Review Questions
MAT V1105 – 001
The final exam is cumulative. Your studying time will be best spent if you work carefully
through this review sheet and go back over previous homeworks and exams.
1. Evaluate the following limits.
(a)
(c)
(e)
(g)
(i)
(k)
lim sin
θ→1
πθ
2
3y 5 − y 2 + 11
5
y→−∞
√1 − y
t2 − 3
lim
t→−2 1 + t + t2 + t3
`2 + ` − 2
lim 2
`→1 ` − 3` + 2
√
2k 2 − (3k + 1) k + 2
lim
k→1
k−1
lim
lim x1/(x−1)
x→1
(b)
(d)
(f )
(h)
(j)
(l)
x−4
− 8x + 16
√
lim w − w2 + 2w
w→∞
√
2z(z − 1)
lim−
z→1
|z − 1|
ln(j 2 + 2j)
lim
j→0+
ln(j)
Z 2x
1
lim
dt
x→∞ x
t
lim
x→4 x2
lim (1 + 2y)1/(2 ln y)
y→∞
2. Consider the piecewise function


if x ≤ 0

2
f (x) = 3 − x if 0 < x ≤ 1


x2 + 1 if x > 1.
Is f continuous at x = 0? Is f continuous at x = 1? Justify your answers.
3. Find a function f that is not continuous, but |f | is continuous.
4. For the following functions f , use the definition of the derivative to determine whether
f is differentiable at 0. If it is differentiable, find f 0 (0).
(
(
(
x2 if x ≤ 0
2x if x ≤ 0
x
if x ≤ 0
(a) f (x) =
(b) f (x) =
(c) f (x) =
3
x if x > 0
x if x > 0
x + 1 if x > 0
5. (a) The cost C (in dollars) to produce g gallons of ice cream can be expressed as
C = f (g). Using units, explain the meaning of the following statements in terms
of ice cream. (i) f (200) = 350; (ii) f 0 (200) = 1.4.
(b) Suppose C(r) is the total cost of paying off a car loan borrowed at an annual
interest rate of r%. What are the units of C 0 (r)? What is the practical meaning
of C 0 (r)? What is its sign?
page 1 of 7
MAT V1105 – 001
Review: page 2 of 7
6. Show how you can represent the following on the graph below.
(a) f (4) (b) f (4) − f (2) (c)
f (5) − f (1)
5−1
(d) f 0 (3)
10
8
6
f (x)
4
2
0
1
2
3
4
5
x
7. Find derivatives for the following functions. Assume a and b are constants.
r
√
√
5
5 √
(a) f (x) = 5x + 5 x + √ −
(b) g(θ) = (cos 1 + tan θ)e
+ 5
x
x
t
−t
(c) h(t) = ee +e
(d) H(z) = ln ln(2z 5 )
eay
(e) T (y) = 2
(f )
u = (v 2 + π)2 (2 − 4v 3 )4
a + yb2
1
(g) M (r) = sin (2r − π)2
(h) P (w) = 2w +e
w
8. Assume that y is a differentiable function of x and find dy/dx. Assume a and b are
constants.
(a)
(c)
x2 y − 2y = 5
ecos(y) = x3 arctan(y)
(b)
(d)
ln x + ln(y 2 ) = 3
sin(ay) + cos(bx) = xy
9. Find the equations of the tangent lines to the following curves at the indicated points.
(a)
(c)
x2
at (4, 2)
xy − 4
x2/3 + y 2/3 = a2/3 at (a, 0)
y2 =
(b)
ln(xy) = 2x at (1, e2 )
(d)
xy + y 2 = 4 where x = 3
10. A spherical mothball is dissolving at a rate of 8π cm3 /h. How fast is the radius of
the mothball decreasing when the radius is 3 cm. (Hint: V = 43 πr3 .)
Review: page 3 of 7
MAT V1105 – 001
11. Find the global extrema for the given functions over the specificed intervals.
(a)
(c)
f (x) = x3 − 12x + 10 on [−10, 10]
√
√
h(y) = y + 1/ y on [1/2, 3/2]
(b)
(d)
g(t) = (t + 1)1/3 on [−2, 7]
q(w) = |w2 − 2w| on [−2, 3]
12. A Norman window is constructed from a rectangular sheet of glass surmounted by a
semicircular sheet of glass. The light that enters through a window is proportional
to the area of the window. What are the dimensions of the Norman window having
a perimeter of 30 feet that admits the most light?
