Question 1

If the function $f(x) = x^2$ is given, what is the derivative of $f(x)$ at $x = 2$?

Accepted Answer

Correct Option: C. Solution: The derivative of $f(x) = x^2$ is $f'(x) = 2x$. At $x = 2$, $f'(2) = 2 	imes 2 = 4$.

Question 2

What is the average velocity of a body dropped from a tall cliff between $t = 1.9$ seconds and $t = 2$ seconds?

Accepted Answer

Correct Option: C. Solution: The average velocity between $t = 1.9$ and $t = 2$ seconds is calculated as $\frac{19.6 - 17.689}{2 - 1.9} = 19.551$ m/s.

Question 3

Using the first principle of derivatives, calculate the derivative of $f(x) = x^3$ at $x = 2$.

Accepted Answer

Correct Option: B. Solution: Using the first principle, $f'(x) = \lim_{h 	o 0} \frac{(x+h)^3 - x^3}{h}$. Expanding $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$, the derivative becomes $\lim_{h 	o 0} (3x^2 + 3xh + h^2) = 3x^2$. At $x = 2$, $f'(2) = 3 	imes 2^2 = 12$.

Question 4

Consider the functions $f(x) = x^2 \sin(\frac{1}{x})$ for $x 
eq 0$ and $f(0) = 0$. Using the Sandwich Theorem, what is $\lim_{x 	o 0} f(x)$?

Accepted Answer

Correct Option: A. Solution: For $x 
eq 0$, $-x^2 \leq x^2 \sin(\frac{1}{x}) \leq x^2$. As $x 	o 0$, both $-x^2$ and $x^2$ tend to $0$. By the Sandwich Theorem, $\lim_{x 	o 0} x^2 \sin(\frac{1}{x}) = 0$.

Question 5

Evaluate the limit $$\lim_{{x 	o 2}} \frac{x^3 - 4x^2 + 4x}{x^2 - 4}$$.

Accepted Answer

Correct Option: B. Solution: The limit can be evaluated by factoring and canceling common terms, resulting in a limit of 1.

Question 6

What is the derivative of the function $f(x) = \frac{1}{x}$ using the first principle?

Accepted Answer

Correct Option: A. Solution: Using the first principle, $f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h} = \lim_{h 	o 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = -\frac{1}{x^2}$.

Question 7

The Sandwich Theorem is used to evaluate limits by comparing a function with two other functions. Which of the following correctly describes the Sandwich Theorem?

Accepted Answer

Correct Option: A. Solution: The Sandwich Theorem states that if $f(x) \leq g(x) \leq h(x)$ for all $x$ in some interval around $a$, except possibly at $a$, and if $\lim_{x 	o a} f(x) = \lim_{x 	o a} h(x) = L$, then $\lim_{x 	o a} g(x) = L$.

Question 8

Consider the function $$f(x) = x^2 \sin \frac{1}{x}$$ for $$x 
eq 0$$ and $$f(0) = 0$$. Use the Sandwich Theorem to find $$\lim_{{x 	o 0}} f(x)$$.

Accepted Answer

Correct Option: A. Solution: We know $$-1 \leq \sin \frac{1}{x} \leq 1$$, thus $$-x^2 \leq x^2 \sin \frac{1}{x} \leq x^2$$. As $$x 	o 0$$, both $$-x^2$$ and $$x^2$$ approach 0, implying $$\lim_{{x 	o 0}} x^2 \sin \frac{1}{x} = 0$$ by the Sandwich Theorem.

Question 9

If $f(x) = x^2 - 4$, what is the limit of $f(x)$ as $x$ approaches 2?

Accepted Answer

Correct Option: B. Solution: As $x$ approaches 2, $f(x) = x^2 - 4$ approaches 4. Therefore, the limit is 4.

Question 10

Find the derivative of the function $f(x) = rac{1}{x}$ using the first principle.

Accepted Answer

Correct Option: A. Solution: Using the first principle, $f'(x) = rac{d}{dx}(rac{1}{x}) = -x^{-2}$.

