Fall 2009 Math 151
Night Before Drill for Exam I
3. A box is held in plae by a able on a fritionfree ramp as shown below. If the mass of the
box is 50kg, nd the magnitude of the tension
in the able.
ourtesy: David J. Manuel
(overing 0.1-3.2 & App D)
1
Setion 0.1
x
1. Given f (x) =
, nd and simplify
x+1
f (f (x)) and state its domain.
2. A tank ontains 2000 liters of pure water. A
brine solution ontaining 20 grams of salt per
liter of water is pumped into the tank at a 4 Setion 1.2
rate of 40 liters per minute. Write a funtion
C(t) whih represents the onentration of the
1. Find the area of the triangle whose versolution, in grams per liter, after t minutes.
ties are at the points A(−1, 2), B(2, 1) and
1
1
3. A box with a square base and no top is to have
C(0, 5). (NOTE: A = bh or A = ab sin θ)
2
2
a volume of 200 ubi entimeters. Find a for2. Let a = −2i + 3j and b =< 1, 2 >. Find the
mula for the surfae area of the box as a funsalar and vetor projetion of a onto b.
tion of the length of one side of the square.
2
3. A 10 kg suitase sits atop the ramp of a ruise
ship. The ramp is 4 meters tall and is attahed 2 meters (horizontally) away from the
dok. Assuming no frition, nd the work
done by gravity in sliding the suitase from
the top of the ramp to the bottom.
Appendix D
1. Convert 240◦ to radians and nd the exat
trigonometri ratios for this angle.
2. Solve for x: cos 2x − sin x = 0.
3
π
3. If sin x = and 0 < x < , nd all other 5 Setion 1.3
4
2
trigonometri ratios of x.
1. Find parametri equations of the line whih
passes through the points (−3, −1) and
(−1, 5) .
3
Setion 1.1
2. Find a Cartesian equation of the urve
parametrized by x = cos t, y = cos 2t and
sketh the graph.
1. Find a unit vetor whih points in the same
diretion as the vetor from the point (−1, 5)
to the point (2, 3).
2. A woman walks due west on a ship at a rate of
4 miles per hour. The ship is moving northeast (N 45◦ E ) at a speed of 20 miles per hour.
Find the diretion and speed of the woman
relative to the water.
1
6
2. Use the limit
√ denition to nd the derivative
of f (x) = 2x − 3
Setion 2.2-2.3
1. Compute the following limits:
3. Given the graph of f passes through the point
(−1, 4) and the equation of the line tangent
to f at this point is y = 5x + 9, ompute
2
2x − 13x + 15
x2 − 3x − 10
1
1
t − 3
i + (2t − 3)j
(b) lim
t→3 t − 3
1
() lim x2 cos 2 + 5
x→0
x
2x
(d) lim+
x→2 4 − x2
(a) lim
x→5
7
lim
x→−1
4. Given the graph of f below, sketh the graph
of f ′
Setion 2.5
1. Determine the values of x for whih the funtion below is not ontinuous. Explain your
answers.
if x ≤ −1
 x+2


|x − 1|



if − 1 < x < 1
x−1
f (x) =
0
if x = 1


 −x2

if 1 < x < 3


−2x − 3 if x ≥ 3
if x < 3
x−c
2. Let f (x) =
Find the
3c − x if x ≥ 3
value of c that makes f ontinuous at x = 3.
10
√
1
x
2. Given f (3) = −2 and f ′ (3) = 4, nd g ′ (3) if
g(x) = x2 f (x)
x3 + 1
3. Given f (x) = 2
, nd the equation of the
x +1
tangent line at the point where x = −1.
4. Find the points on the graph of y = x2 + x
where the tangent line also passes through the
point (2, −10).
Setion 2.6
4x2 + 3x + 5
x→∞ −2 − x + 5x2
p
2. Compute lim x2 + 3x + 1 − x.
1. Compute lim
x→∞
√
4x2 + x − 1
3. Compute lim
.
x→−∞
5x − 3
9
Setion 3.2
1. Find the derivative of f (x) = 3x− 2 x+ √ .
3. Show that the equation x3 − 2x2 + x = 5 has
a solution.
8
f (x) − 4
.
x+1
Setion 2.7, 3.1
1. Given r(t) = (3t)i + (4t − t2 )j, use the limit
denition to nd a vetor tangent to the
graph at the point where t = 1.
2