13. A submarine is traveling due east and heading straight for a point P . A battleship
is traveling due south and heading for the same point P . Both ships are traveling at
a velocity of 30 km/h. Initially, their distances from P are 210 km for the submarine
and 150 km for the battleship. The range of the submarine’s torpedoes is 3 km. How
close will the two vessels come to each other? Does the submarine have a chance to
torpedo the battleship.
14. To sketch the graphs of the functions y = f (x) below, follow these steps:
• Determine the domain of f ;
• Determine (if possible) the x and y-intercepts of f ;
• Calculate the derivative dy/dx = f 0 (x). Find all points at which f 0 (x) = 0 or
f 0 (x) is undefined and determine where the curve is increasing and decreasing.
• Determine local maxima and minima.
• Calculate the derivative d2 y/dx2 = f 00 (x). Find all points at which f 00 (x) = 0
or f 00 (x) is undefined and determine where the curve is concave-up and concavedown.
• Find all inflection points.
• Compute lim f (x) and lim f (x). If either of these is finite, then there are
x→∞
x→−∞
horizontal asymptotes.
• If f is not defined at x0 , determine lim+ f (x) and lim− f (x). If either limit
x→x0
x→x0
equals ±∞ then, the line x = x0 is a vertical asymptote.
(a)
f (x) =
√
3
x2 − 1
(b)
f (x) =
x2 − 4
x2 − 1
15. If f (t) is measured in meters per second squared and t is measured in seconds, what
Rb
are the units of a f (t) dt?
16. Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers,
which become less and less efficient as time goes on. The following measurements,
made at the start of each month, show the rate at which pollutants are escaping (in
tons per month) in the gas:
MAT V1105 – 001
Review: page 4 of 7
Time (months)
0 1 2 3 4 5 6
Rate pollutants escape (in tons per month) 5 7 8 10 13 16 20
(a) Make an overestimate and an underestimate of the total quantity of pollutants
the escaped during the six months.
(b) How often would measurements have to be made to find overestimates and underestimates which differ by less that one ton from the exact quantity of pollutants
that escaped during the first six months?
17. Use the graph below to find the values of
Z 4
Z 0
Z 4
Z 4
(a)
f (x) dx
(b)
f (x) dx
(c)
f (x) dx
(d)
|f (x)| dx
−3
0
−4
−4
2
1
f (x)
0
-1
-2
-3
-4
-2
-1
0
1
2
3
4
x
18. For each function below, find the general antiderivative. Check your answers by
differentiation. Assume a and b are constants.
(a)
2x(1 + x2 )5
(b)
(c)
77
−1
5/3
1 + v12
v3
(d)
(e)
Z x
19. Let g(x) =
(f )
1
√
y(1 + y)
√
(ax2 + bx) 2ax3 + 3bx2
√
f (t) dt. Using the graph below, find
0
(a) g(0)
(b) g 0 (1)
z
3
√
(s4 + 1) s5 + 5s
Review: page 5 of 7
MAT V1105 – 001
(c) The interval where g is concave-up.
(d) The value of x where g takes its maximum on the interval 0 ≤ x ≤ 8.
108
95
64
f (t)
27
0
-17
1
2
3
4
5
6
7
t
20. Compute the following definite integrals:
Z π
(a)
cos(x + π) dx
(b)
0
√
Z 8 3t
e
√
(c)
dt
(d)
3 2
t
1
Z 3
(e)
(w3 + 5w) dw
(f )
−1
Z
2
Z1 2
2
2yey dy
z
dz
(1 + z 2 )2
Z0 π/2
sin(θ)e− cos(θ) dθ
0
21. Find the integrals below. Check your answers by differentiation.
Z
Z
5
dy
(a)
(b)
sin(θ) cos(θ) + π dθ
Z y+5
Z
(ln(z))2
(c)
tan(2r) dr
(d)
dz
z
Z
Z
w cos(w2 )
y+1
p
(e)
dy
dw
(f )
2
2
y + 2y + 19
sin(w )
22. Find the derivatives of the following functions:
Z x
Z π
2
(a) E(x) =
arctan(t ) dt
(b) F (t) =
cos(z 5 ) dz
√
Z 0.5
Zt cos(p)
