Calculus: Limits and Derivatives

Study Notes

Limits

A limit describes the value that a function approaches as the input (often represented by x) approaches a particular value.
Limits are crucial in calculus for understanding continuity, derivatives, and integrals.
Limits are often used to find the behavior of a function at a specific point where the function might be undefined.

Derivatives

The derivative of a function measures the instantaneous rate of change of the function.
Mathematically, it's defined as the limit of the difference quotient as the change in the input approaches zero.
The derivative represents the slope of the tangent line to the function at a given point.
The derivative can be interpreted geometrically as the instantaneous rate of change, or physically as velocity.

Differentiation Rules

Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
- d/dx (cf(x)) = c * d/dx (f(x)).
Sum/Difference Rule: The derivative of the sum or difference of functions is the sum or difference of their derivatives.
- d/dx (f(x) ± g(x)) = d/dx (f(x)) ± d/dx (g(x)).
Power Rule: The derivative of x raised to a power is the power multiplied by x raised to the power minus one.
- d/dx (xⁿ) = nx^n-1.
Product Rule: The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.
- d/dx(f(x)g(x)) = f(x)g'(x) + g(x)f'(x).
Quotient Rule: The derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
- d/dx(f(x)/g(x)) = [g(x)f'(x) - f(x)g'(x)] / [ g(x)²].
Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
- *d/dx(f(g(x))) = f'(g(x))g'(x).

Integrals

An integral is a mathematical concept that represents the area under a curve.
Two main types: indefinite and definite integrals.

Indefinite Integrals

Indefinite integrals find the antiderivative of a function.
The result includes an arbitrary constant of integration (C).
Antiderivatives are the reverse process of differentiation.

Definite Integrals

Definite integrals calculate the exact area under a curve between two specific points.
The result is a numerical value.
Often written using a notation (lower and upper limits)
- Often represented as ∫_a^b f(x) dx.

Integration Rules

Power Rule for Integrals: The integral of xⁿ is (xⁿ⁺¹)/(n+1) + C
Constant Multiple Rule for Integrals: The integral of cf(x) is c∫f(x)dx
Sum/Difference Rule for Integrals: The integral of f(x) ± g(x) is ∫f(x)dx ± ∫g(x)dx.
Basic Integration Formulas for Trigonometric Functions, Exponential Functions, Logarithmic Functions

Applications

Calculus is used in numerous fields, including physics (motion, kinematics), engineering (design, optimization), economics (optimization, growth), and many more.
These include modelling motion, evaluating work done by forces, finding areas and volumes, and solving optimization problems.

Description

This quiz covers key concepts in calculus, focusing on limits and derivatives. You'll explore how limits define function behavior, the meaning of derivatives, and differentiation rules. Test your understanding of these foundational topics in calculus.