Calculus: Limits and Derivatives

Limits and Continuity

• A limit describes the value a function approaches as the input approaches a certain value.
• It's denoted as lim x→c f(x) = L, meaning as x gets closer and closer to c, f(x) gets closer and closer to L.
• Limits can be used to find the slope of a tangent line to a curve.
• Continuity at a point means that the function is defined at that point, the limit of the function exists at that point, and the limit equals the function's value at that point.
• Discontinuities can be classified as removable, jump, or infinite.

Derivatives

• The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
• It's the slope of the tangent line to the function at that point.
• The derivative is calculated using the limit definition, which involves finding the limit of the difference quotient as the change in x approaches zero.
• Common differentiation rules include the power rule, the product rule, the quotient rule, and the chain rule.
• Derivatives have applications in optimization problems, finding the maximum or minimum values of a function, and analyzing the behavior of functions.

Integrals

• The definite integral of a function represents the area under the curve of the function between two points.
• It can be interpreted as the accumulated change of a quantity over an interval.
• The Fundamental Theorem of Calculus connects differentiation and integration.
• Integration techniques include substitution, integration by parts, and partial fraction decomposition.
• Applications of integrals include finding the area between curves, volumes of solids of revolution, and work done by a variable force.

Applications of Calculus

• Optimization problems involve finding the maximum or minimum values of a function that is subject to certain constraints.
• Optimization problems can involve finding the maximum profit, minimum cost, or maximum area.
• Rate problems involve finding the rate of change of a quantity with respect to another variable.
• Modeling real-world phenomena can be done using calculus, providing insights into trends and behaviors.