The function $$f(x) = e^x$$ is known to grow faster than polynomial functions. Which of the following statements is true regarding the growth comparison between $$f(x) = e^x$$ and $$g(x) = x^n$$ for large values of $x$?

Correct Option: B. Solution: The exponential function $f(x) = e^x$ grows faster than any polynomial function $g(x) = x^n$ as $x$ tends to infinity, regardless of the value of $n$. This is a fundamental property of exponential functions compared to polynomial functions.

What is the derivative of the function $f(x) = \cos x$?

Correct Option: B. Solution: The derivative of $\cos x$ is $-\sin x$.

For which of the following functions is the derivative at $x = 0$ equal to zero?

Correct Option: B. Solution: The derivative of $f(x) = x^2$ is $f'(x) = 2x$. Evaluating at $x = 0$, we get $f'(0) = 2(0) = 0$.

If $$f(x) = x^n$$ where $n$ is a positive integer, which of the following is the correct expression for $f'(x)$?

Correct Option: A. Solution: The derivative of $f(x) = x^n$ with respect to $x$ is $f'(x) = nx^{n-1}$, according to the power rule for differentiation.

For the function $f(x) = \ln(x^2 + 1)$, find the derivative $f'(x)$.

Correct Option: A. Solution: Using the chain rule, the derivative of $f(x) = \ln(x^2 + 1)$ is $f'(x) = \frac{d}{dx}[\ln(x^2 + 1)] = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$.

What is the derivative of the function $f(x) = \tan x$?

Correct Option: A. Solution: The derivative of $\tan x$ is $\sec^2 x$.

If $f(x) = \sin x$, what is $f'(x)$?

Correct Option: A. Solution: The derivative of $\sin x$ is $\cos x$.

If $f(x) = \cos x$, what is $f'(x)$?

Correct Option: B. Solution: The derivative of $f(x) = \cos x$ is $f'(x) = -\sin x$.

If $f(x) = x^3$, what is $f'(x)$?

Correct Option: A. Solution: The derivative of $x^3$ is $3x^2$.

Differentiate the function $f(x) = \ln(x^2 + 1)$ with respect to $x$.

Correct Option: A. Solution: Using the chain rule, the derivative of $f(x) = \ln(x^2 + 1)$ is $f'(x) = \frac{d}{dx} [\ln(x^2 + 1)] = \frac{1}{x^2 + 1} \cdot \frac{d}{dx} [x^2 + 1] = \frac{2x}{x^2 + 1}$.

For the function $f(x) = x^n$, where $n$ is a positive integer, what is the derivative $f'(x)$?

Correct Option: A. Solution: The derivative of $f(x) = x^n$ is $f'(x) = nx^{n-1}$.

What is the derivative of the function $f(x) = \sin x$?

Correct Option: A. Solution: The derivative of $\sin x$ is $\cos x$.

Given the function $$f(x) = \ln(x^2 - 1)$$, determine the domain of $f(x)$.

Correct Option: A. Solution: The function $f(x) = \ln(x^2 - 1)$ is defined when $x^2 - 1 > 0$, i.e., $x > 1$ or $x < -1$. Thus, the domain is $(-\infty, -1) \cup (1, \infty)$.

Continuity and Differentiability

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Summary

Chapter 5: Continuity and Differentiability

Summary

Continuation of differentiation study from Class XI.
Introduction of continuity and differentiability concepts.
Differentiation of inverse trigonometric functions.
Introduction of exponential and logarithmic functions.
Fundamental theorems in differential calculus.

Key Concepts

Continuity

A function is continuous at a point if the limit at that point equals the function's value.
Functions can be continuous on their entire domain.
Sum, difference, product, and quotient of continuous functions are also continuous.
Every differentiable function is continuous, but not vice versa.

Differentiability

The derivative of a function at a point is defined using limits:

$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$

Exponential and Logarithmic Functions

Exponential functions grow rapidly; logarithmic functions are their inverses.
Logarithmic differentiation is used for functions of the form $f(x) = [u(x)]^{v(x)}$ .

Important Theorems

If $f$ $f$ and $g$ $g$ are continuous at $c$ $c$ , then:
- $f + g$ is continuous at $c$
- $f - g$ is continuous at $c$
- $f \cdot g$ is continuous at $c$
- $\frac{f}{g}$ is continuous at $c$ if $g(c) \neq 0$

Exercises

Prove that the function $f(x) = 5x - 3$ is continuous at specified points.
Examine the continuity of various functions at given points.
Differentiate functions using the chain rule and logarithmic differentiation.

