Math 171 — Calculus I Spring, 2016
Problem 1.
Practice Questions for Exam III
Differentiate each of the following functions:
1. P (t) = 2t
4. k(x) = ln(7x2 + 2x + 1)
6. g(t) = cosh(at2 + b)
ln(x)
1+x
7. g(t) = sinh(4t + 2)
2
2. E(y) = y ln(1 + ey )
3. l(w) = sinh(cos(w))
5. q(x) =
Problem 2. Find y 0 using logarithmic differentiation of y = xsin
2 (x)
.
Problem 3.
A spy uses a telescope to track a rocket launched vertically from a
launching pad 6 km away, as in the Figure below. At a certain moment, he angle
π
between the telescope and the ground is equal to
and is changing at a rate of 1
4
rad/min. What is the rocket’s velocity at that moment?
Problem 4. A roller coaster has the shape of the graph in the Figure above. Show
that when the roller coaster passes the point (x, f (x)), the vertical velocity of the roller
coaster is equal to f 0 (x) times its horizontal velocity, x0 (t).
Problem 5. Consider the system shown in the figure below. The system consists of
two blocks, labeled A and B, that are connected by a rope that passes over the fixed peg
C. The position coordinates of the blocks are denoted by xA and yB . The constraint on
the system is that the total length of the rope, L, does not change.
a. Use the figure to find
L=
b. Given that L and h don’t depend on time,
find a formula that relates dxdtA , dydtB , xA and
yB . Note, in the diagram, vA = dxdtA and
vB = dydtB .
Problem 6. As Claudia walks away from a 264-cm lamppost, the tip of her shadow
moves twice as fast as she does. What is Claudia’s height?
1
2
Problem 7. A boat is drawn close to a dock by pulling in a rope as shown. Which
of the following choices explains how the rate at which the rope is pulled in is related
to the rate at which the boat approaches the dock?
Dock
a. One is a constant multiple of the
other.
Rope
b. They are equal.
Water
c. It depends on how close the boat is to
the dock.
Boat
Problem 8. 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 9.
(a) Find the linear approximation to f (x) = x cos(x) at a = π.
(b) Use your approximation to estimate (π +
1
) cos(π
100
+
1
).
100
Problem 10. Your physics professor tells you that you can replace tan(x) with x
when x is close to zero. Explain why this is reasonable.
Problem 11. Find the linearization
L(x) to the function f (x) =
√
3
then use the linearization to estimate 8.25.
Problem 12.
√
3
x + 4 at x = 4,
Find all critical points of the following functions (if they exist)
a. f (x) = x1/3 − x
1
b. f (x) = (x2 − 4) 3
c. f (x) = xe−x
Problem 13.
Find the critical points and the intervals on which the function increasing or decreasing. Use the First Derivative test to determine whether the critical
point is a local min or max or neither.
a. y = x5 + x3 + 1
1
b. y = 2
x +1
c. y = x − ln(x),
x>0
d. h(x) = |x2 − 6x|
Problem 14.
Find the absolute maximum and absolute minimum values of the
following functions on the given intervals. Specify the x-values where these extrema
occur.
√
a. f (x) = 2x3 − 3x2 − 12x + 1, [−2, 3]
b. h(x) = x + 1 − x2 , [−1, 1]
Math 171 — Calculus I Spring, 2016
Practice Questions for Exam III
Problem 15. Suppose that g(x) is differentiable for all x and that −5 ≤ 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 16. A trucker handed in a ticket at a toll booth showing that in 2 hours
she had covered 160 miles on a toll road with speed limit 70 mph. The trucker was cited
for speeding. Why did she deserve the ticket?
Problem 17.
Multiple choice: The second derivative, f 00 (x) of a function f (x) is
positive 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 18.
conditions.
Sketch the graph of a function f (x) which has all of the following
f (0) = 0,
f 0 (x) > 0 for x < 0,
f 0 (x) > 0 for x > 0,
lim f (x) = 1,
x→−∞
lim f (x) = 0,
f 00 (x) > 0 for x < 0,
f 00 (x) < 0 for x > 0.
x→∞
Problem 19. The following graph is the graph of f 0 (x). Where do the point(s) of
inflection of f (x) occur and on which interval(s) is f concave up? At which point(s)
does f have a local maximum, if any?
Problem 20.
a. lim+ x ln(x)
x→0
ln(x)
b. lim
x→1 1 − x
Evaluate the following limits.
ex
c. lim
x→0 x
ln(x)
d. lim 2x
x→∞ e
e. lim
x→∞
4
1+
x
x
3
4
Problem 21. The bottom of the legs of a three-legged table are the vertices 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.
B
3
A
3
x
√
x2 + 3 2
√
x2 + 3 2
4−x
C
Problem 22. The power transmitted by a pulley belt system is given by the following
equation
P (v) = (F v − ρAv 2 )(1 − e−µθ )
where P is the power, v is the velocity of the belt, ρ is the density of the belt, A is the
cross sectional area of the belt, F is the maximum force in the belt, µ is the coefficient
of friction, and θ is the contact angle. Assuming the P is a function of v and that all
the other parameters are constant and positive, determine the velocity that makes the
power a maximum, and find a formula for this maximum power.
Problem 23. 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
r
x
Problem 24. 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 above).
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 above.
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.