1. Intermediate Value Theorem (IVT)
If a function f(x) is continuous on the interval [a, b], and N is any number between f(a) and f(b),
then there exists c ∈ (a, b) such that f(c) = N.
➡️ Meaning: The graph of a continuous function never “jumps” — it takes every value between f(a) and f(b).
Example: If f(1)=2 and f(3)=8, then for some c between 1 and 3, f(c)=5.
2. Horizontal Line Test
Used to check if a function is one-one (injective).
➡️ A function passes the test if every horizontal line cuts its graph at most once.
If it passes → the function is one-one and has an inverse.
3. One-One (Injective) Function
A function is one-one if no two different inputs have the same output.
That is, if f(x₁)=f(x₂) then x₁=x₂.
Example: f(x)=2x+3 → one-one.
f(x)=x² → not one-one (since f(1)=f(−1)).
4. Continuity Test
A function f(x) is continuous at x=a if:
[
\lim_{x→a^-} f(x) = \lim_{x→a^+} f(x) = f(a)
]
That means there is no break, jump, or hole at x=a.
5. Sandwich (Squeeze) Theorem
If g(x) ≤ f(x) ≤ h(x) for all x near a, and
[
\lim_{x→a} g(x) = \lim_{x→a} h(x) = L
]
then
[
\lim_{x→a} f(x) = L
]
Example:
[
\lim_{x→0} x^2 \sin(1/x) = 0
]
because −x² ≤ x²sin(1/x) ≤ x² and both outer limits are 0.
6. Chain Rule
Used to differentiate a composite function.
If y=f(g(x)), then
[
\frac{dy}{dx} = f'(g(x))·g'(x)
]
Example: If y=(3x+2)⁵ → dy/dx=5(3x+2)⁴·3=15(3x+2)⁴.
7. Power Rule
If y=xⁿ,
[
\frac{dy}{dx} = n·x^{n−1}
]
Example: d/dx(x³)=3x².
8. Product Rule
If y=u·v,
[
\frac{dy}{dx}=u',v+u,v'
]
Example: d/dx(x²sinx)=2xsinx+x²cosx.
9. Quotient Rule
If y=u/v,
[
\frac{dy}{dx} = \frac{v u' - u v'}{v²}
]
Example: d/dx((x²+1)/x)= (x·2x − (x²+1)·1)/x² = (x²−1)/x².
10. Continuity Extension
If a function has a hole (undefined at one point) but is continuous everywhere else,
we can define its value at that point as the limit to make it continuous.
Example:
[
f(x)=\frac{x²−1}{x−1}, x≠1
]
has a hole at x=1.
[
\lim_{x→1} f(x)=2
]
So define f(1)=2 → now it’s continuous.
11. Removable Discontinuity
A discontinuity that can be “fixed” by redefining the function’s value at that point.
It appears as a hole in the graph.
Example: f(x)=(x²−1)/(x−1) → discontinuous at x=1 but limit exists (2).
So it’s a removable discontinuity.
12. Continuous Function Example
Polynomial, exponential, sine, and cosine functions are continuous everywhere.
Example: f(x)=x²+3x+1 → continuous for all x.