Examples

Example of checking continuity at specific points.
Example of differentiating functions using various methods.

Learning Objectives

Understand the concepts of continuity and differentiability.
Differentiate polynomial, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
Apply the chain rule for differentiation.
Identify and prove the continuity of various functions at specific points.
Utilize logarithmic differentiation for complex functions.
Explore the relationships between continuity and differentiability.
Recognize the importance of fundamental theorems in differential calculus.

Detailed Notes

Chapter 5: Continuity and Differentiability

5.1 Introduction

Continuation of differentiation study from Class XI.
Concepts introduced:
- Continuity
- Differentiability
- Inverse trigonometric functions
- Exponential and logarithmic functions
Fundamental theorems in differential calculus.

5.2 Continuity

Definition: A function is continuous at a point if the limit at that point equals the function's value.
Example: Function defined as:
- f(x) = 1, if x ≤ 0
- f(x) = 2, if x > 0
Graph Analysis: At x = 0, the function is not continuous.

Exercises

Prove that f(x) = 5x - 3 is continuous at x = 0, x = -3, and x = 5.
Examine continuity of f(x) = 2x² - 1 at x = 3.
Examine the following functions for continuity:
- (a) f(x) = 5
- (b) f(x) = x
- (c) f(x) = |x - 5|

5.3 Differentiability

Definition: The derivative of a real function at a point c is defined as:
- lim (f(c+h) - f(c)) / h as h approaches 0.
Key Point: Every differentiable function is continuous, but not vice versa.

5.4 Exponential and Logarithmic Functions

Introduction to exponential functions and logarithmic functions.
Natural Exponential Function: y = e^x.
Logarithm Definition: For b > 1, log_b(a) = x if b^x = a.

Important Observations

Logarithm function is defined from positive real numbers to all real numbers.
Common logarithm (base 10) and natural logarithm (base e).

5.5 Logarithmic Differentiation

Technique for differentiating functions of the form f(x) = [u(x)]^v(x).
Both f(x) and u(x) must be positive for this technique.

5.6 Derivatives of Functions in Parametric Forms

Relationships between variables expressed via a third variable.
Example: x = a cos³(θ), y = a sin³(θ).

5.7 Miscellaneous Exercises

Differentiate various functions and find second-order derivatives.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Continuity: Students often confuse the conditions for a function to be continuous. Remember that a function must be defined at the point, and the limit must equal the function's value at that point.
Ignoring Piecewise Functions: When dealing with piecewise functions, ensure to check continuity at the boundaries where the definition changes.
Limit Calculation Errors: Be careful with limit calculations, especially at points of discontinuity. Ensure to evaluate both left-hand and right-hand limits.
Differentiability vs. Continuity: A common mistake is assuming that if a function is continuous, it is also differentiable. Remember that a function can be continuous but not differentiable at certain points.

Tips for Success

Practice with Examples: Work through various examples of continuity and differentiability to solidify understanding. Pay attention to the definitions and properties of functions.
Graph Functions: Sketching graphs can help visualize continuity and differentiability. Look for breaks, jumps, or corners in the graph that indicate discontinuity or non-differentiability.
Review Definitions: Regularly review the definitions of continuity and differentiability, and practice applying them to different types of functions.
Use Theorems: Familiarize yourself with fundamental theorems related to continuity and differentiability, as they can provide shortcuts in problem-solving.

Practice & Assessment

Multiple Choice Questions

f(x) = \sin(x)

and

f'(x) = \cos(x)

f(x) = \cos(x)

and

f'(x) = \sin(x)

f(x) = \tan(x)

and

f'(x) = \sec(x)

f(x) = x^n

and

f'(x) = nx^{n+1}

Correct Answer: A

Solution:

The derivative of

f(x) = \sin(x)

f'(x) = \cos(x)

, which is correctly matched. The other options are incorrect:

f'(x) = -\sin(x)

for

\cos(x)

f'(x) = \sec^2(x)

for

\tan(x)

, and

f'(x) = nx^{n-1}

for

x^n

g(x)

grows faster than

f(x)

for any

n

f(x)

grows faster than

g(x)

for any

n

g(x)

grows faster than

f(x)

only if

n

is very large.

f(x)

and

g(x)

grow at the same rate.

Correct Answer: B

Solution:

The exponential function

f(x) = e^x

grows faster than any polynomial function

g(x) = x^n

x

tends to infinity, regardless of the value of

n

. This is a fundamental property of exponential functions compared to polynomial functions.