y
200
(1 + u) du
(d) H(p) =
ln(q 3 ) dq
(c) G(y) =
3
ep
8
MAT V1105 – 001
Review: page 6 of 7
23. Evaluate the following integrals.
Z
1
√
dx
(a)
4−x
Z
z cos(z 2 )
p
dz
(c)
sin(z 2 )
Z
p
(e)
(y + 2) 2 + 3y dy
Z
(g)
w2 sin(3w) dw
Z
t3 ln t dt
(i)
Z
et dt
(k)
e2t + 3et + 2
Z
2x3 − 2x2 + 1
dx
(m)
x2 − x
√
Z 3 2
3z + z + 4
(o)
dz
z3 + z
1
Z
1−w
√
(q)
dw
1 − w2
Z
dt
√
(s)
−t2 + 4t − 3
Z
4 dw
(u)
1 + (2w + 1)2
Z
6 dy
(w)
√
y(1 + y)
Z
(y)
(27)2θ+1 dθ
24. Suppose
R1
0
(b)
Z
(d)
et − e−t
dt
et + e−t
(ln w)2 dw
Z
(f )
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(x)
(z)
1
dx
+ 6x + 25
Z
z−1
√
dz
2z − z 2
Z 4
y + 3y 3 + 2y 2 + 1
dy
y 2 + 3y + 2
Z
sin θ dθ
2
cos θ + cos θ − 2
Z
2s + 2
ds
2
(s + 1)(s − 1)2
Z 0
x3 dx
2
−1 x − 2x + 1
Z 1/2
2 − 8y
dy
1 + 4y 2
0
Z
arctan(x) dx
Z eπ/3
dz
z cos(ln(z))
Z1
(1 + ln t) dt
p
t (ln t)(2 + ln t)
Z
sin(2x)ecos(2x) dx
x2
f (t) dt = 3. Calculate the following:
R1
R 1.5
f
(2t)
dt
(b)
f
(1
−
t)
dt
(c)
f (3 − 2t) dt
0
0
1
Rx 1
25. Let F (x) = 2 ln t dt for x ≥ 2.
(a) Find F 0 (x).
(b) Is F increasing or decreasing? What can you say abour the concavity of its
graph?
(c) Sketch a graph of F (x).
(a)
R 0.5
Z
Review: page 7 of 7
MAT V1105 – 001
26. Derive the following formulas:
Z
Z
n x
n x
(a)
x e dx = x e − n xn−1 ex dx
Z
Z
1 n
n
n
(b)
x cos(ax) dx = x sin(ax) −
xn−1 sin(ax) dx
a
a
Z
Z
n−1
1
n−1
n
(c)
cos (x) dx = cos (x) sin(x) +
cosn−2 (x) dx
n
n
27. Calculate the following integrals, if they converge.
Z ∞
Z ∞
1
dx
(b)
te−t dt
(a)
5x
+
2
Z0 ∞
Z1 ∞
dz
dw
(d)
(c)
2
w ln w
2
−∞ z + 25
Z 2
Z 2
dw
1
√
(e)
dx
(f )
4 − x2
1 w ln w
0
Z π
Z 6
1 −√y
dθ
(g)
dy
(h)
√ e
2
y
0
3 (4 − θ)
√
28. The area under 1/ x on the interval 1 ≤ x ≤ b is equal to 6. Find the value of b.
2
29. Calculate the exact area of the region bounded by the graph of y = 21 π3 x and the
graph of y = cos x. The curves intersect at x = ±π/3.
30. Find the volume√of the following solids. The base of a solid is the region between
the curve y = 2 sin x and the interval 0 ≤ x ≤ π on the x-axis. The cross sections
perpendicular to the x-axis are
(a) equilateral triangles with bases running from the x-axis to the curve.
(b) squares with bases running from the x-axis to the curve.
31. Find the volumes of the solids generated by revolving the following regions about the
x-axis.
(a) y = √
x2 , y = 0, x = 2;
(b) y = 9 − x2 , y = 0;
(c) y = e−x , y = 0, x = 0, x = 1.
32. When you have completed all the problems above (and not before), you may view
my solutions at
http://www.math.columbia.edu/~ggsmith/CalcIS/review_solutions.pdf