1. Consider
a.
Approximate the value of at x
= 64.001 and 63.999 – what do your results suggest about ? SOLN:
,
in significant digits: 3.0000
,
in significant digits: 3.0000
Which strongly suggests that .
b. How close does x have to be to 64 to ensure that the function is within 0.1 of it’s limit?
SOLN: We want to
choose a distance δ small enough to
be sure that if then .
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2.
Is there a number a
such that exists?
If not, why not? If so, find the
value of a and the value of the
limit.
SOLN: Since the denominator of has a zero at x = 1, we need to require that the numerator also has a zero at x = 1; that is, 2+a+a = 0 or a = –1.
To be sure,
3. Consider .
a.
What theorem is essential to evaluating this
limit. Why are the conditions of the
theorem met?
The relevant theorem says that if (i.e. the limit exists) and if f is continuous at b (i.e. ) then . Since is a composition of continuous functions, it
is also continuous, thus the conditions of theorem are met.
b.
Use the theorem to evaluate the limit. SOLN:
4. For the function g whose graph is shown, approximate the following, writing “DNE” if the limit doesn’t exist and or , as appropriate.
a. |
b. |
c. |
d. , DNE |
e. |
f. |
5. Suppose the height H of an object (in meters) at time t (in seconds) is given by
a.
What is the average velocity over the interval
SOLN: m/s.
b.
Find an interval over which the average velocity of the
object is a 1000 m/s.
SOLN: Since the jump discontinuity has
an instantaneous change which is infinite, it shouldn’t be to hard to find a
relatively puny rate of change like 1000 m/s.
Fix one point at (0,1) and a variable point (t,0) preceding that point.
Then the average rate of change is .
6.
Let B(t) be the number of Elbonian buffalo per
capita at time t. The table below gives values of B(t)
as of June 30 of the specified year.
What is your best approximation to the value of ?
SOLN: Averaging the immediate before and
after rates of change leads to buffalo per capita per year, but this doesn’t
take into account the data from 1998 and 2002.
Let’s see if we can include these data.
One way is to fit a quartic function to the data, which can be done by
substituting the t, B pairs into the general form .
You could go ahead and solve the 4x4 system this leads to,
or you could get the same result by using the TI85, say, to do “regression
analysis” on the data. Start by
entering the data using the |
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and thus in year 2000 we estimate
buffalo per capita per year, which is about half
the rate of change computed with the previous method…but a whole lot more
trouble! It can be argued that the extra
trouble does produce a more accurate measure, but is there an easier method of
computing it? You bet—and it’s a subject
of intense interest. See, for instance,
the paper at http://webpages.dcu.ie/~carrollj/202-ln-b.pdf
. As a typical starting point, one would
form a forward difference table:
7. Consider the function .
a. Use the definition of the derivative to show that . SOLN:
b.
Find an equation for the line tangent to x(t)
where t = 1.
SOLN: We plug into point-slope form:
c.
Use a linear approximation to approximate x(1.05)
SOLN:
8. For the function whose derivative function is graphed below, find where:
a. is increasing if , which is true for
b. is concave up where is increasing, that is,
c. has a local maximum(?) SOLN: This question turns out to be more interesting than intended. In fact, the graph given contains contradictory information about the derivative function where x = 2: it can’t be true that the derivative function is defined there and yet has a jump discontinuity. That would mean that f has a tangent line at a point where the slope changes abruptly—can’t be! Disregarding the nonsense definition of , f may or may not have a maximum where x = 2, depending on whether or not f(2) is defined and whether that definition leads to a maximum or not.
d. is positive. . SOLN: This is equivalent to part (b). is concave up if is increasing, which is true true for
e. . SOLN: changes from decreasing to increasing where x = 3, and from increasing to decreasing
at x = 5. at both of these points, is smooth/differentiable so . While changes from increasing to decreasing at x = 2, it does so abruptly so that is undefined.