sequence (a_n) is bounded.

Answer

The sequence (a_n) is monotonic.

Question 55

Given a sequence of real numbers {a_n}, which of the following statements is **not** equivalent to the convergence of {a_n} to a limit L?

Accepted Answer

The sequence {a_n} is bounded.

Answer

The sequence {a_n} is Cauchy.

Answer

For every ε > 0, there exists an N ∈ ℕ such that |a_n - L| < ε for all n ≥ N.

Answer

The sequence {a_n} has a unique accumulation point.

Question 56

A function f(x) is continuous at a point x = c if and only if:

Accepted Answer

The limit of f(x) as x approaches c exists and equals f(c).

Answer

The limit of f(x) as x approaches c exists.

Answer

f(c) is defined.

Answer

f(x) is differentiable at x = c.

Question 57

The derivative of a function f(x) at x = c, if it exists, is given by:

Accepted Answer

lim_(h→0) (f(c + h) - f(c))/h

Answer

lim_(x→c) (f(x) - f(c))/(x - c)

Answer

lim_(x→c) (f(x) - f(c))/x

Answer

lim_(h→0) (f(c) - f(c - h))/h

Question 58

Which theorem guarantees that for a continuous function f(x) on the closed interval [a, b], there exists a point c in (a, b) where f(c) = (f(b) - f(a))/(b - a)?

Accepted Answer

Mean Value Theorem

Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Rolle's Theorem

Question 59

If a function f(x) is differentiable at a point x = c, it is also:

Accepted Answer

Continuous at x = c.

Answer

Twice differentiable at x = c.

Answer

Has a local maximum or minimum at x = c.

Answer

Bounded in a neighborhood of x = c.

Question 60

Which of the following statements is **not** necessarily true if f(x) is differentiable at x = c?

Accepted Answer

f'(c) = 0.

Answer

The limit of the difference quotient exists as h approaches 0.

Answer

f(x) is continuous at x = c.

Question 61

A continuous function f(x) on the interval [a, b] is integrable on [a, b] if and only if:

Accepted Answer

f(x) is bounded on [a, b].

Answer

f(x) has a finite number of discontinuities on [a, b].

Answer

f(x) is differentiable on [a, b].

Answer

f(x) has a local maximum or minimum on [a, b].

Question 62

Which of the following functions has a discontinuity at *x* = 0?

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 63

If a function *f(x)* is differentiable at *x* = *a*, which of the following is *always* true?

Accepted Answer

f(x) is continuous at x = a.

Answer

The derivative of f(x) is continuous at x = a.

Answer

f(x) is bounded in a neighborhood of x = a.

Answer

f(x) has a local maximum or minimum at x = a.

Question 64

What is the derivative of *f(x) = x² * sin(x)* with respect to *x*?

Accepted Answer

2x sin(x) + x² cos(x)

Answer

x² cos(x)

Answer

2x sin(x)

Answer

2x cos(x)