It is continuous but not differentiable at

x = 0

It is discontinuous at

x = 0

It is continuous and differentiable at all real numbers.

It is differentiable but not continuous at

x = 0

Correct Answer: C

Solution:

The function

f(x) = e^x

is both continuous and differentiable at all real numbers, including

x = 0

. The derivative of

e^x

e^x

, which is defined everywhere.

Continuous at

x = 3

Discontinuous at

x = 3

Continuous from the right at

x = 3

Continuous from the left at

x = 3

Correct Answer: A

Solution:

The function can be simplified to

f(x) = x + 3

for

x \neq 3

. The limit as

x

approaches 3 is

\lim_{x \to 3} (x + 3) = 6

. Since the function simplifies to a polynomial, it is continuous at

x = 3

Infinity

Undefined

Correct Answer: B

Solution:

The limit

\lim_{x \to 0} \frac{\sin x}{x} = 1

is a standard result in calculus. It can be derived using L'Hôpital's rule or by recognizing it as a fundamental limit.

\sin x

-\sin x

\cos x

\sec^2 x

Correct Answer: B

Solution:

The derivative of

\cos x

-\sin x

f(x) = x^2

f(x) = x^3

f(x) = 10^x

f(x) = x^{10}

Correct Answer: C

Solution:

The function

f(x) = 10^x

is an exponential function and grows faster than any polynomial function.

f(x) = x^3

f(x) = x^2

f(x) = \sin(x)

f(x) = \cos(x)

Correct Answer: B

Solution:

The derivative of

f(x) = x^2

f'(x) = 2x

. Evaluating at

x = 0

, we get

f'(0) = 2(0) = 0

nx^{n-1}

n^2x^{n-1}

nx^{n+1}

x^{n-1}

Correct Answer: A

Solution:

The derivative of

f(x) = x^n

with respect to

x

f'(x) = nx^{n-1}

, according to the power rule for differentiation.

\frac{2x}{x^2 + 1}

\frac{1}{x^2 + 1}

\frac{2}{x^2 + 1}

\frac{x}{x^2 + 1}

Correct Answer: A

Solution:

Using the chain rule, the derivative of

f(x) = \ln(x^2 + 1)

f'(x) = \frac{d}{dx}[\ln(x^2 + 1)] = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}

It grows slower than any polynomial function.

It grows at the same rate as polynomial functions.

It grows faster than any polynomial function.

It grows slower than linear functions but faster than quadratic functions.

Correct Answer: C

Solution:

Exponential functions like

f(x) = 10^x

grow faster than any polynomial function as

x

increases. This is because exponential growth is characterized by a constant multiplicative rate, unlike polynomial growth which is additive.

\sec^2 x

\cos x

\sin x

-\sin x

Correct Answer: A

Solution:

The derivative of

\tan x

\sec^2 x

e^x \sin(x) + e^x \cos(x)

e^x \cos(x) - e^x \sin(x)

e^x (\cos(x) + \sin(x))

e^x (\cos(x) - \sin(x))

Correct Answer: A

Solution:

Using the product rule, if

u(x) = e^x

and

v(x) = \sin(x)

, then

h'(x) = u'(x)v(x) + u(x)v'(x) = e^x \sin(x) + e^x \cos(x).

It is continuous everywhere.

It is discontinuous at

x = 0

It is only defined for

x > 0

It is not differentiable at

x = 0

Correct Answer: A

Solution:

The function

f(x) = \cos x

is continuous everywhere on the real line.

f(x) = \frac{1}{x}

f(x) = e^x

f(x) = \ln(x)

f(x) = \frac{1}{x^2}

Correct Answer: B

Solution:

The function

f(x) = e^x

is continuous for all real numbers, including at

x = 0

. The function

f(x) = \frac{1}{x}

is not defined at

x = 0

f(x) = \ln(x)

is not defined for

x \leq 0

, and

f(x) = \frac{1}{x^2}

is not defined at

x = 0

x = 1

x = 10

x = 2

x = 3

Correct Answer: C

Solution:

The exponential function

g(x) = 10^x

starts growing faster than the polynomial function

f(x) = x^{10}

when

x > 2

. At

x = 2

f(x) = 2^{10} = 1024

and

g(x) = 10^2 = 100

. For

x > 2

g(x)

surpasses

f(x)

in growth rate.

f(x) = |x|

f(x) = x^3

f(x) = \frac{1}{x}

f(x) = \tan x

Correct Answer: B

Solution:

The function

f(x) = x^3

is differentiable at all points, including

x = 0

They grow faster than any polynomial function.

