Tangent Line

Definition

A tangent line is a straight line that touches a curve at a single point, without crossing or intersecting the curve at any other point. It represents the local linear approximation of the curve at the point of tangency, and its slope is equal to the derivative of the function at that point.

5 Must Know Facts For Your Next Test

1. The slope of the tangent line at a point on a curve is equal to the derivative of the function at that point.
2. The tangent line provides a local linear approximation of the curve, which can be used to estimate the value of the function near the point of tangency.
3. The equation of the tangent line can be written in the form $y = mx + b$, where $m$ is the slope of the tangent line and $b$ is the $y$-intercept.
4. Tangent lines are important in the study of rates of change and the behavior of graphs, as they can be used to analyze the local behavior of a function near a particular point.
5. In parametric equations, the tangent line can be used to determine the direction of the curve at a particular point, as the slope of the tangent line is given by the ratio of the derivatives of the parametric equations.

Review Questions

• Explain how the concept of a tangent line is related to the rate of change and behavior of graphs.
  • The tangent line to a curve at a point represents the local linear approximation of the curve at that point. The slope of the tangent line is equal to the derivative of the function at that point, which is a measure of the rate of change of the function. By analyzing the slope and behavior of the tangent line, you can gain insights into the local behavior of the graph, such as the direction of the curve, the rate of change, and the concavity of the function.
• Describe the relationship between the tangent line and parametric equations.
  • In the context of parametric equations, the tangent line can be used to determine the direction of the curve at a particular point. The slope of the tangent line is given by the ratio of the derivatives of the parametric equations, which represent the rates of change of the $x$ and $y$ coordinates with respect to the parameter. By analyzing the slope of the tangent line, you can understand the local behavior of the parametric curve and the direction in which it is moving.
• Analyze how the concept of a tangent line can be used to estimate the value of a function near a particular point.
  • The tangent line provides a local linear approximation of the curve at the point of tangency. This means that near the point of tangency, the curve can be well-approximated by the tangent line. By using the equation of the tangent line, $y = mx + b$, where $m$ is the slope (the derivative) and $b$ is the $y$-intercept, you can estimate the value of the function at points near the point of tangency. This can be particularly useful when the function is difficult to evaluate directly, or when you need a quick estimate of the function's value in a particular region.

Related terms

Derivative: The derivative of a function is a measure of the rate of change of the function at a particular point, and it represents the slope of the tangent line to the function at that point.

Slope: The slope of a line is a measure of the steepness of the line, and it represents the rate of change of the dependent variable with respect to the independent variable.

Point of Tangency: The point of tangency is the single point where the tangent line touches the curve, and it is the point at which the derivative of the function is evaluated to determine the slope of the tangent line.