Question 11

What is the derivative of the function $f(x) = x^2$ at $x = 3$ using the first principles?

Accepted Answer

Correct Option: B. Solution: The derivative of $f(x) = x^2$ using first principles is $f'(x) = \lim_{h 	o 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h 	o 0} \frac{2xh + h^2}{h} = 2x$. At $x = 3$, $f'(3) = 2 	imes 3 = 6$.

Question 12

Evaluate the limit $$\lim_{{x 	o a}} \frac{{x^n - a^n}}{{x-a}}$$ where $n$ is a positive integer.

Accepted Answer

Correct Option: A. Solution: Using the algebraic limit formula, the limit evaluates to $na^{n-1}$.

Question 13

For the function $f(x) = x^3$, find the derivative using the first principle at $x = 1$.

Accepted Answer

Correct Option: B. Solution: The derivative of $f(x) = x^3$ using the first principles is $f'(x) = \lim_{h 	o 0} \frac{(x+h)^3 - x^3}{h} = \lim_{h 	o 0} \frac{3x^2h + 3xh^2 + h^3}{h} = \lim_{h 	o 0} (3x^2 + 3xh + h^2) = 3x^2$. At $x = 1$, $f'(1) = 3 	imes 1^2 = 3$.

Question 14

Consider the function $f(x) = x^2$. Using the first principles, what is the derivative of $f(x)$ at $x = 3$?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = x^2$ using the first principles is $f'(x) = \lim_{h 	o 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h 	o 0} \frac{2xh + h^2}{h} = \lim_{h 	o 0} (2x + h) = 2x$. At $x = 3$, $f'(3) = 2 	imes 3 = 6$.

Question 15

Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. What is the limit of $f(x)$ as $x$ approaches 2?

Accepted Answer

Correct Option: A. Solution: The function $f(x) = \frac{x^2 - 4}{x - 2}$ simplifies to $f(x) = x + 2$ for $x 
eq 2$. As $x$ approaches 2, $f(x)$ approaches $4$. Therefore, the limit is 4.

Question 16

If $f(x) = x^3 - 3x^2 + 2x$, what is the derivative $f'(x)$?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x)$ is found using the power rule: $f'(x) = 3x^2 - 6x + 2$.

Question 17

Consider a function $f(x) = x^2 + 3x + 2$. Calculate the derivative of the function and find the slope of the tangent to the curve at $x = 1$.

Accepted Answer

Correct Option: A. Solution: The derivative of the function $f(x) = x^2 + 3x + 2$ is $f'(x) = 2x + 3$. At $x = 1$, the slope of the tangent is $f'(1) = 2(1) + 3 = 5$.

Question 18

Evaluate the limit $\lim_{{x 	o 1}} \frac{x^{15} - 1}{x^{10} - 1}$ using the algebraic limit formula.

Accepted Answer

Correct Option: C. Solution: Using the algebraic limit formula, $\lim_{{x 	o 1}} \frac{x^{15} - 1}{x^{10} - 1}$ can be evaluated as $\frac{15 \cdot 1^{14}}{10 \cdot 1^9} = \frac{15}{10}$.

Question 19

Find the derivative of the function $f(x) = \sin^2 x$ using the product rule.

Accepted Answer

Correct Option: A. Solution: Using the product rule, the derivative of $f(x) = \sin^2 x$ is $2\sin x \cos x$, which can also be expressed as $\sin 2x$.

Question 20

What is the limit of the function $f(x) = x^2$ as $x$ approaches 0?

Accepted Answer

Correct Option: A. Solution: As $x$ approaches 0, $f(x) = x^2$ approaches 0. Therefore, the limit is 0.

Question 21

What is the limit of the function $$f(x) = \frac{{x^3 - 8}}{{x - 2}}$$ as $x$ approaches 2?

Accepted Answer

Correct Option: A. Solution: Factor the numerator: $$x^3 - 8 = (x - 2)(x^2 + 2x + 4)$$. Cancel the common factor $x - 2$ to get $$\lim_{{x 	o 2}} (x^2 + 2x + 4) = 6$$.

