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🎓 Introduction to Calculus: The Language of
Change
📚 Table of Contents
1. What is Calculus?
2. Limits: The Foundation
3. Derivatives: Instantaneous Change
4. Integrals: Accumulated Change
5. Applications in Real Life
6. Key Formulas & Theorems
1. What is Calculus?
Calculus is a branch of mathematics that deals with continuous change. It has two main
parts:
🔹 Differential Calculus
Focuses on rates of change (derivatives)
Helps answer: "How fast is something changing?"
🔹 Integral Calculus
Focuses on accumulation of quantities (integrals)
Helps answer: "How much has accumulated?"
2. Limits: The Foundation
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Definition:
A limit is the value a function approaches as the input gets closer to some value.
lim f (x) = L
x→c
📊 Example:
lim (x2 + 3x − 1) = 22 + 3(2) − 1 = 9
x→2
Real-world analogy:
A car's speedometer shows instantaneous speed (limit of average speed as time
interval → 0)
3. Derivatives: Instantaneous Change
Definition:
The derivative measures how a function changes as its input changes.
f (x + h) − f (x)
h→0
h
f ′ (x) = lim
📊 Example: Power Rule
If f (x) = xn , then f ′ (x) = nxn−1
f (x) = x3 ⇒ f ′ (x) = 3x2
Common Derivatives:
d
(sin x) = cos x
dx
d
x
x
dx (e ) = e
d
(ln x) = x1
dx
4. Integrals: Accumulated Change
Definition:
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The integral finds the total accumulation of a quantity.
∫ f (x)dx = F (x) + C
Where F ′ (x) = f (x)
📊 Example: Power Rule for Integration
∫ xn dx =
xn+1
+C
n+1
∫ x2 dx =
(neq − 1)
x3
+C
3
Definite Integral (Area under curve):
b
∫ f (x)dx = F (b) − F (a)
a
5. Real-Life Applications
🚗 Motion & Physics:
Velocity = derivative of position
Acceleration = derivative of velocity
Distance traveled = integral of velocity
💰 Economics:
Marginal cost = derivative of total cost
Total revenue = integral of marginal revenue
🏗️ Engineering:
Optimization of structures (minimum material, maximum strength)
Rate of change in electrical currents
🧬 Biology:
Population growth rates
Drug concentration in bloodstream over time
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6. Step-by-Step Examples
Example 1: Finding Maximum Area
Problem: A farmer has 100m of fencing. What rectangle dimensions give maximum area?
Solution:
1. Let length = x, width = 50 − x (perimeter = 2x + 2(50 − x) = 100)
2. Area function: A(x) = x(50 − x) = 50x − x2
3. Derivative: A′ (x) = 50 − 2x
4. Set derivative to 0: 50 − 2x = 0 ⇒ x = 25
5. Maximum area when x = 25m, width = 25m (a square!)
Example 2: Calculating Total Distance
Problem: Car's velocity: v(t) = 3t2 m/s. Find distance traveled from t = 0 to t = 4
seconds.
Solution:
1. Distance = ∫04 3t2 dt
2. Antiderivative: t3
3. Evaluate: [t3 ]40 = 43 − 03 = 64 meters
7. Fundamental Theorem of Calculus
Part 1:
x
d
∫ f (t)dt = f (x)
dx a
Part 2:
b
∫ f (x)dx = F (b) − F (a)
a
Where F ′ (x) = f (x)
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8. Key Formulas Summary
Derivatives:
(cf )′ = cf ′
(f + g)′ = f ′ + g ′
(f g)′ = f ′ g + f g ′ (Product Rule)
′
g
(Quotient Rule)
( fg ) = f g−f
g2
′
′
(f (g(x)))′ = f ′ (g(x)) ⋅ g ′ (x) (Chain Rule)
Integrals:
∫ kdx = kx + C
n+1
∫ xn dx = xn+1 + C
∫ ex dx = ex + C
∫ x1 dx = ln ∣x∣ + C
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