They are always periodic.

They are always decreasing.

They have a constant rate of change.

Correct Answer: A

Solution:

Exponential functions grow faster than any polynomial function.

f(x) = \frac{1}{x}

f(x) = x^2

f(x) = x^3

f(x) = e^x

Correct Answer: A

Solution:

The function

f(x) = \frac{1}{x}

is not defined at

x = 0

, hence it is discontinuous at that point.

(-\infty, -1) \cup (1, \infty)

(-\infty, 1) \cup (1, \infty)

(-\infty, -1) \cup (1, \infty)

(1, \infty)

Correct Answer: A

Solution:

The function

f(x) = \ln(x^2 - 1)

is defined only when

x^2 - 1 > 0

, which implies

x^2 > 1

. Therefore,

x > 1

x < -1

. Thus, the domain of the function is

(-\infty, -1) \cup (1, \infty)

-\sin(x) \ln(x) + \frac{\cos(x)}{x}

-\sin(x) \ln(x) - \frac{\cos(x)}{x}

\sin(x) \ln(x) + \frac{\cos(x)}{x}

\sin(x) \ln(x) - \frac{\cos(x)}{x}

Correct Answer: A

Solution:

Using the product rule, if

u(x) = \cos(x)

and

v(x) = \ln(x)

, then

p'(x) = u'(x)v(x) + u(x)v'(x) = -\sin(x) \ln(x) + \frac{\cos(x)}{x}.

\cos x

-\sin x

\tan x

\sec^2 x

Correct Answer: A

Solution:

The derivative of

\sin x

\cos x

\sin x

-\sin x

\cos x

\sec^2 x

Correct Answer: B

Solution:

The derivative of

f(x) = \cos x

f'(x) = -\sin x

f'(x) = 2x \sin(x) + x^2 \cos(x)

f'(x) = 2x \cos(x) - x^2 \sin(x)

f'(x) = 2x \sin(x) - x^2 \cos(x)

f'(x) = 2x \cos(x) + x^2 \sin(x)

Correct Answer: A

Solution:

Using the product rule,

f'(x) = \frac{d}{dx}(x^2) \cdot \sin(x) + x^2 \cdot \frac{d}{dx}(\sin(x)) = 2x \sin(x) + x^2 \cos(x)

f(x) = \sin x

f(x) = |x|

f(x) = \frac{1}{x}

f(x) = \tan x

Correct Answer: A

Solution:

The function

f(x) = \sin x

is differentiable at every real number.

Continuous at

x = 0

Discontinuous at

x = 0

Continuous everywhere

Not defined at

x = 0

Correct Answer: B

Solution:

The function

f(x)

is discontinuous at

x = 0

because the left-hand limit and right-hand limit at

x = 0

do not equal the function value at

x = 0

. Specifically,

\lim_{x \to 0^-} f(x) = 1

\lim_{x \to 0^+} f(x) = 2

, and

f(0) = 1

. Since these limits are not equal, the function is discontinuous at

x = 0

\infty

Undefined

Correct Answer: A

Solution:

The limit

\lim_{x \to 0} \frac{\sin x}{x} = 1

is a standard result in calculus.

3x^2

x^2

3x

x^3

Correct Answer: A

Solution:

The derivative of

x^3

3x^2

3x^2 + 6x - 1

3x^2 + 6x + 1

3x^2 - 6x - 1

3x^2 - 6x + 1

Correct Answer: A

Solution:

The derivative of

f(x) = x^3 + 3x^2 - x + 5

is found by differentiating each term:

f'(x) = 3x^2 + 6x - 1

f(x) = \frac{1}{x}

f(x) = x^2

f(x) = \frac{1}{x^2}

f(x) = \frac{1}{x-1}

Correct Answer: B

Solution:

The function

f(x) = x^2

is a polynomial function, and polynomial functions are continuous at every real number.

