Differential Calculus

• Calculus is a branch of mathematics that deals with the study of change.
• It is fundamentally concerned with continuous functions and their rates of change.
• Two major branches are differential calculus (concerned with rates of change) and integral calculus (concerned with accumulation of quantities).

Differential Calculus

• Derivatives: The derivative of a function at a point represents the instantaneous rate of change of the function at that point. Geometrically, it corresponds to the slope of the tangent line to the graph at that point.
• Rules for Differentiation: There are various rules for calculating derivatives, including the power rule, product rule, quotient rule, chain rule, and the rules for differentiating trigonometric, exponential, and logarithmic functions. Each rule allows for efficient derivation of complex functions.
• Applications of Derivatives: These include finding maximum and minimum values of functions, determining concavity and inflection points, and solving optimization problems.

Integral Calculus

• Integrals: The definite integral of a function over an interval represents the area under the curve of the function. The function is the integrand, the interval is the region of integration and the result is the area.
• Techniques of Integration: Various techniques exist for evaluating integrals, including substitution (u-substitution), integration by parts, and trigonometric integrals. Specific techniques are required for different types of integrals to facilitate calculations.
• Applications of Integrals: Applying definite integrals allows for finding areas, volumes of solids of revolution, arc length, work done by a force, and other useful applications.

Fundamental Theorem of Calculus

• This fundamental theorem establishes a connection between differentiation and integration.
• Part 1 states that the derivative of a definite integral, with respect to the upper limit, is the integrand evaluated at that upper limit. This relates to the concept of instantaneous rate of change.
• Part 2 demonstrates how to evaluate definite integrals by using antiderivatives in a concise way. It is a powerful tool that allows for the evaluation of complicated integrals without iterative processes.

Limits

• Limits are crucial to the understanding of both differential and integral calculus.
• They describe the behavior of a function as the input approaches a specific value, or approaches positive or negative infinity.
• Calculating limits rigorously often involves algebraic manipulations or various limit theorems.

Continuity

• A function is continuous at a point if its value at that point equals the limit of the function at that point as the input approaches that value.
• Continuity is essential for many theorems and procedures in calculus. Failure in continuity may lead to various problems in the application.
• Discontinuities come in various forms, such as jump discontinuities, removable discontinuities, and infinite discontinuities.

Common Functions and Their Derivatives

• Polynomials: The derivative of a polynomial function can be found using the power rule. The power rule works consistently for various orders of polynomials.
• Trigonometric functions: The derivatives of sine, cosine, tangent, cotangent, secant, and cosecant functions can be derived using known identities. These are fundamental trigonometric relations.
• Exponential and Logarithmic Functions: The derivatives of exponential and logarithmic functions are essential tools for modelling phenomena involving growth and decay. Common applications range from population dynamics to radioactive decay.
• Hyperbolic Functions: The hyperbolic functions and their derivatives often emerge in more complex problems including special functions.

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