The tangent line approximates a smooth curve near a point. Its slope is the derivative. The normal line passes through the same point and is perpendicular to the tangent.
1. Tangent-line formula
For \(y=f(x)\) at \(x=a\), the point is \((a,f(a))\) and the tangent slope is \(f'(a)\). Point-slope form gives
\[y-f(a)=f'(a)(x-a).\]
2. Worked example: tangent to a parabola
Let \(f(x)=x^2\) and find the tangent at \(x=1\).
\[f(1)=1,\qquad f'(x)=2x,\qquad f'(1)=2.\]
Therefore
\[y-1=2(x-1)\Longrightarrow y=2x-1.\]
Graph x^2 and 2*x-1. The line should touch the parabola at \((1,1)\) and have the same local direction.
3. Normal-line formula
If the tangent slope \(m_t\) is nonzero, the normal slope is the negative reciprocal:
\[m_n=-\frac1{m_t}.\]
For the example, \(m_n=-1/2\). Thus
\[y-1=-\frac12(x-1)\Longrightarrow y=-\frac12x+\frac32.\]
Enter -x/2+3/2 to verify the normal line.
4. Horizontal and vertical special cases
- If \(f'(a)=0\), the tangent is horizontal: \(y=f(a)\). The normal is vertical: \(x=a\).
- If the curve has a vertical tangent, the normal is horizontal.
- At a corner or cusp, an ordinary derivative may not exist, so a unique tangent line may fail to exist.
5. Tangent line to an implicit curve
For an implicit relation \(F(x,y)=0\), implicit differentiation often gives
\[\frac{dy}{dx}=-\frac{F_x}{F_y},\]
provided \(F_y\ne0\). For the circle \(x^2+y^2=25\),
\[2x+2y\frac{dy}{dx}=0\Longrightarrow \frac{dy}{dx}=-\frac{x}{y}.\]
At \((3,4)\), the tangent slope is \(-3/4\) and the normal slope is \(4/3\).
6. Local linear approximation
The tangent line also gives the linearization
\[f(x)\approx f(a)+f'(a)(x-a)\]
for x close to a. For \(f(x)=\sqrt{x}\) near \(a=4\),
\[\sqrt{x}\approx2+\frac14(x-4).\]
This estimates \(\sqrt{4.1}\approx2.025\).
Common mistakes
- Using \(f'(x)\) instead of evaluating \(f'(a)\).
- Using the point \((a,f'(a))\) instead of \((a,f(a))\).
- Forgetting the negative sign in the normal slope.
- Applying the reciprocal rule when the tangent is horizontal or vertical.
- Trusting a visual tangent without checking the derivative.
Open the Graphing Calculator and use its analysis tools where available. Continue with concavity and inflection points.