High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize AP Calculus (BC) Textbook > Applications of Differentiation: Contextual

Tangent Lines

Bonus: Logarithmic Differentiation (Important technique in calculus but not usually covered in AP Calculus AB or BC)Previous Section
Normal LinesNext Section
Popular Courses
Find My Course
0:00 / 0:00

Tangent Lines

The derivative of a functions represents the instantaneous rate of change at a point. This rate of change is the same as the slope of the tangent line at a point!

Tangent Lines

A tangent line to f(x)f(x) at x=ax=a is a line that only just touches f(x)f(x) and is locally parallel to f(x)f(x) at x=a.x=a.
Locally parallel means that at x=a,f(x)x=a,f(x) and its tangent line have the same slope.




Equation of Tangent Lines

The equation of the tangent line of f(x)f(x)at x=ax=a is
y=f(a)+f′(a)(x−a)\boxed{y=f(a)+f'(a)(x-a)}


0:00 / 0:00

Example: Tangent Lines


Find the tangent lines to the function y=x3y=x^3 with slope 6.



y′=3x2=6x2=2x=±2→{y(2)=(2)3=22y(−2)=−22\begin{array}{rcl} y'=3x^2=6 & & \\ x^2=2 & & \\ x=\pm\sqrt{2} & \to & \left\{\begin{array}{l} y(\sqrt{2})=(\sqrt{2})^3=2\sqrt{2}\\ y(-\sqrt{2})=-2\sqrt{2} \end{array}\right. \end{array}

ytan⁡,+=22+6(x−2)ytan⁡,−=−22+6(x+2)\begin{array}{l} \bf y_{\tan,+} = 2\sqrt{2}+6(x-\sqrt{2})\\ \bf y_{\tan,-} = -2\sqrt{2}+6(x+\sqrt{2}) \end{array}



Find the equation to the tangent line to y=3x2+4x−1y=3x^2+4x-1 at x=−2.x=-2.