Question 22

Which of the following is true about the limits of $f(x) = |x|$ as $x$ approaches 0?

Accepted Answer

Correct Option: A. Solution: For $f(x) = |x|$, both the left-hand and right-hand limits as $x$ approaches 0 are 0. Therefore, the limit exists and is 0.

Question 23

What is the derivative of the trigonometric function $f(x) = 	an x$?

Accepted Answer

Correct Option: D. Solution: The derivative of $	an x$ is $	ext{sec}^2 x$.

Question 24

What is the limit of the expression $\lim_{{x 	o a}} \frac{x^n - a^n}{x - a}$?

Accepted Answer

Correct Option: A. Solution: The limit of the expression $\lim_{{x 	o a}} \frac{x^n - a^n}{x - a}$ is $na^{n-1}$ as derived from the algebraic limit formula.

Question 25

Consider the function $f(x) = \frac{{x^3 - 4x^2 + 4x}}{{x^2 - 4}}$. Evaluate the limit as $x$ approaches 2.

Accepted Answer

Correct Option: A. Solution: The function evaluates to the form $\frac{0}{0}$ at $x=2$, and after simplification, the limit does not exist.

Question 26

Evaluate the limit: $$\lim_{{x 	o 2}} \frac{{x^2 - 4}}{{x - 2}}$$.

Accepted Answer

Correct Option: A. Solution: The expression can be simplified by factoring the numerator: $$x^2 - 4 = (x - 2)(x + 2)$$. Cancel the common factor $x - 2$ to get $$\lim_{{x 	o 2}} (x + 2) = 4$$.

Question 27

What is the average velocity of a body dropped from a tall cliff between $t = 1$ second and $t = 2$ seconds, given that the distance covered is $S = 4.9t^2$?

Accepted Answer

Correct Option: B. Solution: The average velocity is calculated as the change in distance over the change in time: $$\frac{S(2) - S(1)}{2 - 1} = \frac{19.6 - 4.9}{1} = 14.7 	ext{ m/s}$$.

Question 28

Which of the following is the derivative of $f(x) = rac{1}{x}$?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = rac{1}{x}$ is $f'(x) = -rac{1}{x^2}$.

Question 29

Determine the limit $$\lim_{{x 	o 2}} \frac{x^2 - 4}{x - 2}$$ using factorization.

Accepted Answer

Correct Option: C. Solution: Factor the numerator as $(x - 2)(x + 2)$, then the expression simplifies to $x + 2$. Thus, $$\lim_{{x 	o 2}} \frac{x^2 - 4}{x - 2} = 4$$.

Question 30

If the average velocity of a body between $t = 1.95$ seconds and $t = 2$ seconds is $19.355 	ext{ m/s}$, what does this suggest about the instantaneous velocity at $t = 2$ seconds?

Accepted Answer

Correct Option: B. Solution: As the time interval approaches zero, the average velocity approaches the instantaneous velocity. Since the average velocity is $19.355 	ext{ m/s}$ for an interval close to $t = 2$, the instantaneous velocity is slightly more than $19.355 	ext{ m/s}$.

Question 31

Which of the following statements is true about the derivative of a constant function?

Accepted Answer

Correct Option: A. Solution: The derivative of a constant function is always zero because there is no change in the function value with respect to $x$.

Question 32

Using the Sandwich Theorem, evaluate the limit $$\lim_{{x 	o 0}} \frac{\sin x}{x}$$.

Accepted Answer

Correct Option: B. Solution: The Sandwich Theorem states that if $$f(x) \leq g(x) \leq h(x)$$ and $$\lim_{{x 	o a}} f(x) = \lim_{{x 	o a}} h(x) = L$$, then $$\lim_{{x 	o a}} g(x) = L$$. Here, $$\sin x \leq x \leq 	an x$$ and both $$\lim_{{x 	o 0}} \frac{\sin x}{x}$$ and $$\lim_{{x 	o 0}} \frac{x}{x}$$ equal 1.

