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21-114: Calculus for Architecture
Homework #1 Solutions
November 9 , 2004
Mike Picollelli
1.1 #26. Find the domain of g(u) =
u+
4 u.
Solution: We solve
this by considering the terms in the sum separately: is only
u
dened when u 0, and 4 u is only dened wh
21-114 Calculus for Architecture
SOLUTION OF TEST #1
Problem 1. (15 points)
Find the domain of denition of the function
x+4
f (x) = 2
x +x6
Write your answer as a union of intervals.
Solution:
The function f (x) is dened for x + 4 0 and x2 + x 6 = 0.
x +
21-114: Calculus for Architecture
Homework #5 Solutions
December 9, 2004
Mike Picollelli
4.3 #14. Let
f (x) =
x2
.
x2 + 3
(a) Find the intervals on which f is increasing and decreasing.
Solution: To do this, we note that f is increasing exactly where f (x
21-114: Calculus for Architecture
Homework #2 Solutions
November 18, 2004
Mike Picollelli
2.2 #12. Sketch the graph of the following function to determine the values of a for which
limxa f (x) exists.
2 x
if x < 1
f (x) = x
if 1 x < 1
2
(x 1) if x 1
Solut
21-114: Calculus for Architecture
Homework #4 Solutions
December 2, 2004
Mike Picollelli
3.4 #10. Dierentiate
y=
1 + sin x
.
x + cos x
Solution:
dy
d 1 + sin x
=
dx
dx x + cos x
d
d
(x + cos x) dx (1 + sin x) (1 + sin x) dx (x + cos x)
=
(Quotient Rule)
(
21-114: Calculus for Architecture
Homework #3 Solutions
December 2, 2004
Mike Picollelli
3.1 #20. Dierentiate
f (t) =
Solution: First, since
rules, we have
1
t .
t
t = t1/2 , f (t) = t1/2 t1/2 , so by the Dierence and Power
d
d 1/2
(f (t) =
t t1/2
dt
dt
d