Calculus: Limits and Derivatives

Study Notes

Limits

A limit describes the value a function approaches as the input approaches a certain value.
Formally, lim_x→a f(x) = L if for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
One-sided limits (left and right limits) are crucial for understanding functions with discontinuities.
Calculating limits involves algebraic manipulation, L'Hôpital's rule (for indeterminate forms), and graphical analysis.
Common limit problems include finding limits of rational functions, trigonometric functions, and exponential functions.

Derivatives

The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
Geometrically, the derivative is the slope of the tangent line to the function's graph at the given point.
The derivative is often denoted by f'(x), df/dx, or dy/dx.
Basic differentiation rules allow finding the derivatives of many functions, including power functions, polynomial functions, exponential functions, and trigonometric functions.
The chain rule, product rule, and quotient rule are essential for differentiating composite functions.
Higher-order derivatives represent the rate of change of the rate of change.

Continuous Functions

A function is continuous at a point 'a' if the limit of the function as x approaches 'a' equals the value of the function at 'a'.
Mathematically, lim_x→a f(x) = f(a).
This definition includes three key aspects: the limit must exist, the function must be defined at the point, and the limit value must equal the function's value.
Discontinuities (jump, removable, infinite) are points where a function is not continuous. Identifying the type of discontinuity is important for analysis.
Key properties of continuous functions include the intermediate value theorem and the extreme value theorem.

Applications of Derivatives

Optimization problems involve maximizing or minimizing a function, typically using the first and second derivative test.
Related rates problems involve finding the rate of change of one quantity given the rate of change of another related quantity. These often employ implicit differentiation.
Curve sketching leverages the derivative to determine important features like critical points, increasing/decreasing intervals, concavity, and points of inflection.
Applications in physics and engineering are numerous and involve concepts like velocity, acceleration, and motion.

Integrals

Integrals represent the area under a curve.
Definite integrals calculate the exact area between the curve and the x-axis over a specified interval.
Indefinite integrals find the family of antiderivatives, represented by a general antiderivative with the constant of integration.
The fundamental theorem of calculus establishes a crucial connection between differentiation and integration.
Techniques for evaluating integrals (like substitution and integration by parts) are essential for solving diverse problems.
Applications of integrals encompass determining areas, volumes, lengths of curves, work, and center of mass calculations.

Description

This quiz explores the fundamental concepts of limits and derivatives in calculus. You will learn about one-sided limits, the formal definition of limits, and the basic rules of differentiation. Test your understanding of these key topics essential for further study in calculus.