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Calculus 1 Problems & Solutions – Chapter 2 – Section 2.2 |
2.2
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1. Differentiability Implies Continuity |
We'll show that if a function is differentiable, then it's continuous.
Theorem 1.1
If a function f is differentiable at a point x = a, then f is continuous at x = a.
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Proof
EOP
Note that if we let h = x – a then:
The right-hand side of the above equation looks more familiar: it's used in the definition of the derivative.
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2. Continuity Doesn't Imply Differentiability |
We'll
show by an example that if f
is continuous at x = a, then f may or may not be differentiable at x = a. The converse
to the above theorem isn't true. Continuity doesn't imply differentiability.
Example 2.1
Solution
a.
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Fig. 2.1
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and thus f '(0) don't exist. It follows that f is not differentiable at x = 0.
Remark 2.1
In
handling continuity and differentiability of f, we treat the point x =
0 separately from all other points because
f changes
its formula at that point. We do so because continuity and differentiability
involve limits, and when f
changes its formula at
a point, we must investigate the one-sided limits at both sides of the point to
draw the conclusion about the limit at that
point.
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3. Where Functions Aren't Differentiable |
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Fig. 3.1
f isn't
differentiable at a where it's
discontinuous, at b where its |
Problems & Solutions |
1. Let y = f(x) = x1/3.
a. Sketch a graph of f using graphing technology.
b. Based on the graph, where is f both continuous and differentiable?
c. Based on the graph, where is f continuous but not differentiable?
Solution
a.
b. Based on the graph, f is both continuous and differentiable everywhere except at x = 0.
c. Based on the graph, f is continuous but not differentiable at x = 0.
2.
Let f be defined by
f(x) = |x2
+ 2x – 3|.
a. Show that f is continuous everywhere.
b. Show, using the definition of
derivative, that f is differentiable
everywhere except at x = – 3 and x = 1.
Solution
a. We have f(x) = |(x + 3)(x – 1)|. The following table shows the signs of (x + 3)(x – 1).
So we have:
Similarly, f is also continuous at x = 1. It follows that f is continuous everywhere.
b. |
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Case Where x < – 3 Or x > 1. We have:
Case Where – 3 < x < 1. We have:
So f is differentiable on (– 3, 1).
Case Where x = – 3. We have:
and thus f '(– 3) don't exist. As a consequence, f isn't differentiable at x = – 3.
Case Where x = 1. Similarly, f isn't differentiable at x = 1.
In summary, f is differentiable everywhere except at x = – 3 and x = 1.
Solution
Note
Many other examples are possible, as seen in the figure below.
Solution
5. If possible, give an example of a differentiable function that isn't continuous.
Solution
That's impossible, because if a function is differentiable, then it must be continuous.
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