MATH101 Calculus Workshop Exercise 2 Solutions
W2.1. (i) We have
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(ii) Similarly, we have
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(ii) The Heaviside function is discontinuous at , since there it
jumps from to ; indeed, as , whereas
as The graph of has a step at
The function is continuous as one can draw its graph without lifting pen
from paper. In particular, the right and left halves of the graph meet at since
as and as .
(iii) The switch represented by would be more appropriate
since it increases the current smoothly, whereas the switch
would give a surge of current.
W2.3. Let . Then
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W2.4 With strictly
decreasing, the inverse function is which has domain and codomain .
W2.5 (i) We have , so by the
chain rule
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while
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hence
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(ii) We consider
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while
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so
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(iii) Let , so
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hence
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hence
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hence
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W2.6 We have . Suppose that when , we have , so
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so
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hence
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hence
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W2.7.
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(i) When , , so the intercept with the axis is
at ; also if so
, so or and the intercepts with the
axis are at and .
(ii) By dividing, we obtain
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As , we have ; whereas as
, we have . Hence
is a vetical asymptote to the graph.
(iii) As , , so
the line is close to the graph of . (Hence is an
asymptote to the graph.) Hence
as , and
as .
W2.8. (i) The function
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is defined, provided that the denominator is non-zero. The quadratic
factors as
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so that the roots are ; hence the domian of definition of is
.
(ii) The intercept with the axis is at , and the intercept with the
-axis is at .
Asymptotes are the -axis, since
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and the vertical asymptotes and since
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