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Wize University Calculus 1 Textbook > Derivatives

Normal Lines

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Normal Lines

Since the derivative can give us the tangent line to a curve, it can also tell us the line perpendicular to the tangent line. The line perpendicular to the tangent line is called the Normal Line.

Wize Concept
The equation of the tangent line is ytangent=f(a)+f′(a)(x−a)y_{\text{tangent}}=f(a)+f'(a)(x-a).

Normal Lines

Also, recall that two perpendicular lines with slopes mtangentm_{\text{tangent}} and mnormalm_{\text{normal}} satisfy
mtangent×mnormal=−1\boxed{m_{\text{tangent}} \times m_{\text{normal}}=-1}
Then, the equation of the line normal to f(x)f(x) at x=ax=a is
ynormal=f(a)−1f′(a)(x−a)\boxed{\displaystyle y_{\text{normal}}=f(a)-\frac{1}{f'(a)}(x-a)}

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Example: Normal Lines

Find the normal line to the graph f(x)=xf\left(x\right)=\sqrt{x} at x=9x=9.

f(x)=x=x12  ⟹  f′(x)=12x−12=12x\displaystyle f\left(x\right)=\sqrt{x} = x^{\frac{1}{2}} \implies f'(x)=\frac{1}{2}x^{-\frac{1}{2}}= \frac{1}{2\sqrt x}

f(9)=9=3, f′(9)=129=16\displaystyle f\left(9\right)=\sqrt{9}=3,\ f'\left(9\right)=\frac{1}{2 \sqrt 9}= \frac{1}{6}

mnormal=−61=−6\displaystyle m_{normal}=-\frac{6}{1}=-6

ynormal=3−6(x−9)y_\text{normal}= 3-6(x-9)

=−6x+57=-6x+57

Find the equation of the normal line to f(x)=ln⁡x at the point (e,1)f(x)=\ln{x} \text{ at the point } (e,1)

Note: This questions requires knowledge of derivatives of logarithms
Extra Practice
Slope of a Normal Line
Find all points where the slope of the normal line to the curve y=2x3−5x+1y=2x^3-5x+1 is −119-\frac{1}{19}.
Slope of a Normal Line
Find all points where the slope of the normal line to the curve y=2x3−5x+1y=2x^3-5x+1 is −119-\frac{1}{19}.
Derivatives of Logarithmic Functions: Normal Lines
Find the equation of the normal line to f(x)=ln⁡x at the point (e,1)f(x)=\ln{x} \text{ at the point } (e,1)
Note: This questions requires knowledge of derivatives of logarithms
Finding Tangent and Normal Lines
Find the tangent line and normal line to f(x)=2x2+x−1f(x)=2x^2+x-1 at x=1x=1.
Normal Lines
Find the point on y=x2y=x^2 that is closest to the point (18,0).(18,0).
Hint: The line which connects the closest point on the curve to a point out of the curve is perpendicular to the curve.
Slope of a Normal Line
Find all points where the slope of the normal line to the curve y=2x3−5x+1y=2x^3-5x+1 is −119-\frac{1}{19}.