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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

212 Chapter 2 • The Derivative and Its Properties

NOTE The Earth is not perfectly round; it In Example 9, the acceleration of the ball is constant. In fact, acceleration is
bulges slightly at the equator, and its mass is the same for all falling objects at the same location, provided air resistance is not
not distributed uniformly. As a result, the taken into account. In the sixteenth century, Galileo (1564–1642) discovered this by
acceleration of a freely falling body varies ∗
slightly. experimentation. He also found that all falling bodies obey the law, stating that the
distance s they fall when dropped is proportional to the square of the time t it takes to
fall that distance, and that the constant of proportionality c is the same for all objects.
That is,
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2
s = −ct
The velocity v of the falling object is
ds d 2
v = = (−ct ) = −2ct
dt dt
and its acceleration a is
2
dv d s
a = = = −2c
dt dt 2
1
which is a constant. Usually, we denote the constant 2c by g so c = g. Then
2
1 2
a = −g v = −gt s = − gt
2

The number g is called the acceleration due to gravity. For our planet, g is
2
2
2
approximately 32 ft/s , or 9.8 m/s . On the planet Jupiter, g ≈ 26.0 m/s , and on
2
our Moon, g ≈ 1.60 m/s .
2.4 Assess Your Understanding
Concepts and Vocabulary Skill Building
1. True or False The derivative of a product is the product In Problems 9–40, find the derivative of each function.
of the derivatives. PAGE x 2 x
205 9. f (x) = xe 10. f (x) = x e
′
2. If F(x) = f (x)g(x), then F (x) = .
3
4
2
11. f (x) = x (x − 1) 12. f (x) = x (x + 5)
d
n
3. True or False x = nx n + 1 , for any integer n. PAGE
2
dx 205 13. f (x) = (3x − 5)(3x + 4) 14. f (x) = (3x − 2)(5x + 3)
4. If f and g 6= 0 are two differentiable functions, 5 3
d f (x) 15. s(t) = (2t − t)(t − 2t + 1)
then = .
4
2
2
dx g(x) 16. F(u) = (u − 3u + 1)(u − u + 2)
e x
3
2
5. True or False f (x) = can be differentiated using the 17. f (x) = (x + 1)(ln x + 1) 18. f (x) = (x + 1)(ln x + x)
x 2
e x 2s z + 1
e and using the
Quotient Rule or by writing f (x) = = x −2 x 19. g(s) = 20. F(z) =
x 2 s + 1 2z
Product Rule. 1 − 2u 1 − w 2
d 1 21. G(u) = 22. f (w) =
6. If g 6= 0 is a differentiable function, then = . 1 + 2u 1 + w 2
dx g(x)
2
3
4x − 2 −3x − 1
PAGE
′′ .
7. If f (x) = x, then f (x) = 207 23. f (x) = 24. f (x) =
2
3x + 4 2x + 1
8. When an object moving on a line is modeled by the position
function s = s(t), then the acceleration a of the object PAGE 1 1
207 25. f (w) = w − 1 26. g(v) = v + 5v − 1
2
3
at time t is given by a = a(t) = .
In a famous legend, Galileo dropped a feather and a rock from the top of the Leaning Tower of
∗
Pisa, to show that the acceleration due to gravity is constant. He expected them to fall together,
but he failed to account for air resistance that slowed the feather. In July 1971, Apollo 15
astronaut David Scott repeated the experiment on the Moon, where there is no air resistance.
He dropped a hammer and a falcon feather from his shoulder height. Both hit the Moon’s surface
at the same time. A video of this experiment may be found at the NASA website
(https://moon.nasa.gov).
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