Master Real Analysis Concepts: Quizzes and Learning Resources

Question 1

What is the derivative of the function f(x) = x^3 + 2x^2 - 1 with respect to x?

Accepted Answer

3x^2 + 4x

Answer

3x^2 + 2

Answer

x^3 + 4x

Answer

x^3 + 2x

Question 2

Which of the following improper integrals is convergent?

Accepted Answer

∫[0, ∞) e^(-x) dx

Answer

∫[0, ∞) 1/x^2 dx

Answer

∫[0, ∞) sin(x) dx

Answer

∫[0, ∞) e^x dx

Question 3

Which of the following sequences is convergent?

Accepted Answer

{n / (n + 1)}

Answer

{n^2 / (n + 1)}

Answer

{(-1)^n}

Answer

{n}

Question 4

What is the limit of the function f(x) = (x^2 - 1) / (x - 1) as x approaches 1?

Accepted Answer

1

Answer

0

Answer

2

Answer

Does not exist

Question 5

Find the radius of convergence for the power series ∑(n=0 to ∞) (x - 3)^n / n!

Accepted Answer

∞

Answer

0

Answer

1

Answer

3

Question 6

Which of the following is a necessary condition for uniform continuity of a function f(x) on an interval [a, b]?

Accepted Answer

f(x) must be continuous on [a, b]

Answer

f(x) must be differentiable on [a, b]

Answer

f(x) must be bounded on [a, b]

Question 7

Which of the following is a precise definition of the limit of a function?

Accepted Answer

The real number which the function approaches as its input approaches a specific value

Answer

The value of the function at the specified input value

Answer

The derivative of the function at the specified input value

Answer

The maximum or minimum value of the function over some interval

Question 8

Which of the following is a sufficient condition for a function to be continuous at a point?

Accepted Answer

The left-hand limit and right-hand limit of the function exist and are equal at the point

Answer

The derivative of the function is continuous at the point

Answer

The function is defined at the point

Answer

The function is differentiable at the point

Question 9

Compute the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x + 4

Answer

x + 2

Question 10

Determine the interval of convergence for the power series ∑[n=0,∞] (x - 5)^n / n^3:

Accepted Answer

|x - 5| < 3

Answer

|x - 5| > 3

Answer

|x - 5| <= 3

Answer

|x - 5| >= 3

Question 11

Which of the following functions is NOT uniformly continuous on the interval [0, 1]?

Accepted Answer

f(x) = 1 / x

Answer

f(x) = x^2

Answer

f(x) = |x - 1/2|

Answer

f(x) = x

Question 12

Which condition must be satisfied for a function to be differentiable at a point?

Accepted Answer

The limit of the difference quotient exists at the point

Answer

The function is defined and continuous at the point

Answer

The derivative of the function is continuous at the point

Answer

The function has a finite discontinuity at the point

Question 13

Given the function f(x) = |x|, calculate its derivative at x = 0.

Accepted Answer

It is undefined.

Answer

0

Answer

1

Answer

-1

Question 14

Evaluate the improper integral ∫[0, ∞) e^(-x) dx.

Accepted Answer

1

Answer

0

Answer

∞

Answer

It cannot be evaluated.

Question 15

Find the general term of the series 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

1/2^n

Answer

1/n

Answer

n

Answer

2^n

Question 16

Find the radius of convergence of the power series ∑[n=0, ∞) x^n / n!

Accepted Answer

∞

Answer

0

Answer

1

Answer

Cannot be determined

Question 17

Consider the differential equation y' + 2y = 0. Determine its general solution.

Accepted Answer

y = e^(-2x)

Answer

y = e^(2x)

Answer

y = -e^(-2x)

Answer

y = -e^(2x)

Question 18

Determine the convergence of the series ∑[n=1 to ∞] (n + 1) / (n^2 + 2n). (Hint: Use the limit comparison test.)

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges conditionally

Question 19

Determine the interval of absolute convergence of the series ∑[n=1 to ∞] (-1)^n * n * e^(-n). (Hint: Use the ratio test.)

Accepted Answer

(-∞, ∞)

Answer

[0, ∞)

Answer

(-∞, 0]

Answer

[0, 1]

Question 20

Let f be a continuous function on [0, 1]. Which of the following must be true? (Hint: Use the Extreme Value Theorem.)

