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Calculus Preread 0

The document outlines fundamental concepts in calculus: limits, continuity, differentiability, and differentiation. It explains how limits indicate the value a function approaches, continuity ensures no breaks in a function, differentiability relates to the smoothness of a function, and differentiation is the process of finding the rate of change. Understanding these concepts is essential for modeling real-world behaviors in various fields such as science and engineering.

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16 views3 pages

Calculus Preread 0

The document outlines fundamental concepts in calculus: limits, continuity, differentiability, and differentiation. It explains how limits indicate the value a function approaches, continuity ensures no breaks in a function, differentiability relates to the smoothness of a function, and differentiation is the process of finding the rate of change. Understanding these concepts is essential for modeling real-world behaviors in various fields such as science and engineering.

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sg
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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📘 Pre-Read: Limit, Continuity, Differentiability &

Differentiation

🧠 Why This Matters


Calculus starts with the idea of change—how fast something changes, how smoothly
it changes, and where it might break or jump. The journey begins with limits, the
bridge to understanding continuity, differentiability, and ultimately, differentiation (the
core of derivatives).

These concepts are not just math tools—they are how we model motion, growth,
optimization, and real-world behavior in science, engineering, and economics.

🔹 1. Limit – The Idea of “Getting Close”


The limit tells us what value a function approaches as the input gets closer and
closer to a certain point.

Example: As x approaches 2, what does f(x) = x² approach? → f(2) = 4, and as x


gets closer to 2, f(x) gets closer to 4.

📌 Key Ideas:
​ Left-hand limit (approaching from the left)
​ Right-hand limit (approaching from the right)
​ A limit exists only if both sides agree

🔹 2. Continuity – No Breaks, No Jumps


A function is continuous at a point if:

​ The function is defined at that point.


​ The limit exists at that point.
​ The value of the function equals the limit.

💡 Think of drawing the graph without lifting your pen.


Discontinuity Types:

​ Jump Discontinuity
​ Infinite Discontinuity
​ Removable Discontinuity

🔹 3. Differentiability – Smoothness Counts


A function is differentiable at a point if:

​ It is continuous there, and


​ It has a definable, finite slope at that point.

❗ Note: All differentiable functions are continuous, but not all continuous functions
are differentiable.

Not differentiable when:

​ There’s a sharp corner (e.g., |x| at x = 0)


​ The graph has a cusp or vertical tangent
​ There’s a discontinuity

🔹 4. Differentiation – Finding the Rate of Change


Differentiation is the process of finding the derivative of a function, which tells us the
rate of change or slope at any point.

Basic Rules:

​ Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹


​ Sum Rule: d/dx(f + g) = f' + g'
​ Product & Quotient Rule
​ Chain Rule (for composite functions)

Applications:

​ Velocity from position


​ Slope of tangent line
​ Optimization problems
​ Curve sketching

🔄 Summary Table:
Concept Meaning Graph View
Limit Value approached by Value from both sides match
function

Continuity No breaks or jumps Smooth curve

Differentiability Smoothness + defined No corners or vertical


slope tangents

Differentiation Process to find derivative Gives rate of change / slope

✅ What You Should Know Before Class


​ Understand the basic idea of approaching a number (limit).
​ Be able to identify where a graph is continuous or not.
​ Recognize sharp corners or jumps as signs of non-differentiability.
​ Know at least one rule for finding a derivative.
​

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