Question 33

What is the derivative of the function $f(x) = x^2$ at $x = 1$?

Accepted Answer

Correct Option: B. Solution: The derivative of $f(x) = x^2$ is $f'(x) = 2x$. At $x = 1$, $f'(1) = 2 	imes 1 = 2$.

Question 34

For the function $f(x) = \frac{x^2 - 4}{x - 2}$, what is the derivative $f'(x)$ at $x = 2$ using the limit definition?

Accepted Answer

Correct Option: D. Solution: The function $f(x) = \frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$, hence the derivative does not exist at this point.

Question 35

What is the derivative of the function $f(x) = x^3$ at $x = 1$?

Accepted Answer

Correct Option: B. Solution: The derivative of $f(x) = x^3$ is $f'(x) = 3x^2$. At $x = 1$, $f'(1) = 3 \cdot 1^2 = 3$.

Question 36

Given the function $S = 4.9t^2$, which represents the distance $S$ in meters covered by a body dropped from a tall cliff after $t$ seconds, what is the instantaneous velocity of the body at $t = 2$ seconds?

Accepted Answer

Correct Option: B. Solution: The instantaneous velocity is the derivative of the distance function with respect to time. The derivative of $S = 4.9t^2$ is $v(t) = \frac{dS}{dt} = 9.8t$. At $t = 2$ seconds, $v(2) = 9.8 	imes 2 = 19.6 	ext{ m/s}$.

Question 37

Using the definition of derivative, find the derivative of $f(x) = x^2$ at $x = 3$.

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = x^2$ is $f'(x) = 2x$. Evaluating at $x = 3$, we get $f'(3) = 2 	imes 3 = 6$.

Question 38

Which theorem helps in evaluating the limit of a function that is sandwiched between two other functions?

Accepted Answer

Correct Option: C. Solution: The Sandwich Theorem states that if a function is sandwiched between two other functions that have the same limit at a point, then the function also has that limit.

Question 39

What is the geometric interpretation of the derivative of a function at a point?

Accepted Answer

Correct Option: B. Solution: The derivative of a function at a point represents the slope of the tangent to the curve at that point.

Question 40

What is the derivative of the function $f(x) = 3x$ at $x = 2$ using the first principle?

Accepted Answer

Correct Option: B. Solution: Using the first principle, the derivative of $f(x) = 3x$ is $f'(x) = \lim_{h 	o 0} \frac{3(x+h) - 3x}{h} = \lim_{h 	o 0} \frac{3h}{h} = 3$. At $x = 2$, $f'(2) = 3$.

Question 41

Evaluate the limit: $$\lim_{x 	o 0} \frac{\sin x}{x}$$.

Accepted Answer

Correct Option: B. Solution: The limit $$\lim_{x 	o 0} \frac{\sin x}{x}$$ is a standard limit that evaluates to 1.

Question 42

If $f(x) = x \sin x$, what is $f'(x)$ using the product rule?

Accepted Answer

Correct Option: A. Solution: Using the product rule, $f'(x) = \frac{d}{dx}(x) \sin x + x \frac{d}{dx}(\sin x) = \sin x + x \cos x$.

Question 43

Using the first principle of derivatives, find the derivative of the function $f(x) = \sin x$ at $x = 0$.

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = \sin x$ using the first principles is $f'(x) = \lim_{h 	o 0} \frac{\sin(x+h) - \sin x}{h} = \lim_{h 	o 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h} = \cos x$. At $x = 0$, $f'(0) = \cos 0 = 1$.

Question 44

Which of the following statements best describes the derivative of a function at a point?

Accepted Answer

Correct Option: B. Solution: The derivative of a function at a point is defined as the instantaneous rate of change of the function at that point, represented by the slope of the tangent to the curve at that point.

Question 45

Find the derivative of the function $f(x) = 	an x$ using the quotient rule.

Accepted Answer

Correct Option: D. Solution: Using the quotient rule, the derivative of $f(x) = 	an x = rac{	an x}{1}$ is $f'(x) = rac{1 	imes 	an x - 	an x 	imes 0}{	an^2 x} = rac{	an^2 x}{	an x} - 	an x$.