Accepted Answer

f has a maximum and a minimum on [0, 1].

Answer

f has a unique root on [0, 1].

Answer

f is differentiable on (0, 1).

Answer

f is Riemann integrable on [0, 1].

Question 21

What is the best definition of a limit?

Accepted Answer

The value that a function gets infinitely close to as the input approaches a specific value.

Answer

The value that a function equals at a particular input.

Answer

The rate of change of a function at a particular input.

Answer

The area under the curve of a function between two points.

Question 22

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

Accepted Answer

The function must be differentiable at the point.

Answer

The function must be defined at the point.

Answer

The function must have a limit at the point.

Answer

The graph of the function must have no breaks at the point.

Question 23

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

Accepted Answer

3x^2

Answer

x^2

Answer

x

Answer

1

Question 24

What is the integral of the function f(x) = sin(x)?

Accepted Answer

cos(x) + C

Answer

sin(x) + C

Answer

-cos(x) + C

Answer

-sin(x) + C

Question 25

State the squeeze theorem.

Accepted Answer

If f(x) ≤ g(x) ≤ h(x) for all x in an interval and lim(x->a) f(x) = lim(x->a) h(x) = L, then lim(x->a) g(x) = L.

Answer

If lim(x->a) f(x) = L, then lim(x->a) f(x)^2 = L^2.

Answer

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) + g(x)] = L + M.

Answer

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) * g(x)] = L * M.

Question 26

Find the derivative of the function g(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

4e^(2x + 1)

Answer

(2x + 1)e^(2x + 1)

Answer

e^(2x + 1)

Question 27

Determine the convergence of the sequence a_n = (n + 1) / n as n approaches infinity.

Accepted Answer

1

Answer

does not converge

Answer

infinity

Answer

0

Question 28

Find the sum of the series 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

2

Answer

infinity

Answer

1/2

Answer

1

Question 29

Find the infimum of the set {x ∈ R | 0 ≤ x ≤ 1}.

Accepted Answer

0

Answer

1/2

Answer

does not exist

Answer

1

Question 30

Determine if the following statement is true or false: Every Cauchy sequence in R is convergent.

Accepted Answer

True

Answer

False

Answer

False for unbounded sequences only

Answer

True for bounded sequences only

Question 31

Find the limit of the sequence $$a_n = \frac{n^2 - 1}{n^2 + 1}$$.

Accepted Answer

1

Answer

\infty

Answer

0

Answer

-1

Question 32

Determine if the function $$f(x) = \frac{x^2 - 4}{x - 2}$$ is continuous at $$x = 2$$.

Accepted Answer

No

Answer

Yes

Answer

Cannot be determined

Answer

Only if $x \neq 2$

Question 33

Find the radius of convergence of the power series $$\sum_{n=0}^\infty \frac{(x - 2)^n}{n!}$$.

Accepted Answer

\infty

Answer

2

Answer

1

Answer

0

Question 34

Find the Taylor series expansion of the function $$f(x) = \sin(x)$$ about $$x = 0$$.

Accepted Answer

$$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

Answer

$$x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$$

Answer

$$x - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

Question 35

Determine if the function $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.

Accepted Answer

No

Answer

Cannot be determined.

Answer

Yes

Answer

It depends on the value of $x$.

Question 36

Find the derivative of the function $f(x) = e^{2x} + \sin(3x)$.

Accepted Answer

2e^{2x} + 3\cos(3x)

Answer

e^{2x} - \sin(3x)

Answer

2xe^{2x} + 3x\cos(3x)

Answer

e^{2x} + \cos(3x)

Question 37

Determine if the series $\sum_{n=1}^{\infty} \frac{n}{n^2 + 1}$ converges or diverges.

Accepted Answer

Converges

Answer

Cannot be determined.

Answer

Diverges

Question 38

Find the supremum and infimum of the set $A = \{x \in \mathbb{R} : 1 \leq x \leq 5\}$.

Accepted Answer

Supremum: 5, Infimum: 1

Answer

Supremum: 6, Infimum: 1

Answer

Supremum: \infty, Infimum: 0

Answer

Supremum: 5, Infimum: 0

Question 39

Determine the open intervals on which the function $f(x) = \frac{1}{x - 1}$ is increasing.

