● Limits : The concept of a limit represents the behavior of a function as the input
values approach a specific point.
○ Basic Limit Properties:
■ Linearity: lim(x→a) [f(x) ± g(x)] = lim(x→a) f(x) ± lim(x→a) g(x)
■ Homogeneity: lim(x→a) [c * f(x)] = c * lim(x→a) f(x)
○ Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) and lim(x→a) f(x) = lim(x→a) h(x) = L,
then lim(x→a) g(x) = L
● Continuity : A function f(x) is continuous at x = a if lim(x→a) f(x) = f(a)
○ Types of Discontinuities:
■ Removable Discontinuity: lim(x→a) f(x) exists, but f(a) is not defined
or is not equal to the limit
■ Infinite Discontinuity: lim(x→a) f(x) = ∞ or -∞
■ Essential Discontinuity: lim(x→a) f(x) does not exist
Unit 2: Differentiation
● Derivatives : The derivative of a function f(x) represents the rate of change of the
function with respect to x.
○ Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
○ Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
○ Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
● Implicit Differentiation : A technique used to find the derivative of an implicitly
defined function.
○ Example: Find dy/dx for the equation x^2 + y^2 = 4
● Logarithmic Differentiation : A technique used to find the derivative of a function by
first taking the logarithm of both sides.
○ Example: Find dy/dx for the equation y = x^x
Unit 3: Applications of Derivatives
● Optimization : The process of finding the maximum or minimum value of a function.
○ Example: Find the maximum area of a rectangle with a fixed perimeter.
● Motion Along a Line : The study of the motion of an object along a straight line.
○ Position, Velocity, and Acceleration:
■ Position: s(t)
■ Velocity: v(t) = s'(t)
■ Acceleration: a(t) = v'(t) = s''(t)
● Related Rates : The study of the rates of change of two or more related quantities.
○ Example: A spherical balloon is being inflated. Find the rate of change of the
volume with respect to time.
