📘 Class Notes: Introduction to Calculus
1. What is Calculus?
Calculus is the branch of mathematics that studies change. It has two main branches:
● Differential Calculus: Concerned with the rate of change (e.g., slopes, velocity).
● Integral Calculus: Concerned with accumulation (e.g., area under curves, total
distance).
2. Key Concepts
a. Limits
● A limit describes the value a function approaches as the input approaches a certain
value.
● Notation:
limx→af(x)\lim_{x \to a} f(x)x→alimf(x)
● Essential for defining derivatives and integrals.
b. Derivatives
● Measures how a function changes as its input changes.
● Represents the slope of the tangent line at any point on a curve.
● Notation:
f′(x)=ddxf(x)f'(x) = \frac{d}{dx}f(x)f′(x)=dxdf(x)
● Rules:
○ Power Rule:
ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}dxdxn=nxn−1
● ]
○ Sum Rule:
(f+g)′=f′+g′(f + g)' = f' + g'(f+g)′=f′+g′
● ]
○ Product Rule:
(fg)′=f′g+fg′(fg)' = f'g + fg'(fg)′=f′g+fg′
● ]
○ Quotient Rule:
(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}(gf)′=g2f′g−fg′
● ]
○ Chain Rule:
ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot
g'(x)dxdf(g(x))=f′(g(x))⋅g′(x)
● ]
c. Integrals
● Opposite of derivatives; they add up small pieces to find total value.
● Represent area under a curve.
● Indefinite Integral (antiderivative):
∫f(x)dx\int f(x)dx∫f(x)dx
● Definite Integral:
∫abf(x)dx\int_a^b f(x)dx∫abf(x)dx
(gives area under the curve from x = a to x = b)
● Fundamental Theorem of Calculus:
○ Connects derivatives and integrals:
∫abf′(x)dx=f(b)−f(a)\int_a^b f'(x)dx = f(b) - f(a)∫abf′(x)dx=f(b)−f(a)
3. Applications of Calculus
● Physics (motion, forces)
● Economics (cost/revenue analysis)
● Biology (population growth models)
● Engineering (optimization, fluid flow)
● Computer Graphics (curves and motion)
4. Tips for Success
● Practice limit problems to build intuition.
● Master derivative rules—they’re used everywhere.
● Understand the graphical meaning of integrals and derivatives.
● Use visuals and graphs to deepen understanding.
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