Key takeaway: The equation of a tangent line to y = h(x) at x = k is y = h(k) + h′(k)(x − k), a straight line that shares the curve’s height and slope at that one point.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Equation of a Tangent Line
The equation of a tangent line describes the straight line that touches a curve at one chosen point and heads in the same direction as the curve there. Two facts fix it: where the line touches, and how steep it is.
Where it touches comes from evaluating the function. How steep it is comes from the derivative. Near the touching point the line hugs the curve closely, so tangent lines also serve as quick linear approximations.
Formula and notation
For a function \(h\) and a chosen input \(k\):
\[ y=h(k)+h^{\prime}(k)\,(x-k). \]
The constant term \(h(k)\) is the height of the touching point, and the number \(m=h^{\prime}(k)\) multiplies how far \(x\) has moved from \(k\). Both are plain numbers once \(k\) is fixed.
Key facts
- Evaluate the function for the height; evaluate its derivative at the same input for the steepness.
- A horizontal tangent has slope 0, while a vertical tangent has an undefined slope.
- The line crossing it at a right angle at that point has the opposite reciprocal slope.
- Near the touching point, outputs of the tangent line estimate outputs of f.
Equation of a Tangent Line example
Write the tangent line to \(g(x)=2x^3-5x\) at \(x=3\).
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Height: \(g(3)=54-15=39\), so the touching point is \((3,39)\).
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Steepness: \(g^{\prime}(x)=6x^2-5\), so \(g^{\prime}(3)=54-5=49\).
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Assemble: \(y=39+49(x-3)=49x-108\).
The tangent line is \(y=49x-108\).
Common mistake
Watch out
A student writes \(y=39+(6x^2-5)(x-3)\), leaving the letter \(x\) inside the slope.
Why it is wrong
A line has one constant slope, so the derivative must be evaluated at \(x=3\) before it is used. Keeping \(6x^2-5\) turns the answer into a cubic curve. Evaluate, then substitute, as in the equation of a tangent line above.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus