Chapter 3: Applications of Differentiation
Section 3.1: Tangent and Normal Lines
Example 3.1.1
At , obtain the equations of the lines tangent and normal to the graph of .
On the same set of axes, graph and the two lines.
Solution
Mathematical Solution
Figure 3.1.1(a) provides a graph of , along with lines tangent and normal to this graph at .
The slope of at is . The point of contact has coordinates .
The equation of the tangent line at is therefore
The equation of the normal line at is therefore
Figure 3.1.1(a) Graph of tangent and normal lines
Tangent Line by Task Template
The task template, whose use is illustrated in Table 3.1.1(a), provides a complete solution for the tangent line.
Tools≻Tasks≻Browse: Calculus - Differential≻Applications≻Tangent Line
Tangent Line
(Default value: )
Table 3.1.1(a) Solution via the Tangent Line task template
Normal Line by Task Template
The task template, whose use is illustrated in Table 3.1.1(b), provides a complete solution for the normal line.
Tools≻Tasks≻Browse: Calculus - Differential≻Applications≻Normal Line
Normal Line
Normal Line:
Table 3.1.1(b) Solution via the Normal Line task template
Complete Solution from First Principles
Define only if not already defined
Control-drag (or copy/paste)
Context Panel: Assign Function
Obtain equations of tangent and normal lines
Using Table 3.1.1, write the equation of the tangent line and press the Enter key.
Using Table 3.1.1, write the equation of the normal line and press the Enter key.
Graph
Type and press the Enter key.
Context Panel: Plots≻Plot Builder Adjust plot range to
2-D Options≻scaling≻constrained
Onto the graph of , copy the expressions for each line, and paste that expression on the existing graph.
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