Question 46

A ball is dropped from a height and its distance from the ground at any time $t$ is given by $S = 4.9t^2$. What is the instantaneous velocity of the ball at $t = 2$ seconds?

Accepted Answer

Correct Option: B. Solution: The instantaneous velocity is the derivative of the distance function $S = 4.9t^2$. Differentiating, we get $v(t) = \frac{dS}{dt} = 9.8t$. At $t = 2$ seconds, $v(2) = 9.8 	imes 2 = 19.6$ m/s.

Question 47

What is the derivative of $f(x) = 	an x$ using the quotient rule?

Accepted Answer

Correct Option: A. Solution: Using the quotient rule, $f'(x) = \sec^2 x$ since $	an x = \frac{\sin x}{\cos x}$ and the derivative of $	an x$ is $\sec^2 x$.

Question 48

Which of the following statements is true about the left-hand and right-hand limits of a function?

Accepted Answer

Correct Option: B. Solution: The left-hand and right-hand limits are equal only if the limit exists.

Question 49

Consider the function $f(x) = x^2$. What is the derivative of $f(x)$ at $x = 1$, and what does it represent?

Accepted Answer

Correct Option: D. Solution: The derivative of $f(x) = x^2$ is $f'(x) = 2x$. At $x = 1$, $f'(1) = 2 	imes 1 = 2$. This represents the slope of the tangent line to the curve at $x = 1$.

Question 50

What is the derivative of the function $f(x) = x^n$ with respect to $x$, where $n$ is a positive integer?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = x^n$ is $nx^{n-1}$ as per the power rule of differentiation.

Question 51

Using the product rule, what is the derivative of $f(x) = x^2 \cdot \sin x$?

Accepted Answer

Correct Option: A. Solution: The product rule states $(uv)' = u'v + uv'$. Here, $u = x^2$ and $v = \sin x$. Thus, $u' = 2x$ and $v' = \cos x$. Therefore, $f'(x) = 2x \cdot \sin x + x^2 \cdot \cos x$.

Question 52

Find the left-hand limit of the function $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the left.

Accepted Answer

Correct Option: C. Solution: As $x$ approaches 0 from the left, $\frac{1}{x}$ approaches $-Infinity$. Thus, the left-hand limit is $-Infinity$.

Question 53

The function $f(x) = \frac{1}{x}$ is given. What is the derivative of $f(x)$ using the first principles?

Accepted Answer

Correct Option: A. Solution: Using the first principles, the derivative of $f(x) = \frac{1}{x}$ is $f'(x) = \lim_{h 	o 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h 	o 0} \frac{-h}{x(x+h)h} = -\frac{1}{x^2}$.

Question 54

Given $f(x) = \frac{x^2 + 1}{x}$, find the derivative $f'(x)$ using the quotient rule.

Accepted Answer

Correct Option: B. Solution: Using the quotient rule, $f'(x) = \frac{(x)(2x) - (x^2 + 1)(1)}{x^2} = \frac{x^2 - 1}{x^2}$.

Question 55

Consider the function $f(x) = 3x^2 + 2x - 5$. What is the derivative of $f(x)$ at $x = 1$?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x)$ is $f'(x) = 6x + 2$. At $x = 1$, $f'(1) = 6(1) + 2 = 8$.

Question 56

If $f(x) = x^3 - 3x^2 + 5x - 7$, what is the derivative of $f(x)$ using the sum and difference rule?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = x^3 - 3x^2 + 5x - 7$ is $f'(x) = 3x^2 - 6x + 5$ using the power rule for each term.

Question 57

Evaluate the limit $$\lim_{{x 	o 0}} \frac{1 - \cos x}{x^2}$$ using the Sandwich Theorem.

Accepted Answer

Correct Option: D. Solution: Using the identity $$1 - \cos x = 2 \sin^2 \frac{x}{2}$$, we have $$\lim_{{x 	o 0}} \frac{1 - \cos x}{x^2} = \lim_{{x 	o 0}} \frac{2 \sin^2 \frac{x}{2}}{x^2} = \lim_{{x 	o 0}} \frac{2 \left(\frac{x}{2}ight)^2}{x^2} = \frac{1}{2}$$.

