Fall 2009 Math 151
Night Before Drill for Exam III
2. Compute lim cos
x→−∞
3. Find and
simplify the derivative of f (x) =
6
arccos
.
ourtesy: David J. Manuel
x
(overing 4.3 - 6.2)
5
1
1. Compute lim
1. Evaluate log5 10 + log5 20 − 3 log5 2.
x→0
x→0
lim − ln(cos x).
x
2
3. Compute lim 1 +
.
x→∞
x
x→π/2
Setion 4.4
6
1. Given f (x) = x ln x, nd f (e)
3
′
Setion 5.1
1. The graph of the DERIVATIVE of a funtion is shown below. On whih interval(s) is
the original funtion f ...
2. Find
and simplify the derivative of f (x) =
√
x ln x
3. Use logarithmi dierentiation to nd the
(a) dereasing?
(b) onave down?
(x3 + 1)4 sin2 x
√
.
derivative of f (x) =
3
x
3
sin x − x
.
x3
2. Compute lim tan(4x) cot(7x).
2. Solve for x: log(3 − x) + log(x + 4) = 1.
2
Setion 4.8
Setion 4.3
3. Compute
!
√
ex + 3 e−x
.
2ex + 2e−x
−1
Setion 4.5
1. Money is invested in an aount whih earns
interest ompounded ontinuously. Twenty
years later, the investment has tripled in
value. Find the interest rate.
2. A tank ontains 500 liters of salt solution with
10kg of salt. Pure water enters the tank at a
rate of 20 liters per minute, and the solution
is ompletely mixed before leaving the tanka
tthe same rate. When will the solution have
only 1 kg of salt remaining?
2. Sketh a graph with the following properties:
• f ′ (x) > 0 when x < 1; f ′ (x) < 0 when
x>1
4
• f ′′ (x) > 0 when x < −2 and x > 2,
f ′′ (x) < 0 when −2 < x < 2
Setion 4.6
1. Find the value of sin−1 sin
7π
6
.
•
1
lim f (x) = −2, lim f (x) = 0
x→−∞
x→∞
7
11
Setion 5.2
1. Find the ritial numbers of f (x) = x(x −
1)1/3 .
1. Compute
n→∞
Setion 5.3
12
n
i=1
n
Setion 6.2
1. Approximate the area under the graph of
f (x) = 16 − x2 between x = 0 and x = 4
using a partition P = {0, 2, 3, 4} and taking
x∗i = left endpoint. Graph the funtion and
illustrate your approximation on the graph.
2. Determine the intervals where f (x) =
2
e(−1/2)x is inreasing or dereasing and onave up or onave down.
2. Approximate the area under the graph of
f (x) = ln x between x = 1 and x = 7 using 3
equal subintervals and taking x∗i = midpoint.
Setion 5.5
3. Find the exat area under the graph of f (x) =
x2 − 1 between x = 1 and x = 3 using n equal
subintervals (n → ∞) and taking x∗i =right
endpoint.
1. Find the x-oordinate of the point on the line
y = 2x whih is losest to the point (3, 7).
2. A wire 30m long is to be ut into at most 2
piees. The rst piee is bent into a square;
the seond piee is bent into an equilateral
triangle. How should the wire be divided so
that the total area enlosed by the gures is
a maximum?
3. A right irular ylinder is insribed in a
sphere of radius 3 m. Find the volume of
the largest possible ylinder.
10
(i + 2).
2. Use the formulas
on p367
to alulate
"
#
2
n
X
2
2i
lim
1+
−1 .
1. Find the x-oordinates of the inetion points
of f (x) = x5 + 10x4 .
9
4
X
i=1
2. Find the absolute maximum and absolute
minimum of f (x) = sin x + cos x on the interval [0, π]
8
Setion 6.1
Setion 5.7
1. Find the most general antiderivative of
√
1
f (x) = 2 x + 2 .
x
2
2. Given f ′ (x) =
and f (1) = 0, nd
1 + x2
f (x).
3. The aeleration of a partile is given by
a(t) = (1 + et )i + (cos t)j. If the initial veloity is i and the initial position is j, nd the
position of the partile at any time t.
2