Math 1310 Lab 1.
Name/Unid:
Lab section:
1. (Algebra Review)
(a) Factor the following
x2 − 5x + 6 (1 pt)
(b) Evaluate:
(−32)(3/5) (1 pt)
(c) Write the following expression without any denominator
1/2 1/3
8x
(1 pt)
y −6
(d) Solve for x. Remember when multiplying by a negative, the inequality switches
direction.
x+3
≥ 1 (2 pts)
3x + 3
Solution:
(a) (x − 2)(x − 3)
(b) -8
(c) 2x1/6 y 2
(d) If x > −1, then
x + 3 ≥ 3x + 3
0 ≥ x
so x ∈ (−1, 0].
If x < −1, then we get 2x ≥ 0, which cannot happen, so we only get the one
interval.
2. (Balancing a Spring) When you pull a spring for a length x, it stores an energy which
is proportional to x2 , i.e. we have an elastic energy Eel = 12 kel x2 , where kel is a constant
kg
depending on the material and on the shape of the spring (its unit of measure is ms
2,
i.e. kilograms over meters times seconds squared). It is also known that a body of mass
m has a gravitational energy Eg = mg (remember, g = 9.81 sm2 ).
Now suppose a body of mass m is hung to a spring with elastic constant kel . The mass
is going to pull the spring until the two energies balance each other, i.e. when
Eel = Eg
kg
(a) Suppose that you hang a body with mass 10kg to a spring with constant kel = 4 ms
2.
How much is the spring pulled (i.e., how much is x)? (2 pts)
(b) Now suppose that you have a body of mass 5kg which gives an elongation (i.e. x)
of 0.2m to the spring. How much is kel ? (2 pts)
Remember to include the units of measure!
Solution:
(a) 7.00m
kg
(b) 2452.5 ms
2
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3. (Logarithms and Exponentials) Solve for x in the following equations.
(a) ln(x) = 7 (1 pt)
(b) e2x+1 = 2e2 (1 pt)
(c) ln(x2 − 1) − ln(x − 1) = 1 (1 pt)
(d) ln(x − 1) + ln(x + 2) = ln(10) (2 pts)
Solution:
(a) x = e7
(b)
e−2 e2x+1 = 2
e2x−1 = 2
2x = ln(2) + 1
1
1
x =
ln(2) +
2
2
(c)
ln((x + 1)(x − 1)) − ln(x − 1)
ln(x + 1)
x+1
x
=
=
=
=
1
1
e
e−1
(d)
ln(x2 − x − 2) = ln(10)
x2 + x − 12 = 0
(x + 4)(x − 3) = 0
x = 3, x = −4 are solutions of the quadratic equation. x = 3 is the only solution
of the problem, since x = −4 is not admissible for the domain of the logarithms.
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4. (Fun With Functions)
(a) Consider the following graph of f (x) = x2 . Without using technology, sketch the
graph of f (x + 3) and f (x) − 1. (2 pts)
(b) Consider the following function
(
x2
f (x) =
−x + 3
if x ≤ 3
if x > 3.
Evaluate f (0), f (3) and f (4). (2 pts)
(c) Let f (x) = x3 − 1, and g(x) = 3x + 2. Find f ◦ g(x), and evaluate f ◦ g(0). (2 pts)
Solution:
(a) The graph of f (x + 3) will be the original graph of f (x) shifted to the left by
three units.
The graph of f (x) − 1 will be the original graph shifted down by one unit.
(b) f (0) = 0, f (3) = 9 and f (4) = −1.
(c)
f ◦ g = (3x + 2)3 − 1
= 27x3 + 54x2 + 36x + 7
and we have f ◦ g(2)=7.
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5. (Trigonometry Review)
(a) Convert from degrees to radians (1 pt)
• 180 degrees
• 25 degrees
Convert from radians to degrees (1 pt)
• 1
• π/4
(b) Draw the unit circle, label the following angles and write their cos, sin and tan (1
pt):
•
•
•
•
3π/4
5π/6
−π/2
7π/6
(c) Find all values of x such that cos(2x) = cos(x) for 0 ≤ x ≤ 2π. (2 pts)
Solution:
(a) Convert from degrees to radians
• 180 degrees = π
• 25 degrees = 25 ·
2π
360
Convert from radians to degrees
180
π
• π/4 = 45
• 1=
(b)
(c) x = 0, 2π/3, 4π/3, 2π
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