Here are some partial solutions to the questions on Review for Midterm #2. For most of the problems, I’ve just provided the answer, so that you can check to make sure that you are doing the problem correctly. In some cases, I’ve provided some of the intermediate steps to help guide you.

- The function

- The function

- The function

- Use appropriate derivative formulas to get the following.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

- Use

- Use

- First, find the derivative using the product
rule:
*x*-values that make this function 0 determine the horizontal tangents. These values are*x*-values where this function is undefined determine the__possible__vertical tangents. These values are

- Use implicit differentiation to get the following.

(a)

(b)

- First, find the derivative:

- First, find the derivative:

- The answer is

- See Book.

- See Book.

- In each case, draw a number line and put 2 down
as the only critical number. Then figure out whether the function
is increasing or decreasing on either side of

(a) Local min at .

(b) Local max at .

(c) Neither.

- In each case, check to see if the 3 hypotheses of Rolle’s Theorem are satisfied.

(a)
No, *g *is not
continuous
at
,
which is in the interval
.

(b)
Yes, *h* is continuous
on
(the only discontinuities are at
,
which are not in the interval),
is differentiable on
(to see this, find
and observe that the only
discontinuities of this function are at
,
as well), and
. Note: According to the book, we
should
have
,
but this is unnecessary.

16.
*f* satisfies the
hypotheses of the MVT since (i) *f*
is continuous on
(the only discontinuity of *f* is at
,
which is not in the interval), and (ii) *f* is differentiable on
. To find all points that satisfy the
conclusion of the MVT, solve for *x*
in

.

You will find out that the answer is . Note that is not in the interval.

17.
If we implicitly differentiate with respect to *t*, we get
. Plug in
(decreasing!) and
,
and then solve for
meters/sec.

18.
Here is the equation you need to use:
. At some moment in time, we have
,
(since distance between observer and
point on the ground below the nugget is decreasing).
Differentiating with respect to *t*, we get:
. We can see that we are also going to
need an *x*-value.
To get this, plug
into
and then solve for *x*. You
should get
. Now, plug in all of the information
that you have and solve:
.

19. To determine the intervals of increase and decrease, do the usual number line thing with the first derivative. To determine the intervals of concave up and down, do the number line thing with the second derivative.

(a)
For
,
notice that the domain is
. *f*
is decreasing on
,
*f* is increasing on
,
and *f* is concave up on
.

(b)
*g* is decreasing on
and
,
*g* is increasing on
,
*g* is concave up on
and
,
and *g* is concave down
on
.

20. Same game as #19.

(a) Local min at , inflection points at .

(b)
Along the way, you should get
and
. By setting
equal to 0 and solving for *x*, you should get that the only *x*-value in the interval is
. Doing the usual number line thing,
you
determine that *g* has a
local min
at
. Next set
equal to 0 and solve for *x* by factoring.
You will encounter
and
,
which become
and
,
respectively. Solving for *x* in each, you will get that the only *x*-values in the interval are
and
. Using a calculator, you can pick test
points on the number line. You
will find out that *g*
has
inflection points at
.

21.
Along the way you will get the following: *x*-intercept:
,
*y*-intercept:
,
vertical asymptote: None, horizontal asymptote: None, *g* is increasing on
,
*g* is decreasing on
,
,
,
*g* is concave down on
and
,
*g* is concave up on
,
and
. Use this information to sketch graph.

22. Find the second derivative: . Then plug in 2: .

23. Here, you are just supposed to find the equation of the tangent line to at . The answer is .

24. First, get the linear approximation for using . You get . Then .

25. To find , just find the derivative, then tack on .

(a)

(b)