Continuous at

x = 1

Discontinuous at

x = 1

Continuous from the left at

x = 1

Continuous from the right at

x = 1

Correct Answer: A

Solution:

To check the continuity at

x = 1

, we need to verify if

\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)

. We have

\lim_{x \to 1^-} f(x) = 1^2 = 1

and

\lim_{x \to 1^+} f(x) = 2(1) + 1 = 3

. Since

f(1) = 1

, the function is not continuous at

x = 1

f(x) = |x|

f(x) = x^3

f(x) = \sqrt{x}

f(x) = \frac{1}{x}

Correct Answer: B

Solution:

The function

f(x) = x^3

is differentiable at

x = 0

because it is a polynomial function, which is differentiable everywhere. The derivative at

x = 0

f'(x) = 3x^2

, and

f'(0) = 0

\frac{2x}{x^2 + 1}

\frac{x}{x^2 + 1}

\frac{1}{x^2 + 1}

\frac{2}{x^2 + 1}

Correct Answer: A

Solution:

Using the chain rule, the derivative of

f(x) = \ln(x^2 + 1)

f'(x) = \frac{d}{dx} [\ln(x^2 + 1)] = \frac{1}{x^2 + 1} \cdot \frac{d}{dx} [x^2 + 1] = \frac{2x}{x^2 + 1}

It is continuous for all real numbers.

It is continuous for all positive real numbers.

It is continuous for all non-negative real numbers.

It is discontinuous for all real numbers.

Correct Answer: B

Solution:

The natural logarithm function

f(x) = \ln(x)

is defined and continuous for all positive real numbers

x > 0

. It is not defined for

x \leq 0

. Therefore, it is continuous for all positive real numbers.

Yes,

f(x)

is continuous at

x = 2

No,

f(x)

is not continuous at

x = 2

f(x)

is continuous for

x \neq 2

f(x)

is discontinuous for all

x

Correct Answer: B

Solution:

The function

f(x) = \frac{x^2 - 4}{x - 2}

simplifies to

f(x) = x + 2

for

x \neq 2

. However,

f(x)

is not defined at

x = 2

, making it discontinuous at this point.

f(x) = x^3

f(x) = \frac{1}{x}

f(x) = \cos x

f(x) = e^x

Correct Answer: B

Solution:

The function

f(x) = \frac{1}{x}

is not continuous at

x = 0

because it is not defined there.

It is continuous at all real numbers.

It is discontinuous at

x = \pi

It is continuous only at integer multiples of

\pi

It is discontinuous at

x = 0

Correct Answer: A

Solution:

The function

f(x) = \sin(x)

is continuous for all real numbers, including at

x = 0

and

x = \pi

. Therefore, it is continuous everywhere.

Continuous at all real numbers

Continuous at

x = 0

and

x = 1

only

Continuous at

x = 0

only

Continuous at

x = 1

only

Correct Answer: A

Solution:

Polynomial functions are continuous at all real numbers. Therefore, the function

f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1

is continuous for all real numbers.

nx^{n-1}

x^n

n^x

x^{n+1}

Correct Answer: A

Solution:

The derivative of

f(x) = x^n

f'(x) = nx^{n-1}

\cos x

-\sin x

\tan x

-\cos x

Correct Answer: A

Solution:

The derivative of

\sin x

\cos x

Continuous at

x = 0

Discontinuous at

x = 0

Continuous everywhere except at

x = 0

Continuous at

x = 0

and

x = 1

Correct Answer: B

Solution:

To determine the continuity at

x = 0

, we check if

\lim_{x \to 0} f(x) = f(0)

. Calculating the limit,

\lim_{x \to 0} (x^3 + 3) = 3

. Since

f(0) = 1

, the limit does not equal the function value at

x = 0

, hence the function is discontinuous at

x = 0

(-\infty, -2) \cup (2, \infty)

(-\infty, 2) \cup (2, \infty)

(-2, 2)

(0, \infty)

Correct Answer: A

Solution:

The function

g(x) = \ln(x^2 - 4)

is defined when

x^2 - 4 > 0

. Solving the inequality, we get

x^2 > 4

, which implies

x > 2

x < -2

. Therefore, the domain of

g(x)

(-\infty, -2) \cup (2, \infty)

f(x) = e^x

g(x) = x^n

Both grow at the same rate

It depends on the value of

n

Correct Answer: A

Solution:

The exponential function

f(x) = e^x

grows faster than any polynomial function

g(x) = x^n

x \to \infty

. This is because the rate of growth of exponential functions is proportional to their current value, leading to a rapid increase compared to polynomials.

f'(x) = 2e^{2x} \sin(x) + e^{2x} \cos(x)

f'(x) = 2e^{2x} \cos(x) + e^{2x} \sin(x)

f'(x) = 2e^{2x} \sin(x) - e^{2x} \cos(x)

f'(x) = e^{2x} (2\sin(x) + \cos(x))

Correct Answer: A

Solution:

Using the product rule,

f'(x) = \frac{d}{dx}(e^{2x}) \cdot \sin(x) + e^{2x} \cdot \frac{d}{dx}(\sin(x)) = 2e^{2x} \sin(x) + e^{2x} \cos(x)

The function is continuous everywhere except at

x = 0

The function is continuous for all real numbers

The function is discontinuous only at

x = 1

The function is continuous only for

x > 0

Correct Answer: A

Solution:

The function

f(x) = \frac{1}{x}

is not defined at

x = 0

, hence it is discontinuous at

x = 0

. It is continuous for all other real numbers.

