Chapter 3: Applications of Differentiation
Section 3.4: Differentials and the Linear Approximation
Example 3.4.3
Over what interval would the tangent line at x=3 approximate fx=x e−x with an error no greater than 0.1?
Solution
Mathematical Solution
The line tangent to the graph of fx=x e−x at any point x=b is
Yb
= f′bb−3+f3
= −2 ⅇ−3b−3+3 ⅇ−3
= −2 b−9e−3
The difference between the tangent line and the function is to be no greater than 0.1, so the following inequality must be solved.
Figure 3.4.3(a) Graph of fb−Yb and y=0.1
fb−Yb = b e−b+2 b−9e−3≤0.1
Figure 3.4.3(a) contains a graph of fb−Yb and the line y=0.1. Use this graph to obtain b≐0.49 and b≐5.22, the two values of b where the line y=0.1 intersects the graph of fb−Yb. By continuity, the interval where the inequality is satisfied is then approximately 0.49,5.22.
Maple Solution
Define the function f
Type fx=…, being sure to use the exponential "e".
Context Panel: Assign Function
fx=x ⅇ−x→assign as functionf
Obtain the equation of the tangent line
Yb=f′3 b−3+f3→assign as functionY
Graph fb−Yb and the line y=0.1
Figure 3.4.3(a) can be obtained interactively by applying the Plot Builder to the expression
fb−Yb = bⅇ−b+2ⅇ−3b−3−3ⅇ−3
setting the domain to 0≤b≤6 and then copying/pasting 0.1 onto the resulting graph to add the horizontal line y=0.1.
Figure 3.4.3(a) suggests two values of b.
Solve fb−Yb=0.1 for b
Write the equation fb−Yb=0.1 Press the Enter key.
Context Panel: Solve≻Numerically Solve
fb−Yb=0.1
bⅇ−b+2ⅇ−3b−3−3ⅇ−3=0.1
→solve
5.221046886
Context Panel: Solve≻Numerically Solve from point≻b=0.1
0.4878338773
On the closed interval 0.4878338773,5.221046886, the tangent line approximates fx = xⅇ−x with an error no greater than 0.1.
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