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Problem Set 2 ANSWERS
Due 9:00AM Sep. 2
1. Graph the following functions
Please come to my office hours if you have any questions on this.
(a)
f (x) =
−1 if 0 ≤ x ≤ 1
2x if 1 < x < 3
(b)
h(x) =
x − 3 if 0 ≤ x < 2
2 − 3x if −3 < x ≤ −1
2. Compute the indicated values of the given function
(a) f (x) = 3x2 + 5x − 2; f (1), f (0), f (−1)
f (1) = 6, f (0) = −2, f (−1) = −4
(b) g(x) = x + x1 ; g(1), g(2)
g(1) = 2, g(2) = 52
(c) h(x) = x − |x − 2|; h(1), h(2), h(3)
h(1) = 0, h(2) = 2, h(3) = 2
(d) t(x) = x2 + 3x + 5; t(0), t(3), t(−3)
t(0) = 5, t(3) = 23, t(−3) = 5
(e) f (x) = 3x2 + x1 + 5; f (x − 1)
1
f (x − 1) = 3(x − 1)2 + x−1
+5
3. We will practise some multiplication and division:
(a) Multiply 5x2 y 4 by 4yx6
= 20x8 y 5
(b) Divide 6x2 y 3 by 2yx5
2
= 3y
x3
(c) Divide
= 6x
y4
3x2
y
(d) Simplify
=
by
xy 3
2 .
12xy 3
.
2x2 y 2
6y
x
1
4. Factorize the following, simplifying wherever possible:
(a) 3x + 6xy
= 3x(1 + 2y)
(b) 2y 2 + 7y
= 2y(2y + 7)
(c) x(x2 + 8) + 2x2 (x − 5) − 8x
= x2 (3x − 10)
(d) 3x(x + x4 ) − 4(x2 + 3) + 2x
= x(2 − x)
5. Factorize the following quadratics:
(a) x2 + 5x + 6
= (x + 3)(x − 2)
(b) y 2 − y − 12
= (y − 4)(y + 3)
(c) 2x2 − 5x − 12
= (x − 4)(2x + 3)
(d) x2 − 49
= (x − 7)(x + 7)
6. Fun with square roots:
√
√
(a) Show that 2 × 18 = 6
=
=
√
√
2 × 18
36
=6
(b) Show that
√
√
245 = 7 5
√
49 × 5
√
=7 5
=
(c) Show that
15
√
3
√
=5 3
√
15 3
=
√3
=5 3
(d) Simplify
= x2
√
3
√2x .
8x
7. Laws of Indices and Logs
2
2
3
(a) Without using a calculator, simplify ( 27
8 )
9
=4
(b) Without using a calculator, evaluate log2 64
=6
(c) Without using a calculator, evaluate log10 100
=3
(d) Simplify log10 a2 + 31 log10 b − 2 log10 ab
= − 35 log10 b
8. Equations and Inequalities. Solve the following:
(a) 5(3 − y) < 2y + 3
12
7 >y
(b) |9 − 2x| = 11
x = −1, 10
(c) |x + a| < 2, where a is a parameter and 0 < a < 2
−2 − a < x < 2 − a
(d) x2 − 8x + 12 < 0
2<x<6
9. Find the inverse function for the relationship between Celcius and Farenheight, where f (x) =
9
5 x + 32
f −1 (x) = 95 (x − 32).
10. S&B 2.8 (a) and (e) (p 21)
(a) Use substitution to identify the y intercept: y = 2x + 3
Since the slope is 2, then we have m = 2 immediately. Since the y-intercept is (0, 3), we plug
in this point to y = mx + b to obtain 3 = 2(0) + b, so b = 3.
(e) Use y1 − y2 = m(x1 − x2 ) to identify the y intercept: y = x + 1
Plugging in points (2, 3) and (4, 5) to y = mx + b, we have 3 = 2m + b and 5 = 4m + b.
Subtracting the first equation from the second,
5 − 3 = 4m + b − (2m − b)
2 = 2m
m = 1.
Using (2, 3) in y = 1x + b, we have 3 = 1(2) + b, which yields b = 1.
11. Find the value of r such that vA = vB
3
(1)
(2)
(3)
zA = 1 − δvB
zB = 1 − δvA
1
vA = pzA + prδvA + qδvA
2
vB = qzB + 2p(1 − r)δvB .
Solving for vA and vB , we obtain
p
1 + pδ − prδ − 2q δ
q
vB =
.
1 + qδ + 2pδ(r − 1)
vA =
Setting these two values equal, we have
vA = vB
q
p
q =
1 + pδ − prδ − 2 δ
1 + qδ + 2pδ(r − 1)
Now cross multiply so you get:
q
p[1 + qδ + 2pδ(r − 1)] = q[1 + pδ − prδ − δ]
2
And now solving for r is trivial
q − p + 2p2 δ −
r=
2p2 δ + pqδ
12. Solve the following equation (calculate a numeric answer):
(ex )2 = 3e4
e2x = 3e4
e2x−4 = 3
2x − 4 = ln 3
ln 3 + 4
x =
= 2.549
2
4
q2 δ
2
.