FINAL REVIEW
(2) Compute the area of the region between
the vertical lines x = 0 and x = 1,
and between the x-axis and the graph of
y = x3 .
(1) True or false
(a) If f and g are differentiable, then
d
[f (x) + g(x)] = f 0 (x) + g 0 (x).
dx
(b) If f is differentiable, then
f 0 (x)
dp
f (x) = p
.
dx
2 f (x)
(c) If f and g are differentiable, then
(3) Find lim
d
[f (x)g(x)] = f 0 (x)g 0 (x).
dx
x→1+
(d) If f and g are continuous on [a, b],
then
Z b
Z b
Z b
f (x)g(x)dx =
f (x)dx
g(x)dx .
a
a
x2 −9
x2 +2x−3
(4) Find the absolute maximum and absolute minimum values of f (x) = x4 −2x2 +
3 on [−2, 3].
a
(e) If f is continuous on [a, b], then
Z b
Z b
xf (x)dx = x
f (x)dx.
a
a
(5) Find the area of the region bounded by
the curves y = x2 − x − 6 and y = 0
(f) If f and g are continuous and f (x) ≥
g(x) for a ≤ x ≤ b, then
Z b
Z b
f (x)dx ≥
g(x)dx.
a
a
1
(6) Find the points on the graph y = x2 at
which the tangent line is parallel to the
line y = 6x − 1.
(8) Find the volume of the solid obtained by
rotating the region bounded by x2 −y 2 =
a2 and x = a + h about the y-axis (where
a and h are positive constants).
(7) Find the local extrema of f (x) = 2x6 −
6x4 . Find the intervals on which the
graph of f is concave up or concave
down, and find the x-coordinates of the
points of inflection. Sketch the graph.
(9) Find lim
v→4+
4−v
|4−v|
(10) Find an antiderivative for f (x) =
(11) Find the derivative of xecot x
(12) Evaluate
2
R
sin π+x
5 dx
√ 1
1−x2
(16) Find the area of the region bounded by
the curves y = ex − 1, y = x2 − x, and
x = 1.
(13) Find the length of the curve
3
1
y = (x2 + 4) 2
6
for 0 ≤ x ≤ 3.
(17) If y = x3 + 2x and
x = 2.
(14) Find lim
x→∞
dx
dt
= 5, find
dy
dt
when
√ ln x
x+ln x
(18) The altitude of a triangle is increasing
at a rate of 1cm/min while the area of
the triangle is increasing at a rate of
2cm2 /min. At what rate is the base of
the triangle changing when the altitude
is 10cm and the area is 100cm2 ?
(15) Find the local extrema of f (x) = x ln x.
Determine where f is increasing or decreasing. Discuss the concavity, find the
points of inflection, and sketch the graph
of f .
3
1−2x2 −x4
4
x→−∞ 5+x−3x
(b) The y-axis
(19) Find lim
(20) Set up, but do not evaluate, an integral
for the volume of the solid obtained by
rotating the region bounded by y = x3
and y = x2 about y = 1.
(c) y = 2
(24) Find the absolute maximum and absolute minimum values of f (x) = xe−
[−1, 4]
(21) Differentiate h(θ) = etan 2θ
(22) Evaluate
R1
0
(1 − x9 )dx
1
(23) Find the volumes of the solids obtained
by rotating the region bounded by the
curves y = x and y = x2 about the following lines:
(a) The x-axis
(25) Find the derivative of f (x) = (ln x) 5
4
x2
8
,
(26) Two cars start moving from the same
point. One travels south at 60mi/h and
the other travels west at 25mi/h. At
what rate is the distance between the
cars increasing two hours later?
(29) Find the equation for the line tangent to
8
the curve y = 4+x
2 at the point (2, 1)
(30) Evaluate
R1
0
y(y 2 + 1)5 dy
(31) What is the 99th derivative of xex ?
1
(27) Differentiate y = xe x
(28) Find the volume of the solid obtained by
rotating the region bounded by y = 2x
and y = x2 about the x-axis.
t3
3
t→0 tan 2t
(32) Find lim
(33) Evaluate
5
R1
0
eπt dt
√
R √
3
(38) Compute ( x− x4 +
(34) Find the derivative of f (x) = xln x
(35) if A is a positive constant, then find
e2x +ex
lim Ae
2x −ex
7
√
3 2
x
−6ex +1)dx
(39) Find the volume of the solid obtained by
rotating the region bounded by x = 0
and x = 9 − y 2 about x = −1.
x→−∞
(36) √
Determine the area under the curve y =
a2 − x2 and between the lines x = 0
and x = a.
