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Practice Questions for Exam 4
Problem 1. Multiple Choice. The second derivative, f 00 (x) of a function f (x) is negative
everywhere. We know that f (0) = 0 and f 0 (0) = 0. What must be true about f (1)?
a. f (1) is negative
b. f (1) is positive
c. f (1) is zero
d. Not enough information to conclude anything about f (1).
Problem 2. Find the interval(s) on which f is concave up or down, the points of inflection,
the critical points, and minima and maxima for f (x) = ln(x)
x for x > 0.
Problem 3.
(a) Find the linear approximation to f (x) = x cos(x) at a = π.
(b) Use your approximation to estimate (π +
1
100 ) cos(π
+
1
100 ).
Problem 4. The following graph is the graph of f 0 (x). Where do the points of inflection of
f (x) occur and on which intervals is f concave up? Does f have a local max or min at x = 0?
Problem 5. Recall that Newton’s method is used to find approximate solutions to f (x) = 0
n)
by using the iteration: xn+1 = xn − ff0(x
(xn ) . Show how to setup Newton’s method to find the
cube root of b, i.e. show how would to setup Newton’s method to compute a solution of x3 = b.
Problem 6.
Multiple choice: Suppose that f 00 (x) < 0 for x near a point a. Then the
linearization of f at a is
1. an over approximation
2. an under approximation
3. unknown without more information.
Problem 7. Let (a, f (a)) be the point where the tangent line to the graph of f (x) = x cos(x)
is horizontal. We use Newton’s method to find x1 , x2 , x3 , . . . , the successive approximations to
a. Give the formula that we use to compute xn+1 from xn .
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Problem 8. We will use each of the xn below as the starting point for Newton’s method. For
which of them do you expect Newton’s method to work and lead to the root of the function?
x1
x2
x3
x4
1. x1 and x2 only.
2. x2 only.
3. x1 , x2 and x3 only.
4. All four.
Problem 9. Write the formula for Newton’s method applied to f (x) = x1/5 , and use the
given initial approximation x0 = 1 to compute x1 and x2 . What happens here? Does the
method converge?
Problem 10.
a. f (x) =
Find all critical points of the following functions (if they exist)
√
x2
−33x
2
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b. f (x) = (x2 − 4) 3
c. f (x) = xe−x
d. f (x) =
x2
1+x
Problem 11.
x−π
x→π sin(x − π)
1
2
lim x sin
x→∞
2x2
ln(x)
lim
x→∞ x
√
x+1
lim √
x→∞
2x + 1
1 − sin(x)
lim
x→0 cos(x)
a. lim
b.
c.
d.
e.
Evaluate the following limits.
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Problem 12. The bottom of the legs of a three-legged table are the vertices’s of an isosceles
triangle with sides 5,5,6. The legs are to be braced at the bottom by three wires in the shape
of a Y. What is the minimum length of wire needed? Show it is a minimum.
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B
A
3
x
√
√
x2 + 3 2
x2 + 3 2
4−x
C
Problem 13. The amount of energy used by a trout to swim at speed v against a current
with speed U a distance L is given by
E(v) = L
v3
v−U
as long as v > U . Find the speed that minimizes the energy E. Be sure to show that you do
have a min.
Problem 14.
An architect plans to build a rectangular window under a arch that is
a semi-circle. Find the maximum area of a rectangle inscribed in semi-circle of radius r.
y
0
x
r
Problem 15. Sketch the graph of a function for which f 0 (x) < 0 for x < −2,
f 0 (x) > 0 for x > −2,
f 00 (x) > 0 for x < 0,
and f 00 (x) < 0 for x > 0.
Problem 16. Sketch the graph of a continuous function for which f 00 (x) < 0 on
(0, 1) and f 00 (x) > 0 everywhere else.
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Problem 17. Given that f (x) = x2x+1 , f 0 (x) = x22x+1 , and f 00 (x) = (x2−6x
2 +1)3 . Make
a sketch of the graph the function f . Be sure to include local extrema and points of
inflection.