Question 58

What is the derivative of the function $f(x) = x^2$ at $x = 3$?

Accepted Answer

Correct Option: A. Solution: The derivative of $f(x) = x^2$ is $f'(x) = 2x$. At $x = 3$, $f'(3) = 2 	imes 3 = 6$.

Question 59

Evaluate the limit $$\lim_{x 	o 0} \frac{1 - \cos x}{x}$$.

Accepted Answer

Correct Option: A. Solution: The limit $$\lim_{x 	o 0} \frac{1 - \cos x}{x}$$ is 0, using the identity $$1 - \cos x = 2 \sin^2 \left(\frac{x}{2}ight)$$.

Question 60

Which of the following is the correct derivative of $f(x) = \sin x \cdot \cos x$?

Accepted Answer

Correct Option: A. Solution: Using the product rule, $(\sin x \cdot \cos x)' = \cos x \cdot \cos x - \sin x \cdot \sin x = \cos^2 x - \sin^2 x$.

Question 61

Using the first principles, what is the derivative of $f(x) = \sin x$ at $x = 0$?

Accepted Answer

Correct Option: B. Solution: Using the first principles, $f'(x) = \lim_{h 	o 0} \frac{\sin(x+h) - \sin(x)}{h} = \lim_{h 	o 0} \frac{\sin h}{h} \cos x + \lim_{h 	o 0} \frac{\cos h - 1}{h} \sin x$. At $x = 0$, $f'(0) = 1$.

Question 62

What is the limit of $$\frac{\sin x}{x}$$ as $x$ approaches 0?

Accepted Answer

Correct Option: B. Solution: The limit of $$\frac{\sin x}{x}$$ as $x$ approaches 0 is 1, as shown by the Sandwich Theorem.

Question 63

The derivative of the polynomial function $f(x) = x^3$ at $x = 1$ is $3$.

Accepted Answer

Answer: True. Solution: The derivative of $f(x) = x^3$ is $f'(x) = 3x^2$. At $x = 1$, $f'(1) = 3(1)^2 = 3$.

Question 64

The limit $$\lim_{{x 	o a}} \frac{{x^n - a^n}}{{x - a}} = na^{n-1}$$ can be used for any positive integer $$n$$.

Accepted Answer

Answer: True. Solution: The formula $$\lim_{{x 	o a}} \frac{{x^n - a^n}}{{x - a}} = na^{n-1}$$ is valid for any positive integer $$n$$, as shown by dividing $$x^n - a^n$$ by $$x - a$$ and evaluating the limit.

Limits and Derivatives

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Intuitive Idea of Derivatives

Limits and Their Algebra

Derivative from First Principles

Algebra of Derivatives

Derivative of Polynomial and Trigonometric Functions

Sandwich Theorem

Limits of Trigonometric Functions

Limits of Polynomial and Rational Functions

Important Algebraic Limit Formula

Tangent and Rate-of-Change Applications

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Intuitive Idea of Derivatives

Limits and Their Algebra

Derivative from First Principles

Algebra of Derivatives

Derivative of Polynomial and Trigonometric Functions

Sandwich Theorem

Limits of Trigonometric Functions

Limits of Polynomial and Rational Functions

Important Algebraic Limit Formula

Tangent and Rate-of-Change Applications

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Multiple Choice Questions

Q1.If the function f(x)=x2f(x) = x^2f(x)=x2 is given, what is the derivative of f(x)f(x)f(x) at x=2x = 2x=2?

Correct Answer: C

Q2.What is the average velocity of a body dropped from a tall cliff between t=1.9t = 1.9t=1.9 seconds and t=2t = 2t=2 seconds?

Correct Answer: C

Q3.Using the first principle of derivatives, calculate the derivative of f(x)=x3f(x) = x^3f(x)=x3 at x=2x = 2x=2.