The growth of

f(x)

is linear.

The growth of

f(x)

is exponential.

The growth of

f(x)

is polynomial and depends on

n

The growth of

f(x)

is logarithmic.

Correct Answer: C

Solution:

The growth of

f(x) = x^n

is polynomial, and it depends on the degree

n

. As

n

increases, the function grows faster for the same increment in

x

f(x) = x^3

g(x) = 10^x

Both grow at the same rate

Cannot be determined

Correct Answer: B

Solution:

Exponential functions like

g(x) = 10^x

grow faster than polynomial functions like

f(x) = x^3

x

increases.

f(x) = \sin x

f(x) = \frac{1}{x}

f(x) = \tan x

f(x) = \sqrt{x}

Correct Answer: A

Solution:

The sine function

f(x) = \sin x

is continuous at every real number. The other functions have points of discontinuity:

f(x) = \frac{1}{x}

is discontinuous at

x = 0

f(x) = \tan x

is discontinuous at odd multiples of

\frac{\pi}{2}

, and

f(x) = \sqrt{x}

is undefined for

x < 0

x = 0

x = 1

x = -1

x = 2

Correct Answer: A

Solution:

The function

f(x) = \frac{1}{x}

is not defined at

x = 0

, hence it is not continuous there.

(-\infty, -1) \cup (1, \infty)

(-\infty, 1) \cup (1, \infty)

(-\infty, -1) \cup (-1, 1) \cup (1, \infty)

(-1, 1)

Correct Answer: A

Solution:

The function

f(x) = \ln(x^2 - 1)

is defined when

x^2 - 1 > 0

, i.e.,

x > 1

x < -1

. Thus, the domain is

(-\infty, -1) \cup (1, \infty)

x = 0

x = 1

x = -1

The function is continuous everywhere.

Correct Answer: A

Solution:

The function is discontinuous at

x = 0

because the left-hand limit and right-hand limit at this point do not match.

True or False

Correct Answer: True

Solution:

The derivative of

\sin x

with respect to

x

is indeed

\cos x

, as per standard differentiation rules.

Correct Answer: False

Solution:

The function

f(x) = \tan x

is not continuous at points where

\cos x = 0

, such as

x = \frac{\pi}{2} + n\pi

, where

n

is an integer.

Correct Answer: True

Solution:

A constant function

f(x) = k

is continuous at every real number because the limit of

f(x)

x

approaches any real number

c

k

, which equals

f(c)

Correct Answer: True

Solution:

The derivative of

\sin x

with respect to

x

\cos x

, as shown in standard differentiation rules.

Correct Answer: True

Solution:

The derivative of

\sin(x)

\cos(x)

, which is defined for all real numbers, indicating that

\sin(x)

is differentiable everywhere.

Correct Answer: True

Solution:

Trigonometric functions like

f(x) = \cos x

are continuous at every real number.

Correct Answer: True

Solution:

The derivative of

\arcsin(x)

is indeed

\frac{1}{\sqrt{1-x^2}}

, which is a standard result in calculus for differentiating inverse trigonometric functions.

Correct Answer: True

Solution:

The derivative of

\cos x

is indeed

-\sin x

, as per standard differentiation rules.

Correct Answer: True

Solution:

The function

f(x) = x^3

is a polynomial function, which is continuous at every real number, including

x = 0

Correct Answer: True

Solution:

Polynomial functions, such as

f(x) = x^3

, are continuous at every point in their domain, which is all real numbers.

Correct Answer: False

Solution:

While the function

f(x) = x^n

grows faster with increasing

n

, it is still a polynomial function. There are functions, such as exponential functions, that grow faster than any polynomial function.

Correct Answer: True

Solution:

The derivative of

\sin x

with respect to

x

\cos x

, as given by the standard differentiation rules.

Correct Answer: True

Solution:

For the identity function

f(x) = x

, the limit as

x

approaches any real number

c

c

, which equals

f(c)

. Hence, it is continuous everywhere.

