Chapter 3: Applications of Differentiation
Section 3.1: Tangent and Normal Lines
In Section 2.1 the "tangent line" was introduced as the line touching the graph of a function at a single point. The slope of this line is the value of the derivative of the function at the point of contact.
The "normal line" is a line that is perpendicular to the tangent line at the point of contact. Hence, its slope is the negative reciprocal of the derivative of the function at the point of contact.
Table 3.1.1 lists the equations of tangent and normal lines drawn at x=a on the graph of fx.
Table 3.1.1 Equations for tangent and normal lines
At x=2, obtain the equations of the lines tangent and normal to the graph of fx= 3 x2−5 x+7.
On the same set of axes, graph fx and the two lines.
Which point on the graph of gx=7 x+9−x2 is closest to the point 3,10?
What is that minimal distance?
Let C1 be the ellipse defined by x=uθ=5 cosθ,y=vθ=3 sinθ,0≤θ≤2 π, and let C2 be the curve defined parametrically by x=Uθ=165cos3θ, y=Vθ=−163sin3θ, 0≤θ≤2 π.
Graph C1 and C2 on the same set of axes.
Show that for each value of θ, the slope of the line normal to C1 is the same as the slope of the line tangent to C2.
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