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 calculation of limits.
- Determine the continuity of functions.
- Define and calculate derivatives.
- Analyze the relationship between continuity and differentiability.
Common exam trap
Confusing continuity with differentiability.
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 it is defined at that point, the limit exists at that point, and the limit equals the function's value at that point.
- Term
Differentiability
- Meaning
A function is differentiable at a point if its derivative exists at that point, implying the function is smooth and has a unique tangent line there.
- Term
Derivative
- Meaning
The instantaneous rate of change of a function with respect to its variable, or the slope of the tangent line to the function's graph.
Learning objectives
Understand the definition and calculation of limits.
Determine the continuity of functions.
Define and calculate derivatives.
Analyze the relationship between continuity and differentiability.
Interpret the derivative as a rate of change.
Formulae
- Name
Limit Definition
- Note
As x approaches c, f(x) approaches L.
- Expression
limx→c f(x) = L
- Name
Continuity Condition
- Note
For a function to be continuous at x=c.
- Expression
limx→c f(x) = f(c)
- Name
Derivative Definition (First Principles)
- Note
The instantaneous rate of change of f(x).
- Expression
f'(x) = limh→0 \frac{f(x+h) - f(x)}{h}
- Name
Left-Hand Limit (LHL)
- Note
Limit as x approaches c from values less than c.
- Expression
limx→c^- f(x)
- Name
Right-Hand Limit (RHL)
- Note
Limit as x approaches c from values greater than c.
- Expression
limx→c^+ f(x)
Prerequisites
Algebraic manipulation
Understanding of functions and their graphs
Basic trigonometry
Common mistakes
Confusing continuity with differentiability.
Assuming a function is differentiable just because it is continuous.
Errors in calculating limits, especially with indeterminate forms (0/0, ∞/∞).
Incorrectly applying limit properties.
Keywords
Limit
Continuity
Differentiability
Derivative
Rate of Change
Tangent Line
LHL
RHL
Practice preview
If f(x) = sin(x^2), what is the first derivative, f'(x)?…
easy
Find the minimum value of the function f(x) = x^2 - 4x + 5.…
medium
Evaluate the indefinite integral: integral (x^3 + 2x) dx.…
easy