Correct Answer: False

Solution:

Exponential functions like

f(x) = 10^x

grow faster than polynomial functions like

f(x) = x^n

for large values of

x

Correct Answer: True

Solution:

Trigonometric functions like

\sin(x)

are continuous at every real number.

Correct Answer: True

Solution:

The function

f(x) = \sin x

is differentiable at every real number, as it is a standard trigonometric function with a well-defined derivative

f'(x) = \cos x

Correct Answer: False

Solution:

Exponential functions like

f(x) = 10^x

grow faster than polynomial functions like

f(x) = x^n

x

increases.

Correct Answer: True

Solution:

For a function to be continuous on a closed interval

[a, b]

, it must be continuous at every point in the interval, including the endpoints

a

and

b

Correct Answer: False

Solution:

The differentiation of inverse trigonometric functions is indeed covered in the study of continuity and differentiability.

Correct Answer: True

Solution:

The derivative of the tangent function is the square of the secant function, so

f'(x) = \sec^2(x)

for

f(x) = \tan(x)

Correct Answer: False

Solution:

The function is not continuous at

x = 0

because the left-hand limit and right-hand limit at

x = 0

do not match the value of the function at

x = 0

Correct Answer: True

Solution:

If a function is defined only at one point, it is considered continuous there because there are no other points to compare.

Correct Answer: True

Solution:

A constant function

f(x) = k

is continuous at every real number because

\lim_{x \to c} f(x) = k = f(c)

for any real number

c

Correct Answer: False

Solution:

The function

f(x) = x^2

is a polynomial and is continuous at all real numbers, including

x = 0

Correct Answer: False

Solution:

The function is not continuous at

x = 0

because the limit of the function as

x

approaches 0 from the left is 1, and from the right is 2, which do not equal the function's value at

x = 0

Correct Answer: True

Solution:

Polynomial functions like

f(x) = x^3

are continuous at every real number.

Correct Answer: False

Solution:

The function

f(x) = \frac{1}{x}

is not defined at

x = 0

, and therefore it is not continuous at

x = 0

Correct Answer: False

Solution:

The function

f(x) = \tan x

is not defined at

x = \frac{\pi}{2}

, and thus it cannot be continuous at this point.

Correct Answer: True

Solution:

Exponential functions grow faster than polynomial functions for large values of

x

. As

x

increases,

10^x

will surpass any polynomial function

x^n

Correct Answer: True

Solution:

The function

\ln(x)

is continuous and differentiable for all

x > 0

, as it is defined and smooth in this domain.

Correct Answer: True

Solution:

The identity function

f(x) = x

is continuous at every real number, as the limit of

f(x)

x

approaches any real number

c

is equal to

c

, which is

f(c)

Correct Answer: True

Solution:

The identity function

f(x) = x

is defined and continuous at every real number because for any real number

c

\lim_{x \to c} f(x) = c = f(c)

Correct Answer: True

Solution:

Exponential functions like

f(x) = e^x

grow faster than polynomial functions as

x

increases.

Correct Answer: True

Solution:

The derivative of the sine function is the cosine function, so

f'(x) = \cos(x)

for

f(x) = \sin(x)

Correct Answer: True

Solution:

The function

f(x) = x^3

is a polynomial, and polynomials are differentiable everywhere on the real line.

Correct Answer: False

Solution:

The function

f(x) = \tan(x)

is not continuous at points where

\cos(x) = 0

, such as

x = \frac{\pi}{2} + k\pi

, where

k

is an integer.

Correct Answer: False

Solution:

The identity function

f(x) = x

is continuous at every real number because

f(c) = c

for every real number

c

, and the limit as

x

approaches

c

is also

c

Correct Answer: True

Solution:

Exponential functions like

y = 10^x

increase more rapidly than polynomial functions such as

y = x^n

x

becomes larger.

Correct Answer: False

Solution:

The function

f(x) = \tan x

is not differentiable at

x = \frac{\pi}{2}

because it has a vertical asymptote at this point.

Correct Answer: False

Solution:

Polynomial functions, such as

f(x) = x^3

, are continuous everywhere on the real line, including at

x = 0

Correct Answer: False

Solution:

The function

\tan(x)

is not continuous at points where

\cos(x) = 0

, i.e., at

x = (2n+1)\frac{\pi}{2}

for any integer

n

, due to vertical asymptotes.

Correct Answer: True

Solution:

Exponential functions like

y = 10^x

grow faster than polynomial functions like

y = x^4

x

increases.

Correct Answer: True

Solution:

Exponential functions like

f(x) = e^x

are continuous at every real number.

