Tangent

Definition

A tangent to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point.

5 Must Know Facts For Your Next Test

1. The equation of the tangent line to a curve $y = f(x)$ at a point $(a, f(a))$ is $y - f(a) = f'(a)(x - a)$.
2. The slope of the tangent line represents the instantaneous rate of change of the function at that point.
3. To find the tangent line, you need to compute the derivative of the function and evaluate it at the given point.
4. In calculus, tangents are used to approximate functions near specific points using linearization.
5. If a function is differentiable at a point, then it has a unique tangent line there.

Review Questions

• What is the geometric significance of a tangent line in relation to a curve?
• How do you determine the equation of a tangent line for a function $y = f(x)$ at $x = a$?
• Why is finding the derivative important when calculating the slope of the tangent line?

Related terms

Derivative: The derivative of a function represents its rate of change and is used to calculate slopes of tangent lines.

Linear Approximation: A method for approximating values of functions near known points using their tangents.

$\lim$ (Limit): $\lim_{x \to c}f(x)$ describes what value $f(x)$ approaches as $x$ gets arbitrarily close to $c$.