Chapter 3: Applications of Differentiation
Section 3.1: Tangent and Normal Lines
Example 3.1.1
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.
Solution
Mathematical Solution
Figure 3.1.1(a) provides a graph of fx, along with lines tangent and normal to this graph at x=2.
The slope of fx at x=2 is f′2=7. The point of contact has coordinates 2,f2=2,9.
The equation of the tangent line at x=2 is therefore
y
=7x−2+9
=7 x−5
The equation of the normal line at x=2 is therefore
=−17x−2+9
=65−x/7
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
fx= x0= (Default value: x0=0)
f′x=
m=f′x0=
y0=fx0=
y=mx−x0+y0=
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
fx = x0= (Default value: x0=0)
Normal Line:
Table 3.1.1(b) Solution via the Normal Line task template
Complete Solution from First Principles
Define fx only if not already defined
Control-drag (or copy/paste) fx=…
Context Panel: Assign Function
fx= 3 x2−5 x+7→assign as functionf
Obtain equations of tangent and normal lines
Using Table 3.1.1, write the equation of the tangent line and press the Enter key.
y=f′2 x−2+f2
y=7x−5
Using Table 3.1.1, write the equation of the normal line and press the Enter key.
y=−1f′2 x−2+f2
y=−17x+657
Graph
Type fx and press the Enter key.
Context Panel: Plots≻Plot Builder Adjust plot range to 1≤x≤3
2-D Options≻scaling≻constrained
Onto the graph of fx, copy the expressions for each line, and paste that expression on the existing graph.
fx
3x2−5x+7
→
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