21-120 Dierential and Integral Calculus
Questions for Quiz 5:
1. Evaluate: log4
2. Evaluate: log3
1
128
1
81
3. Evaluate: log5 (125)
4. Determine the derivative and factor out any common factors: y = e
x2
sin(ln(x).
5. Determine the derivative and factor
An approach to solving applied extrema problems:
1. Read the problem carefully to gain an understanding of the
problem and the quantities involved. Decide what is being asked.
2. Draw a diagram if possible, and identify given and unknown
quantities on the
1. (16 points) Determine the derivative of each of the following and factor out any
common factor:
(a) f (5”) 2 (134-2?) tame?) :2 2 2x
4‘04) :— (4 {3:54am x?) + (xi x) a SQ ﬁx 3
2 4 4
M‘Q‘W‘Cﬁ h
QICXB : (K2 “2%
i \< gAB HAM/AL 3. (12 points) Determine
Suppose that c is a constant and that the limits
lim f (x) and
x!a
lim g(x)
x!a
exist. Then:
The rst six rules say the limits behave as expected under algebraic combination:
1. lim [f (x) + g(x)] = lim f (x) + lim g(x)
x!a
2. lim [f (x)
x!a
x!a
x!a
g(x)]
Inverse functions: A function has an inverse if it is a one-to-one function, i.e. if it passes
the horizontal line test.
The graph of a function and of its inverse are reections of each other in the graph of the
line y = x:
y
.
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21-120 Dierential and Integral Calculus
Questions for Quiz 2:
Some of the questions on this quiz will be useful in word problems to come later.
5
1. If sec() = , determine sin(2).
3
2. Determine the equation of the line through (2, 5) and perpendicular to
Everything you need about exponentials and logarithms:
An exponential function:
For a a positive constant, an = a a a a (n factors)
a0 =
a
n
a1 =
m
an =
=
ax+y =
ax
y
(ax)y =
(ab)x =
=
If a > 1, then lim ax =
and lim ax =
If 0 < a < 1, then lim ax =
and l
21-120 Dierential and Integral Calculus
Questions for Quiz 3:
Factor out any common factors from your answers:
1. Determine the derivative: f (x) = (x2 + 4x + 2)6
2. Determine the derivative: g(x) =
x2
x+4
3. Determine the derivative: f (x) = (2x2
4. Dete
Test 2 Review
21-120 Test 2 will be Wednesday, November 4 in McConomy Auditorium.
Exit via the left front.
Linear approximation (Actually relative change relationships): 3.10 no. 39, 40
plus three more below:
Stevens Law as applied to the sensation of l
05X -.)(,&,o~.
god-9 (1w) =32 éM/L
ll: (,L "l of" j! 7 M
m (1&4 ? u A Mqu w)
a?
5
=9
55 N
:9
:9
59,
=9 06x
=9
3
3
5
a
3
Q
Q
a
9
9
9
9 ,
9
D
D
D
E
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Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-120 Dierential and Integral Calculus
Test 1 will be Wednesday, September 30.
The test will be in McConomy Auditorium at the usual lecture time of 8:30. There will be
plenty of room - spre