1
Practice Questions for Final
Problem 1.
1.
2.
3.
4.
5.
6.
Compute the derivative of the given function:
2
7. f (x) = e−x cos(x)
f (θ) = cos(2θ2 + θ + 2)
g(u) = ln sin2 (u)
r(y) = arctan(y 3 + 1)
s(t) = e−kt sin(βt)
r(θ) = θ3 sin(θ)
s(t) = tan(t) + sec(t)
8. g(x) =
ex
tan(x)
9. x(t) = e−kt cos(ln(t))
10. G(s) =
s2
k 2 + s2
Problem 2. Compute the following integrals:
Z
Z π
1.
cos(2t + 1)dt
8.
sec2 (t/4) dt
0
Z
Z 1
1
2
dx
2.
2
9.
xe−x dx
1+x
Z
0
Z 1
3.
x(1 + x)6 dx
10.
x(1 + x)6 dx
Z
0
4.
e−kx dx
Z π/4
Z 1
11.
sin(2x) cos(2x) dx
0
5.
(1 + t2 ) dt
Z −1 −1
2
Z e
2
dx
12.
3x +
1
x
6.
dx
−2
1 x
Z π
Z ln(2)
4 cos(x) dx
13.
7.
e−x dx
0
0
Problem 3.
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 4.
Given F (x) =
1
2
+
√1
π
Rx
0
2
e−t dt find the linearization of F at a = 0, L(x) =?
Problem 5. Energy lost from the earth due to radiation into space depends on the current
temperature of the Earth T , and is approximated as
E = 4πr2 σT 4
where is the emissivity of the Earth’s atmosphere, which represents the Earth’s tendency to
emit radiation energy. This constant depends on cloud cover, water vapor as well as greenhouse
gas concentration in the atmosphere, such as carbon dioxide and methane levels. σ is a physical
constant, the Stephan-Boltzmann constant.
Without solving for T as a function E, find
with respect to E.)
dT
dE .
(Hint: differentiate both side of the equation
2
Problem 6. For a fish swimming at a speed v relative to the water, the energy expenditure
per unit time is proportional to v 2 . It is believed that migrating fish try to minimize the total
energy required to swim a fixed distance. If the fish are swimming against a current with speed
s (where s < v), then the time required to swim a distance L is L/(v − s), and the total energy
E required to swim the distance is given by
E(v) = aL
v2
,
(v − s)
for v > s.
Here a, s and L are positive constants. Find the critical point(s) of E(v) in terms of these
constants.
Problem 7.
A popcorn box was created from a 1200 ×1200 square with the finished
box shown on the right.
a. Express the volume of the box, V , as a function of x.
12−x
x/2
6−x
b. What is value of x maximizes the Volume?
x
Problem 8.
Water pours into a conical tank of height 10 m and radius 5 m at
a rate of 6 m3 /min. The Volume of a conical tank of radius r and
height h is given by V = π3 r2 h.
At what rate is the water level rising when the level is 4 m high?
5m
r
10m
h
Problem 9.
The planet Quirk is flat. The satellite in the figure travels above the
planet. The radar tracks the satellite and computes the distance
r, the angle θ as functions of time and then numerical computes θ0
and r0 .
a. In terms of θ, θ0 , r, and r0 find the ground speed x0 of the
satellite.
b. What is the ground speed, x0 , when θ = π/2, θ0 =
−2 rad/sec, r = 100 km and r0 = 10 km/sec?
A
x
3
Problem 10. The height of the automobile jack shown is controlled by rotating a screw ABC
which is single-threaded at each end, the pitch is 2.5 mm. If the screw ABC is rotated at a
rate of 10 rpm clockwise at C, then one can show that the length of AB DECREASES at the
constant rate of 2.5 cm/min, i.e., dx/dt = −2.5. (B is the midpoint of AC)
5cm
2
A
x
D
y
B
When the length AB is 20 cm, at what rate is the length BD changing? [i.e when x = 20 what
is dy/dt]? (Note: 252 = 202 + 152 ).
Problem 11.
Three students were given the following question:
R
dx
Find the following indefinite integral:
Unfortunately we cannot make out the original problem. However, we know that Casey’s answer
was tan2 x + C, Jack’s answer was sec2 x + C, and Veronica’s answer was − sec2 x + C. Two of
the students got the correct answer the other one got it
R wrong. The wrong answer was given
by
and the original question was compute f (x)dx for f (x) =
.
Problem 12. First find x0 then find x if x00 (t) = −10, x0 (1) = 0 and x(1) = 6.
0
Problem 13.R Below is pictured the graph of the
R function f (x), its derivative f (x), and an
0
antiderivative f (x) dx. Identify f (x), f (x) and f (x) dx.
4
Problem 14.
Rx
Let A(x) = 0 f (t)dt, with f (x) as in the figure below
1. A00 (x) at x = P is:
2. Where does A(x) have a local minimum?
3. Where does A(x) have a local maximum?
4. A(x) < 0 for all x between 0 and P , True or false?
Problem 15. When you slice a loaf of bread, you change its volume. Let x be the length of
the loaf from one end to the place where you cut off the last slice. Let V (x) be the volume of
the loaf of length x (see figure). For each x, dV
dx is
1. the volume of a slice of bread.
2. the area of the cut surface of the loaf
where the last slice was removed.
3. the volume of the last slice divided by
the thickness of the slice.
Problem 16.
a. Given the graph of y = 1+x, indicate on the graph below the
rectangles used Zto calculate the right endpoint approximation
3
to the integral
4
1 + x dx. with three subintervals (R3 ).
0
3
b. Calculate R3 .
2
1
c. Will this estimate be larger or smaller than the actual value
of definite integral? Circle one: Smaller Larger
−1
0
1
2
3
5
Problem 17.
a. Suppose that g(x) is differentiable for all x and that 1 ≤ 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).
b. Note it follows from the FTC-1 that
Z
g(4) − g(0) =
4
g 0 (x)dx.
0
So if 0 ≤ g 0 (x) ≤ 3 for all x then
Z
≤
4
g 0 (x)dx ≤
0
(fill in the blanks). Show that you get the same estimate you got in part (a) above using
the comparison theorem for integrals instead of the MVT.
Problem 18. Recognize each of the sums as a Riemann sum, express the limit as an integral
and use the Fundamental Theorem to evaluate the limit.
!
q
q
n
i
X
i
i 2
n
n
1
3
+
X
X
2
+
1
+
n
n
n
2. lim
1. lim
3. lim
n→∞
n 1 + i22
n→∞
n→∞
n
n
i=1
n
i=1
i=1
Problem 19. Evaluate the following integrals given the graph of f and that the two parts
of the graph below are semicircles.
Z 2
(a)
f (x)dx =
0
Z
6
(b)
f (x)dx =
2
Z
6
(c)
f (x)dx =
0
Z
(d)
4
f (x)dx =
1
Problem 20. Even before the invention of calculus, Archimedes using an ingenious method,
discovered that the area A under a parabolic arch is two-thirds the base b times the height h:
2
A = bh
3
Use calculus to verify by finding the area under the curve y = h(1 −
4 2
x )
b2
for −b/2 ≤ x ≤ b/2.
6
Problem 21.
d
a.
dx
Z
x4
2x
Find the following derivatives:
!
sin(t)
t
c.
d 

