Calculus
What is Calculus?
The value that a function or sequence takes as the input or index approaches some value.
Key formula / rule: Limit Definition
Key points
- Understand the definition and evaluation of limits.
- Determine the continuity of functions at a point and over an interval.
- Define and calculate the derivative of a function.
- Relate continuity and differentiability.
Common exam trap
Confusing the limit of a function with its value at a point.
Definitions
- Term
Limit
- Meaning
The value that a function or sequence takes as the input or index approaches some value.
- Term
Continuity
- Meaning
A function is continuous at a point if its graph is unbroken at that point, meaning the limit exists, the function value exists, and they are equal.
- Term
Differentiability
- Meaning
A function is differentiable at a point if its derivative exists at that point, implying the existence of a unique tangent line.
- Term
Indeterminate Form
- Meaning
An expression, such as 0/0 or ∞/∞, that arises in the evaluation of limits and requires further analysis (e.g., L'Hôpital's Rule) to determine the limit's value.
Learning objectives
Understand the definition and evaluation of limits.
Determine the continuity of functions at a point and over an interval.
Define and calculate the derivative of a function.
Relate continuity and differentiability.
Apply limit, continuity, and differentiability concepts to solve problems.
Formulae
- Name
Limit Definition
- Note
The value f(x) approaches as x approaches c.
- Expression
\limx \to c f(x) = L
- Name
Continuity Condition
- Note
For a function to be continuous at x=c.
- Expression
\limx \to c f(x) = f(c)
- Name
Derivative Definition (First Principles)
- Note
The instantaneous rate of change of f at c.
- Expression
f'(c) = \limh \to 0 \frac{f(c+h) - f(c)}{h}
- Name
L'Hôpital's Rule
- Note
Requires f and g to be differentiable and g'(x) ≠ 0 near c.
- Expression
\text{If } \limx \to c \frac{f(x)}{g(x)} \text{ is } \frac{0}{0} \text{ or } \frac{\∞}{\∞}, \text{ then } \limx \to c \frac{f(x)}{g(x)} = \limx \to c \frac{f'(x)}{g'(x)}
Prerequisites
Basic algebra (functions, equations, inequalities).
Understanding of graphs of functions.
Basic trigonometry.
Common mistakes
Confusing the limit of a function with its value at a point.
Assuming differentiability from continuity without checking the derivative's existence.
Errors in algebraic manipulation when evaluating limits, especially with indeterminate forms.
Incorrectly applying limit laws.
Keywords
Limit
Continuity
Differentiability
Derivative
Rate of Change
L'Hôpital's Rule
Indeterminate Form
Practice preview
If y = sin(x^2), find dy/dx.…
medium
Evaluate the indefinite integral: integral (2x * e^(x^2)) dx.…
medium
If f(x) = 3x^2 + 2x - 5, find the first derivative, f'(x).…
easy