Calculating Limits and Derivatives

This section delves deeper into limit calculations and introduces derivative problems. Students are asked to compute limits using techniques such as rationalization and to derive equations involving exponential and trigonometric functions.

Definition: Derivative - The rate of change of a function with respect to a variable, typically denoted as f'$x$.

Highlight: The page includes a problem on finding the derivative of a composite function: f$x$ = e²ˣ · tan(3x), which requires the application of the chain rule and product rule.

Tangent Line Equation

This page focuses on a specific application of derivatives: finding the equation of a tangent line. The problem presented involves finding the tangent line to the curve y = 3sin$x$ + cos$x$ at the point (0,3).

Vocabulary: Recta tangente (Tangent line) - A line that touches a curve at a single point and has the same slope as the curve at that point.

Example: The equation y = 3sin$x$ + cos$x$ is given, and students must find its tangent line at x = 0.

Detailed Solutions

This page provides comprehensive solutions to the limit and derivative problems presented earlier. It demonstrates step-by-step approaches to solving complex limits, including the use of L'Hôpital's rule for indeterminate forms.

Highlight: The solution for lim arctan$(2x² - 6x) / (x - 3)$ as x approaches 3 is particularly noteworthy, showing the application of limit properties to simplify a complex expression.

Example: The derivative of g$x$ = √$√x + 1$ is calculated using the chain rule, demonstrating how to handle nested square roots.

Tangent Line Calculation

The final page concludes the tangent line problem, showing the complete process of finding the equation. It includes verifying the point's existence on the curve, calculating the derivative, finding the slope at the given point, and formulating the final equation.

Definition: Pendiente de la recta tangente (Slope of the tangent line) - The derivative of the function at the point of tangency, representing the rate of change at that specific point.

Example: The final equation of the tangent line is derived as y = -3x + 3, demonstrating how to use the point-slope form of a line equation.

Introduction to Limits and Derivatives

This page introduces a series of exercises focused on límites y derivadas. Students are presented with multiple limit problems to solve, ranging from simple evaluations to more complex scenarios involving infinity and undefined results. The exercises are designed to test understanding of limit behavior in various function types.

Vocabulary: Límites (Limits) - The value that a function approaches as the input (usually x) gets closer to a specific value.

Example: lím F$x$ = 0 as x approaches 1 from the left side, demonstrating a one-sided limit. x→1-