Correct Answer: True

Solution:

If a function is defined only at one point, it is continuous there by definition, as there are no other points to consider for limits.

Continuity and Differentiability

AI Learning Assistant

Summary

Chapter 5: Continuity and Differentiability

Summary

Key Concepts

Continuity

Differentiability

Exponential and Logarithmic Functions

Important Theorems

Exercises

Examples

Learning Objectives

Learning Objectives

Detailed Notes

Chapter 5: Continuity and Differentiability

5.1 Introduction

5.2 Continuity

Exercises

5.3 Differentiability

5.4 Exponential and Logarithmic Functions

Important Observations

5.5 Logarithmic Differentiation

5.6 Derivatives of Functions in Parametric Forms

5.7 Miscellaneous Exercises

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.Which of the following derivatives is correctly matched with its function?

Correct Answer: A

Q2.The function f(x)=exf(x) = e^xf(x)=ex is known to grow faster than polynomial functions. Which of the following statements is true regarding the growth comparison between f(x)=exf(x) = e^xf(x)=ex and g(x)=xng(x) = x^ng(x)=xn for large values of xxx?

Correct Answer: B

Q3.Which of the following statements is true about the function f(x)=exf(x) = e^xf(x)=ex?

Correct Answer: C

Q4.Given the function f(x)=x2−9x−3f(x) = \frac{x^2 - 9}{x - 3}f(x)=x−3x2−9​, determine its continuity at x=3x = 3x=3 after simplification.

Correct Answer: A

Q5.Evaluate the limit lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}limx→0​xsinx​.

Correct Answer: B

Q6.What is the derivative of the function f(x)=cos⁡xf(x) = \cos xf(x)=cosx?

Correct Answer: B

Q7.Which of the following functions grows faster than any polynomial function?

Correct Answer: C

Q8.For which of the following functions is the derivative at x=0x = 0x=0 equal to zero?

Correct Answer: B

Q9.If f(x)=xnf(x) = x^nf(x)=xn where nnn is a positive integer, which of the following is the correct expression for f′(x)f'(x)f′(x)?

Correct Answer: A

Q10.For the function f(x)=ln⁡(x2+1)f(x) = \ln(x^2 + 1)f(x)=ln(x2+1), find the derivative f′(x)f'(x)f′(x).

Correct Answer: A

Q11.Consider the exponential function f(x)=10xf(x) = 10^xf(x)=10x. What can be said about its rate of growth compared to polynomial functions?

Correct Answer: C

Q12.What is the derivative of the function f(x)=tan⁡xf(x) = \tan xf(x)=tanx?

Correct Answer: A

Q13.For the function h(x)=exsin⁡(x)h(x) = e^x \sin(x)h(x)=exsin(x), find the derivative using the product rule.

Correct Answer: A

Q14.Which of the following statements is true about the function f(x)=cos⁡xf(x) = \cos xf(x)=cosx?

Correct Answer: A

Q15.Which of the following functions is continuous at x=0x = 0x=0?

Correct Answer: B

Q16.Given the polynomial function f(x)=x10f(x) = x^{10}f(x)=x10 and the exponential function g(x)=10xg(x) = 10^xg(x)=10x, for which value of xxx does g(x)g(x)g(x) start growing faster than f(x)f(x)f(x)?

Correct Answer: C

Q17.Which of the following functions is differentiable at x=0x = 0x=0?

Correct Answer: B

Q18.Which of the following is a property of exponential functions?

Correct Answer: A

Q19.Which of the following functions has a point of discontinuity at x=0x = 0x=0?

Correct Answer: A

Q20.Consider the function f(x)=ln⁡(x2−1)f(x) = \ln(x^2 - 1)f(x)=ln(x2−1). What is the domain of this function?

Correct Answer: A

Q21.Determine the derivative of the function p(x)=cos⁡(x)ln⁡(x)p(x) = \cos(x) \ln(x)p(x)=cos(x)ln(x) using the product rule.

Correct Answer: A

Q22.If f(x)=sin⁡xf(x) = \sin xf(x)=sinx, what is f′(x)f'(x)f′(x)?

Correct Answer: A

Q23.If f(x)=cos⁡xf(x) = \cos xf(x)=cosx, what is f′(x)f'(x)f′(x)?

Correct Answer: B

Q24.For the function f(x)=x2sin⁡(x)f(x) = x^2 \sin(x)f(x)=x2sin(x), find the derivative using the product rule.

Correct Answer: A

Q25.Which of the following functions is differentiable at every real number?