dx
d
d.
dx
Z 2x
sin(t)
d
2
b.
x
dt
dx
t
1
Problem 22.
defined by

sin(t)
dt

t
.
2
x
R
2x
1
x5
Z
2
x5
!
sin(t)
dt .
t
Give the interval(s) for which the function F is increasing. The function F is
Z
F (x) =
0
x
t−3
dt
+ 10
t2
Problem 23.
Recently, a team of Mechanical Engineers give an argument that the huge
stones that make up parts of China’s Forbidden City were transported along artificial ice paths
lubricated with water. Part of their argument involves a calculation of the amount of heat due
to friction that diffuses into the wooden sledge or the ice-covered ground to see if friction alone
would maintain an ice-water interface or if they needed to pour water onto the road to maintain
the slick water-ice interface. The heat diffusing into the ice-covered road per unit area from
time t = 0 to t = τ , is given by
τ
k(T0 − T1 )
√
dt
παt
0
where k, T0 , T1 , α and of course π are all constants. τ is the contact time.
Z
Q=
Compute this integral above integral.
Problem 24. An ant moves in a a straight line with the velocity v(t) = cos(t) (m/s). Find
the displacement and distance traveled over the time interval [0, 3π] seconds.
Problem 25. An oil storage tank is leaking oil at a rate of r(t) = 4π(t − 2)2 liters per hour.
How much oil leaks out between t = 1 and t = 2 hours?
Problem 26. A population of rabbits at time t increases at a rate of 40 − 12t + 3t2 rabbits
per year. Find the population after 2 years if there are 2 rabbits at t = 0.
Problem
R 427. If the units for t are hours and the units of Q are gallons per hour, then the
units for 0 Q(t)dt are
.
√
Problem 28. Water flows into an empty tank at a rate of 100 + 16 t gals/hour. What is
the quantity of water in the tank after 4 hours?
Z
Problem 29.
Let F (x) =
x
sin2 (t) dt. Evaluate the limit
0
F (x)
.
x→0 x2
lim
Problem 30.
Find f given that
f 0 (x) = sin(x) − sec(x) tan(x),
f (π) = 1.