Tangent Line
Definition
A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.
Analogy
Imagine you are walking along a curvy path in a park, and suddenly you want to know your exact direction at one particular spot on the path. You can extend your arm forward until your finger just touches the path without going inside it - that's like drawing a tangent line!
Related terms
Secant Line: A secant line intersects a curve at two points, helping us estimate average rates of change between those points.
Derivative: The derivative gives us the slope (rate of change) of a function at any given point on its graph. It helps us find tangent lines and analyze functions' behavior.
Concavity: Concavity describes whether a curve bends upward (concave up) or downward (concave down). We determine concavity by analyzing the second derivative of a function.
"Tangent Line" appears in:
Study guides (7)
AP Calculus AB/BC - 2.2 Defining the Derivative of a Function and Using Derivative Notation
AP Calculus AB/BC - 2.3 Estimating Derivatives of a Function at a Point
AP Calculus AB/BC - 4.6 Approximating Values of a Function Using Local Linearity and Linearization
AP Calculus AB/BC - 5.1 Using the Mean Value Theorem
AP Calculus AB/BC - 7.5 Approximating Solutions Using Euler’s Method
AP Calculus AB/BC - 9.1 Defining and Differentiating Parametric Equations
AP Calculus AB/BC - 9.4 Defining and Differentiating Vector-Valued Functions
Additional resources (1)
Practice Questions (5)
What is a tangent line?
What is the purpose of using a tangent line to approximate a point on a function?
What is the relationship between the tangent line and the function at the point of tangency?
Given the polar equation r=2θ, determine the polar coordinates of the point where the tangent line is horizontal.
How do we find the slope of the tangent line for polar functions?
Resources
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