Chapter 3: Applications of Differentiation
Section 3.1: 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.
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
The equation of the normal line at x=2 is therefore
Figure 3.1.1(a) Graph of tangent and normal lines
Tangent Line by Task Template
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
fx= x0= (Default value: x0=0)
Table 3.1.1(a) Solution via the Tangent Line task template
Normal Line by Task Template
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
fx = x0= (Default value: x0=0)
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.
Using Table 3.1.1, write the equation of the normal line and press the Enter key.
Type fx and press the Enter key.
Context Panel: Plots≻Plot Builder
Adjust plot range to 1≤x≤3
Onto the graph of fx, copy the expressions for each line, and paste that expression on the existing graph.
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