Calculus

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Practice 8 questions Back to syllabus~12 min

What is Calculus?

The value that a function or sequence 'approaches' as the input or index approaches some value.

Key formula / rule: Limit Definition

Key points

Understand the concept of a limit and how to evaluate it.
Define and check for continuity of a function at a point and over an interval.
Define and check for differentiability of a function at a point.
Relate continuity and differentiability.

Common exam trap

Confusing limits with function values.

Definitions

Term: Limit
Meaning: The value that a function or sequence 'approaches' as the input or index approaches some value.

Term: Continuity
Meaning: A function is continuous at a point if its graph can be drawn through the point without lifting the pen; formally, if the limit exists, the function is defined at the point, and the limit equals the function's value.

Term: Differentiability
Meaning: A function is differentiable at a point if its derivative exists at that point, meaning the graph has a unique, non-vertical tangent line.

Term: Derivative
Meaning: The instantaneous rate of change of a function with respect to a variable; geometrically, the slope of the tangent line to the function's graph.

Learning objectives

Understand the concept of a limit and how to evaluate it.
Define and check for continuity of a function at a point and over an interval.
Define and check for differentiability of a function at a point.
Relate continuity and differentiability.
Interpret the derivative as a rate of change and slope.

Formulae

Name: Limit Definition
Note: f(x) approaches L as x approaches c.
Expression: lim_x→c f(x) = L

Name: Continuity Condition
Note: Requires f(c) to be defined and the limit to exist and be equal to f(c).
Expression: lim_x→c f(x) = f(c)

Name: Derivative Definition (Limit Form)
Note: Represents the instantaneous rate of change at c.
Expression: f'(c) = lim_h→0 \frac{f(c+h) - f(c)}{h}

Name: L'Hôpital's Rule
Note: Used for indeterminate forms.
Expression: \lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)} \quad \text{(if form is } \frac{0}{0} \text{ or } \frac{\∞}{\∞})

Prerequisites

Basic algebra (functions, equations, inequalities).
Understanding of graphs of functions.
Familiarity with basic trigonometric and exponential functions.

Common mistakes

Confusing limits with function values.
Assuming continuity implies differentiability.
Incorrectly applying limit rules, especially with indeterminate forms (0/0, ∞/∞).
Errors in algebraic manipulation when evaluating limits.

Keywords

Limit
Continuity
Differentiability
Derivative
Rate of Change
Tangent Line
L'Hôpital's Rule
Indeterminate Forms

Practice preview

Evaluate the indefinite integral: integral (2x + 3) dx…
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If z = x^2y + 3xy^4, find the partial derivative of z with respect to x, denoted as partial z / partial x.…
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For the function f(x) = x^2 on the interval [0, 2], find a value 'c' that satisfies the Mean Value Theorem.…
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Practice 8 questions