Problem 18.
Find the min and max of the function f (x) =
x
ln(x)
√
on [ e, e4 ]
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Problem 19.
Suppose you are driving along a highway in a car with a broken
speedometer at a nearly constant speed and you record the number of seconds it
takes to travel between two consecutive mile markers. If is takes 60 seconds, then
your average speed is 1 mi/60 s or 60 mi/hr. Now suppose that you travel one mile
in 60 + x seconds; for example if it takes 62 seconds then x = 2 and if it takes 57
seconds, then x = −3. In this case, your speed over one mile is 1 mi / (60 + x) 63.16
mi/hr.seconds. Because there are 3600 seconds in 1 hour, the function
s(x) =
3600
60 + x
gives your average speed in mi/hr if you travel one mile in x seconds more or less
the 60 seconds. For example, if x = 2 then your average speed is s(2) ≈ 58.06
mi/hr. If you travel one mile in 57 seconds, then x = −3 and your average speed is
s(−3) ≈ 63.16 mi/hr. Because you don’t want to use your phone or calculator while
driving, you need an easy approximation to this function. Use linear approximation
to derive such a formula.
Problem 20. The fastest drag racers can reach a speed of 300 mi/hr over a quartermile strip in 4.45 seconds (from a standing start), Complete the following sentence
about such a drag racer: At some point during the race, the maximum acceleration
of the drag racer is at least
mi/hr/sec.
Problem 21. Suppose that g(x) is differentiable for all x and that 0 ≤ g 0 (x) ≤ 3
for all x. Assume also that g(0) = 4. Based on this information, use the Mean Value
Theorem to determine the largest and smallest possible values for g(2).
Problem 22. In 1613, Kepler bought a barrel of wine for his wedding but questioned
the method the wine merchant used to measure the volume of the barrel and thus
determine the price. The price for a barrel of wine was determined solely by the
length L of a dipstick that was inserted diagonally through a centered hole in the top
of barrel to the edge of the base of barrel (see figure).
In consequence, afterwards Kepler set out to find the proportions that optimize the
volume of such a barrel. Suppose that the wine barrel is a cylinder as shown. Determine the ratio of the radius r to the height h of the barrel that maximize the Volume
of the barrel subject to the wetted length L of the dipstick is fixed.
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Problem 23. The Reaction R(x) of a patient to a drug dose of size x depends on
the type of drug. For a certain drug, it was determined that a good description of
the relationship is:
R(x) = Ax2 (B − x)
where A and B are positive constants. The Sensitivity of the patients body to the
drug is defined to be R0 (x).
a. For what value of x is the reaction a maximum, and what is that maximum
reaction value?
b. For what value of x is the sensitivity a maximum? What is that maximum
sensitivity?
Problem 24.
The Hill function is given by
h(x) =
xn
,
k n + xn
x≥0
Here, n ≥ 1 and can be large and k > 0 It is easy to show that
h0 (x) =
and that
h00 (x) =
nk n xn−1
(k n + xn )2
nk n xn−2 (k n (n − 1) − (n + 1)xn )
.
(k n + xn )3
a. We see that for small x, h00 (x) > 0 provided n > 1 and for large x, h00 (x) < 0.
Assuming that n > 1, find the x-coordinate of the point of inflection: (c, h(c)).
This number will depend on n and k. c is called the point of diminishing return,
for it is the value of x at which h0 (x) is maximal.
b. Is c less than or greater than k?
c. Compute lim c.
n→∞
d. Sketch a graph of h for several values of n.
Problem 25. A snow-making machine makes snow for Bridger Bowl. Let C(x)
denote the cost per hour of running the machine on a setting that produces x kilograms per hour. Resurfacing boot hill require a total of 10,000 kg of snow. Illustrate
graphically the most economical production rate x (in kg/hr).