Correct Answer: B

Q4.Consider the functions f(x)=x2sin⁡(1x)f(x) = x^2 \sin(\frac{1}{x})f(x)=x2sin(x1​) for x≠0x \neq 0x=0 and f(0)=0f(0) = 0f(0)=0. Using the Sandwich Theorem, what is lim⁡x→0f(x)\lim_{x \to 0} f(x)limx→0​f(x)?

Correct Answer: A

Q5.Evaluate the limit lim⁡x→2x3−4x2+4xx2−4\lim_{{x \to 2}} \frac{x^3 - 4x^2 + 4x}{x^2 - 4}limx→2​x2−4x3−4x2+4x​.

Correct Answer: B

Q6.What is the derivative of the function f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ using the first principle?

Correct Answer: A

Q7.The Sandwich Theorem is used to evaluate limits by comparing a function with two other functions. Which of the following correctly describes the Sandwich Theorem?

Correct Answer: A

Q8.Consider the function f(x)=x2sin⁡1xf(x) = x^2 \sin \frac{1}{x}f(x)=x2sinx1​ for x≠0x \neq 0x=0 and f(0)=0f(0) = 0f(0)=0. Use the Sandwich Theorem to find lim⁡x→0f(x)\lim_{{x \to 0}} f(x)limx→0​f(x).

Correct Answer: A

Q9.If f(x)=x2−4f(x) = x^2 - 4f(x)=x2−4, what is the limit of f(x)f(x)f(x) as xxx approaches 2?

Correct Answer: B

Q10.Find the derivative of the function f(x) = rac{1}{x} using the first principle.

Correct Answer: A

Q11.What is the derivative of the function f(x)=x2f(x) = x^2f(x)=x2 at x=3x = 3x=3 using the first principles?

Correct Answer: B

Q12.Evaluate the limit lim⁡x→axn−anx−a\lim_{{x \to a}} \frac{{x^n - a^n}}{{x-a}}limx→a​x−axn−an​ where nnn is a positive integer.

Correct Answer: A

Q13.For the function f(x)=x3f(x) = x^3f(x)=x3, find the derivative using the first principle at x=1x = 1x=1.

Correct Answer: B

Q14.Consider the function f(x)=x2f(x) = x^2f(x)=x2. Using the first principles, what is the derivative of f(x)f(x)f(x) at x=3x = 3x=3?

Correct Answer: A

Q15.Consider the function f(x)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2}f(x)=x−2x2−4​. What is the limit of f(x)f(x)f(x) as xxx approaches 2?

Correct Answer: A

Q16.If f(x)=x3−3x2+2xf(x) = x^3 - 3x^2 + 2xf(x)=x3−3x2+2x, what is the derivative f′(x)f'(x)f′(x)?

Correct Answer: A

Q17.Consider a function f(x)=x2+3x+2f(x) = x^2 + 3x + 2f(x)=x2+3x+2. Calculate the derivative of the function and find the slope of the tangent to the curve at x=1x = 1x=1.

Correct Answer: A

Q18.Evaluate the limit lim⁡x→1x15−1x10−1\lim_{{x \to 1}} \frac{x^{15} - 1}{x^{10} - 1}limx→1​x10−1x15−1​ using the algebraic limit formula.

Correct Answer: C

Q19.Find the derivative of the function f(x)=sin⁡2xf(x) = \sin^2 xf(x)=sin2x using the product rule.

Correct Answer: A

Q20.What is the limit of the function f(x)=x2f(x) = x^2f(x)=x2 as xxx approaches 0?

Correct Answer: A

Q21.What is the limit of the function f(x)=x3−8x−2f(x) = \frac{{x^3 - 8}}{{x - 2}}f(x)=x−2x3−8​ as xxx approaches 2?

Correct Answer: A

Q22.Which of the following is true about the limits of f(x)=∣x∣f(x) = |x|f(x)=∣x∣ as xxx approaches 0?

Correct Answer: A

Q23.What is the derivative of the trigonometric function f(x)=anxf(x) = an xf(x)=anx?