2x
x −2x
3
x→∞
(40) Find lim
(41) Evaluate
(37) Each side of a square is increasing at a
rate of 8cm/s. At what rate is the area
of the square increasing when that area
of the square is 16cm2 ?
6
R
√ x+2 dx
x2 +4x
(42) Verify that the function satisfies the hypotheses of the Mean Value Theorem on
the given interval. Find all numbers c
that satisfy the conclusion of the Mean
Value Theorem.
(44) RUse a
sin 2x
√
substitution
dx
1+cos 2x
to
evaluate
f (x) = e−2x , [0, 3]
ex −x2
2x +e−x
e
x→∞
(45) Find lim
(43) Find the local extrema of f (x) = 5 −
7x − 4x2 and the intervals on which f is
increasing or is decreasing, and sketch a
graph of f .
(46) Find the extrema and sketch the graph
2
of f (x) = x +2x−8
x+3
7
√
(47) Evaluate
R
e√ x
dx
x
(49) Differentiate H(v) = v arctan v
(50) The interior of a half-mile track consists
of a rectangle with two semicircles at two
opposite ends. Find the dimensions that
will maximize the area of the rectangle.
(48) A man wishes to put a fence around a
rectangular field and then subdivide this
field into three smaller rectangular plots
by placing two fences parallel to one of
the sides. If he can afford only 1000 yards
of fencing, what dimensions will give him
the maximum area?
√
x
√e x
x→∞ e +1
(51) Find lim
(52) Evaluate
R
x3
dx
1+x4
(53) Find the derivative of f (x) = x2 ex
8
Answers
9
(22) 10
(23) (a) 2π
15
(b) π6
(c) 8π
15
(24) f (2) =
(1) True or false
(a) True
(b) True
(c) False
(d) False
(e) False.
(f) True.
(2) 0.25
(3) −∞
(4) f (3) = 66, f (±1) = 2
(5) 125
6
(6) Since the slope of y = 6x − 1 is 6, the
slope of the tangent line must be 6. Thus
the derivate 2x = 6, x = 3. The desired
point is (3, 9).
√
(7) f (0) = 0 is a maximum, f (± 2) =
−8) are
q minima, qconcave up on
(−∞, −
6
5)
and (
q q
down on (− 65 , 65 ).
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
4
3 π(2ah
6
5 , ∞),
(25)
f 0 (x)
√2 ,
e
f (−1) = −
1
=
1
1
e8
4
5x(ln x) 5
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
concave
3
+ h2 ) 2
−1
arcsin x
ecot x (1 − x csc2 x)
−5 cos π+x
5 +C
15
2
0
Min: f (e−1 ) = −e−1 , increasing on
[e−1 , ∞), decreasing on (0, e−1 ), concave
up on (0, ∞), no points of inflection.
(16) e − 11
6
(17) 70
(18) −1.6cm/min
(19) 31
R1
(20) 0 π[(1 − x3 )2 − (1 − x2 )2 ]dx
(21) h0 (θ) = 2 sec2 (2θ)etan 2θ
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
9
65mi/h.
1
y 0 = e− x (1 + x1 )
64π
15
y = − 12 x + 2
21
4
xex + 99ex
1
8
1 π
π (e − 1)
2xln x−1 ln x
−1
πa2
4
2
48cm
√ /s 3 √
√
3
2
x
3
3
7
3 x − 7 x + 21 x − 6e + x + C
1656π
5
0
√
x2 + 4x + C
c = − 12 ln( 61 (1 − e−6 ))
Max: f (− 78 ) = 129
16 , increasing on
7
(−∞, − 8 ), decreasing on [− 87 , ∞)
√
− 1 + cos 2x + C
0
no√extrema
2e x + C
125 yards by 250 yards
v
H 0 (v) = 1+v
2 + arctan v
1
radius is 8π mi and length is 18 mi.
1
1
4
4 ln |1 + x | + C
f 0 (x) = x(x + 2)ex