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

It is slightly more than 19.355 m/s.

It is slightly less than 19.355 m/s.

It cannot be determined from the given information.

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Limits and Derivatives

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Algebra of Derivatives

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Derivative from First Principles

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Learning Objectives

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Learning Objectives

Understand the concept of derivatives as the instantaneous rate of change, using average velocity as an example.
Calculate limits using formal definitions, including left-hand and right-hand limits, and perform algebraic operations involving limits.
Derive the definition of the derivative using the limit of the difference quotient and understand its geometric interpretation.
Apply rules for derivatives including sum, difference, product, and quotient rules.
Find derivatives of polynomial functions and basic trigonometric functions using standard rules.
Use the Sandwich Theorem to evaluate limits, especially for trigonometric functions.
Evaluate limits involving trigonometric functions using geometric proofs and identities.
Apply direct substitution for polynomial limits and factorization/cancellation for rational limits, especially in 0/0 forms.
Utilize important algebraic limit formulas such as $\lim_{x \to a} \frac{x^n - a^n}{x-a} = na^{n-1}$ and related rational/surd limits.
Use derivatives to determine the slope of tangents and the rate of change in simple motion/function problems.

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Revision Notes & Summary

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Chapter Notes

Intuitive Idea of Derivatives

Instantaneous Rate of Change: The derivative represents the instantaneous rate of change of a function.
Example: Consider the distance $S = 4.9t^2$ for a body dropped from a cliff. The velocity at $t = 2$ seconds is approximated by calculating average velocities over smaller intervals.
Conclusion: As intervals decrease, the average velocity approaches the instantaneous velocity.

Limits and Their Algebra

Formal Definition: The limit of $f(x)$ as $x$ approaches $a$ is denoted as $\lim_{x \to a} f(x) = L$ .
Left-Hand and Right-Hand Limits: Defined by approaching $a$ from the left ( $x \to a^-$ ) and right ( $x \to a^+$ ).
Algebraic Operations: Limits respect addition, subtraction, multiplication, and division.

Derivative from First Principles

Definition: The derivative of $f(x)$ at $x = a$ is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ .
Geometric Interpretation: The derivative is the slope of the tangent to the curve at a point.

Algebra of Derivatives

Sum Rule: $(f + g)' = f' + g'$ .
Difference Rule: $(f - g)' = f' - g'$ .
Product Rule: $(fg)' = f'g + fg'$ .
Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ .

Derivative of Polynomial and Trigonometric Functions

Polynomials: The derivative of $x^n$ is $nx^{n-1}$ .
Trigonometric Functions: Derivatives include $(\sin x)' = \cos x$ and $(\cos x)' = -\sin x$ .

Sandwich Theorem

Theorem: If $f(x) \leq g(x) \leq h(x)$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} g(x) = L$ .

Limits of Trigonometric Functions

Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ using the Sandwich Theorem.

Limits of Polynomial and Rational Functions

Direct Substitution: For polynomials, $\lim_{x \to a} f(x) = f(a)$ .
Factorization: For rational functions, factor and cancel common terms to evaluate limits, especially for $\frac{0}{0}$ forms.

Important Algebraic Limit Formula

Formula: $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$ .

Tangent and Rate-of-Change Applications

Tangent: The derivative at a point gives the slope of the tangent to the curve at that point.
Rate of Change: Derivatives can be used to calculate rates of change in various contexts, such as velocity in physics.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

-1

\cos 0

Correct Answer: B

Solution:

Using the first principles,

f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h} = \lim_{h \to 0} \frac{\sin h}{h} \cos x + \lim_{h \to 0} \frac{\cos h - 1}{h} \sin x

. At

x = 0

f'(0) = 1

Chapter Concept:

Derivative from First Principles

Correct Answer: B

Solution:

The derivative of

f(x) = x^2

f'(x) = 2x

. At

x = 1

f'(1) = 2 \times 1 = 2

Chapter Concept:

Limits of Polynomial and Rational Functions

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Limits and Derivatives

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True or False

Correct Answer: True

Solution:

The derivative measures the rate of change, and a constant function does not change, hence its derivative is zero.

Chapter Concept :

Limits and Their Algebra

Correct Answer: False

Solution:

The limit of

\sin x

x

approaches 0 is actually 0, not 1.

Chapter Concept :

Limits of Trigonometric Functions

Correct Answer: False

Solution:

The derivative of the product of two functions is given by the product rule:

(uv)' = u'v + uv'

, not simply the product of their derivatives.

Chapter Concept :

Algebra of Derivatives

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Limits and Derivatives

Study Action Plan (Sorted by Difficulty)

Algebra of Derivatives

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Derivative from First Principles

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Derivative of Polynomial and Trigonometric Functions

Learning Objectives

Learning Objectives

Practice, Analyze & Improve 🚀

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

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Using the first principles, what is the derivative of f(x)=sin⁡xf(x) = \sin xf(x)=sinx at x=0x = 0x=0?

Correct Answer: B

Q2.What is the derivative of the function f(x)=x2f(x) = x^2f(x)=x2 at x=1x = 1x=1?

Correct Answer: B

Q3.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)?

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True or False

Q1.The derivative of a constant function is always zero.

Correct Answer: True

Q2.The limit of the function f(x)=sin⁡xf(x) = \sin xf(x)=sinx as xxx approaches 0 is 1.

Correct Answer: False

Q3.The derivative of the product of two functions is the product of their derivatives.

Correct Answer: False

High

Intuitive Idea of Derivatives

High

Limits and Their Algebra

High

Limits of Polynomial and Rational Functions

High

Limits of Trigonometric Functions

High

Sandwich Theorem

High

Tangent and Rate-of-Change Applications

High

Important Algebraic Limit Formula

High

Correct Answer: A

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Q4.What is the derivative of the function f(x)=x3f(x) = x^3f(x)=x3 at x=1x = 1x=1?

Correct Answer: B

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Q5.Given the function S=4.9t2S = 4.9t^2S=4.9t2, which represents the distance SSS in meters covered by a body dropped from a tall cliff after ttt seconds, what is the instantaneous velocity of the body at t=2t = 2t=2 seconds?

Correct Answer: B

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Q6.Using the definition of derivative, find the derivative of f(x)=x2f(x) = x^2f(x)=x2 at x=3x = 3x=3.

Correct Answer: A

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Q7.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

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Q8.If the average velocity of a body between t=1.95t = 1.95t=1.95 seconds and t=2t = 2t=2 seconds is 19.355 m/s19.355 \text{ m/s}19.355 m/s, what does this suggest about the instantaneous velocity at t=2t = 2t=2 seconds?

Correct Answer: B