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Chapter 2: Differentiation
Section 2.1: What Is a Derivative?
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Example 2.1.2
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Show that at a unique slope cannot be assigned to the graph of . Consequently, a tangent line does not exist at on this curve.
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Solution
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In Figure 2.1.2(a), the green dot on the graph of is at the point where no unique slope can be assigned.
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To see this, consider secant lines through and , the slopes of which are
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The limiting values for these slopes are then, the positive value when the limit is taken from the right; the negative, from the left.
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Consequently, no limit exists for these slopes, and therefore a unique slope cannot be assigned to the graph of at .
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>
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p1:=plot(sqrt(abs(x^2-4)+9),x=-3..3):
p2:=plot([[2,3]],style=point,symbol=solidcircle,symbolsize=15,color=green):
plots:-display(p1,p2,labels=[x,y],view=[-3..3,0..5],scaling=constrained);
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Figure 2.1.2(a) Graph of
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The "corner" at on the graph of is an example of a point where a tangent line does not exist.
The following details demonstrate this computationally.
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Define the function
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Control-drag
Context Panel: Evaluate and Display Inline
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Limiting slopes of the secant lines
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Expression palette: Limit operator
Limit from the left
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=
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Expression palette: Limit operator
Limit from the right
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=
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Stepwise computation of the limits:
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When computing the limit from the right, , so ; but from the left, , so . The fraction
tends to as , so the limit from the right is but the limit from the left is .
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