Accepted Answer

(-∞, 1) U (1, ∞)

Answer

(-1, 1)

Answer

(0, 1)

Answer

(-∞, -1) U (1, ∞)

Question 40

Determine if the sequence $a_n = \frac{(-1)^n}{n}$ is monotonic.

Accepted Answer

It is oscillating, not monotonic.

Answer

It is increasing.

Answer

It is decreasing.

Answer

It is neither increasing nor decreasing.

Question 41

Let f(x) = |x|. Which of the following is true about the derivative of f(x) at x = 0?

Accepted Answer

It does not exist

Answer

It exists and is equal to 1

Answer

It exists and is equal to -1

Answer

It exists and is equal to 0

Question 42

Which of the following functions is NOT uniformly continuous on the interval [0, 1]?

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 43

What is the Cauchy criterion for the convergence of a sequence in a metric space?

Accepted Answer

For any ε > 0, there exists an integer N such that d(a_m, a_n) < ε for all m, n > N

Answer

For any ε > 0, there exists an integer N such that d(a_m, a_n) = ε for all m, n > N

Answer

For any ε > 0, there exists an integer N such that d(a_m, a_n) > ε for all m, n > N

Question 44

What is the limit of the function *f(x) = (x² - 4) / (x - 2)* as *x* approaches 2?

Accepted Answer

4

Answer

Undefined

Answer

2

Answer

0

Question 45

Calculate the definite integral of the function *f(x) = 2x + 1* over the interval [0, 2].

Accepted Answer

6

Answer

8

Answer

10

Answer

4

Question 46

Does the sequence *aₙ = (n² + 1) / (n² - 1)* converge or diverge? If it converges, what is its limit?

Accepted Answer

Converges to 1

Answer

Diverges to infinity

Answer

Diverges to -1

Answer

None of the above

Question 47

Is the function *f(x) = e^x* uniformly continuous on the interval [0, 1]?

Accepted Answer

Yes

Answer

No

Answer

None of the above

Answer

Cannot be determined

Question 48

Which of the following statements is **NOT** true about a continuous function *f(x)* on a closed interval [*a, b*]?

Accepted Answer

The function *f(x)* must be differentiable on [*a, b*].

Answer

The function *f(x)* must be bounded on [*a, b*].

Answer

The function *f(x)* must attain a maximum and minimum value on [*a, b*].

Answer

The function *f(x)* must be uniformly continuous on [*a, b*].

Question 49

What is the limit of the sequence {*a_n*} = {1/*n*} as *n* approaches infinity?

Accepted Answer

0

Answer

∞

Answer

Does not exist

Answer

1

Question 50

What is the derivative of the function *f(x)* = *x*² at *x* = 2?

Accepted Answer

4

Answer

2

Answer

8

Answer

1

Question 51

Which statement accurately describes the integral of a continuous function *f(x)* over a closed interval [*a, b*]?

Accepted Answer

The integral represents the area under the curve of *f(x)* between *x* = *a* and *x* = *b*.

Answer

The integral is always equal to the area of a rectangle with base (*b* - *a*) and height *f(a)*.

Answer

The integral is always positive.

Question 52

The series ∑(1/*n*) is:

Accepted Answer

Divergent

Answer

Conditionally convergent

Answer

Absolutely convergent

Answer

Convergent

Question 53

The function *f(x)* = |*x*| is:

Accepted Answer

Continuous but not differentiable at *x* = 0

Answer

Differentiable at *x* = 0

Answer

Both continuous and differentiable at *x* = 0

Answer

Not continuous at *x* = 0

Question 54

The Intermediate Value Theorem states that if a function *f(x)* is continuous on a closed interval [*a, b*] and *k* is a number between *f(a)* and *f(b)*, then:

Accepted Answer

There exists a number *c* in [*a, b*] such that *f(c)* = *k*.

Answer

The function *f(x)* must be bounded on [*a, b*].

Answer

The function *f(x)* must be differentiable on [*a, b*].

Question 55

What is the limit of (sin(*x*))/*x* as *x* approaches 0?

Accepted Answer

1

Answer

0

Answer

∞

Answer

Does not exist

Question 56

What is the integral of the function *f(x)* = 1/*x* from 1 to *e*?

Accepted Answer

1

Answer

ln(*e*)

Answer

*e*

Answer

0