Summary pMathematics Notes: Calculus

pMathematics Notes: Calculus




1. Introduction to Calculus



Definition: Calculus is a branch of mathematics that studies continuous change. It is divided into two
main parts: Differential Calculus and Integral Calculus.



Importance: Calculus is fundamental in various fields such as physics, engineering, economics, and
biology. It helps in modeling and solving real-life problems.




2. Limits

Understanding Limits: A limit is a value that a function approaches as the input approaches some value.



Calculating Limits:



Example:



Methods: Direct substitution, factoring, rationalizing.




Properties of Limits:



Limit of a sum:

, Limit of a product:




3. Derivatives

Definition of Derivative: The derivative of a function measures how the function value changes as its
input changes. It is defined as:




f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}



- Power Rule: 👦\frac{d}{dx} x^n = n x^{n-1}👦

- Product Rule: 👦\frac{d}{dx} [u \cdot v] = u'v + uv'👦

- Quotient Rule: 👦\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}👦



Applications of Derivatives:



Finding the slope of a curve at a point.



Solving optimization problems (maxima and minima).