Study Notes: Calculus and Its Applications
Reference Text: Bittinger & Ellenbogen | Subject: Mathematics / Applied Calculus
1. Functions, Graphs, and Limits
In applied calculus, a function models the relationship between independent and dependent variables.
Understanding rate of change begins with analyzing limit behaviors as variables approach boundary conditions.
• Limit Definition: Lim(x→a) f(x) = L means as x approaches a, f(x) approaches L.
• Continuity: A function is continuous at a point if the limit exists and equals the function evaluation at that
point.
2. The Derivative & Differentiation Rules
The derivative measures the instantaneous rate of change or slope of a tangent line at a given point on a curve.
Rule Type
Formula
Application
Power Rule
d/dx [x^n] = n * x^(n-1)
Polynomial differentiation
Product Rule
d/dx [u*v] = u'v + uv'
Multiplying two functions
Quotient Rule
d/dx [u/v] = (u'v - uv') / v^2
Dividing two functions
Chain Rule
d/dx [f(g(x))] = f'(g(x)) * g'(x)
Composite functions
3. Applications of the Derivative
Optimization is one of the primary practical applications of calculus in business, economics, and biology. To find
maximum or minimum values, determine the critical points where f'(x) = 0 or is undefined, then apply the
second derivative test.
• Marginal Cost/Revenue: Represented as derivative functions C'(x) and R'(x).
• Relative Extrema: If f''(c) > 0, the point is a local minimum. If f''(c) < 0, it is a local maximum.
4. Integration & Accumulation
Integration is the inverse process of differentiation. Definite integrals calculate accumulated totals such as total
cost, consumer surplus, or area under a curve.
• Fundamental Theorem of Calculus: ∫[a to b] f(x) dx = F(b) - F(a), where F'(x) = f(x).