Tangent Line Math Example 1

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Example 1

easy
Find the equation of the tangent line to f(x)=x2+3f(x) = x^2 + 3 at x=2x = 2.

Solution

  1. 1
    Find the point of tangency: f(2)=4+3=7f(2) = 4 + 3 = 7. Point: (2,7)(2, 7).
  2. 2
    Find the slope: f′(x)=2xf'(x) = 2x, so f′(2)=4f'(2) = 4.
  3. 3
    Write the tangent line using point-slope form: y−7=4(x−2)y - 7 = 4(x - 2).
  4. 4
    Simplify: y=4x−8+7=4x−1y = 4x - 8 + 7 = 4x - 1.

Answer

y=4x−1y = 4x - 1
Two pieces of information define a line: a point and a slope. The point is (a,f(a))(a, f(a)) and the slope is f′(a)f'(a). Plug both into the point-slope formula y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a).

About Tangent Line

A line that touches a curve at exactly one point and has the same slope as the curve there.

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More Tangent Line Examples

Example 2 medium

Find the equation of the tangent line to [formula] at [formula], and use it to approximate [formula]

Example 3 easy

Find the equation of the tangent line to [formula] at [formula].

Example 4 hard

At what point(s) on [formula] does the tangent line have slope [formula]?

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Related Concepts

SlopeDerivative