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20. a. 34.3 m>
s
b. 39.2 m>
s
c. 54.2 m>
s
21. 0.29 min and 1.71 min
22. 20 ms >
23. 11, 32 and 11, 32
y
24.
25. a. i.
26.
B10, 02
ii.
3
2
1
y
–1
0
1 2 3 4
–1
–2
–3
a 1 5 , 1 5 b
3
2
1
–3 –2 –1
0
1 2 3
–1
–2
–3
a 1 4 , 13 4 b
(0, 3)
iii. a 1 and 15, 472
3 , 103
27 b
b. At these points, the slopes of the
tangents are zero, meaning that the
rate of change of the value of the
function with respect to the domain
is zero. These points are also local
maximum and minimum points.
x y 1
P1a, b2 is on the curve; therefore,
a 0, b 0.
y 1 x
y 1 2x x
dy
dx 1 2 Q2x1 2 1R
At x a. Slope is
1 1 a
1 .
a a
But, a b 1
b a 1
Therefore, slope is b
a b a .
x
x
27. The x-intercept is 1 1 as n Sq,
n ,
1
and the x-intercept approaches 1.
n S 0,
As n Sq, the slope of the tangent at
(1, 1) increase without bound, and the
tangent approaches a vertical line
having equation x 1 0.
28. a.
2x, if x 6 3
f ¿1x2 e
1, if x 3
f ¿132 does not exist.
b.
9
8
7
6
5
4
3
2
1
y
–2 –1
0
1 2 3 4
6x, if x 6 2 or x 7 2
f ¿1x2 e
6x, if 2 x 2
f ¿Q2R and f ¿Q2R do not exist.
7
6
5
4
3
2
1
–3 –2 –1
0
1 2 3
–1
1, if x 7 1
1, if 0 6 x 6 1
c. f ¿1x2 µ
1, if 1 6 x 6 0
1, if x 6 1
f ¿102, f ¿112, and f ¿112 do not exist.
y
x
x
Section 2.3, pp. 90–91
1. a. 2x 4
b. 6x 2 2x
c. 12x 17
d. 45x 8 80x 7 2x 2
e.
f.
8t 3 2t
6
1x 32 2
2. a. 15x 12 3 1515x 12 2 1x 42
b. 15x 2 13x 2 4213 x 3 2 4
6x13 x 3 2 5
c. 8x11 x 2 2 3 12x 62 3
611 x 2 2 4 12x 62 2
d. 61x 2 92 4 12x 12 2
8x1x 2 92 3 12x 12 3
3. It is not appropriate or necessary to
use the product rule when one of the
factors is a constant or when it would
be easier to first determine the product
of the factors and then use other rules
to determine the derivative. For
example, it would not be best to use
the product rule for f 1x2 31x 2 12
or g1x2 1x 121x 12.
4. F ¿1x2 3b1x243c¿1x24
3b¿1x243c1x24
5. a. 9 d. 36
b. 4
e. 22
c. 9
f. 671
6. 10x y 8 0
7. a.
b.
114, 4502
11, 02
8. a. 31x 12 2 1x 421x 32 2
1x 12 3 1121x 32 2
1x 12 3 1x 42321x 324
b. 2x13x 2 42 2 13 x 3 2 4
x 2 3213x 2 4216x2413 x 3 2 4
x 2 13x 2 42 2
3413 x 3 2 3 13x 2 24
9. 4.84 Lh >
10. 30; Determine the point of tangency,
and then find the negative reciprocal
of the slope of the tangent. Use this
information to find the equation of the
normal.
3
2
1
–3 –2 –1 0 1 2 3
–1
y
x
NEL
Answers 631