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Wize University Calculus 2 Textbook > Review: Derivatives

Tangent Lines

Derivative RulePrevious Section
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Tangent and Normal Lines

Recall: The slope of the tangent line to a curve is the
derivative
of that curve.

Equation of the Tangent Line at a Given Point

1. Find a point on the tangent line:
  • They will give you this point (𝑥1, 𝑦1) or
  • They will give you the value of one of 𝑥1 or 𝑦1—solve for the missing value
2. Determine the slope of the line 𝑚 by finding the derivative and substituting the point (𝑥1, 𝑦1)
3. Substitute 𝑚 and (𝑥1, 𝑦1) into the line formula y−y1=m(x−x1)y-y_1=m\left(x-x_1\right)
4. Simplify and rearrange into the more familiar 𝑦 = 𝑚𝑥 + 𝑏 form (only if the question requires it)


Equation of the Normal Line at a Given Point:

Same steps as above, but the slope will be the negative reciprocal of the slope of the tangent
Example: If the slope of tangent line is 3/2→\rightarrowslope of normal line is
-2/3
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  3. View the solution and report whether you got it right or wrong.

Practice: Equation of Tangent Line

Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.

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}.
Extra Practice
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Slope of a Tangent Line
Practice: Slope of a Tangent Line
Find all values of xx such that the tangent line to the curve y=2x3−x22−x−5y=2x^3-\frac{x^2}{2}-x-5 is horizontal.
Slope of a Tangent Line
Practice: Slope of a Tangent Line
Find all values of xx such that the tangent line to the curve y=2x3−x22−x−5y=2x^3-\frac{x^2}{2}-x-5 is horizontal.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Tangent Line Intercept
Let f(x)=3x2−4x+9f(x)=3x^2-4x+9 and let L denote the tangent line to the graph of f(x)f(x) at the point on the graph where x=2x=2. Find the point where L intersects the x-axis.