(a) Use both the first and second derivative tests to show that f x = sin 2 x has a relative minimum at x = 0 . (b) Use both the first and second derivative tests to show that g x = tan 2 x has a relative minimum at x = 0 . (c) Give an informal verbal argument to explain without calculus why the functions in parts (a) and (b) have relative minima at x = 0 .
(a) Use both the first and second derivative tests to show that f x = sin 2 x has a relative minimum at x = 0 . (b) Use both the first and second derivative tests to show that g x = tan 2 x has a relative minimum at x = 0 . (c) Give an informal verbal argument to explain without calculus why the functions in parts (a) and (b) have relative minima at x = 0 .
(a) Use both the first and second derivative tests to show that
f
x
=
sin
2
x
has a relative minimum at
x
=
0
.
(b) Use both the first and second derivative tests to show that
g
x
=
tan
2
x
has a relative minimum at
x
=
0
.
(c) Give an informal verbal argument to explain without calculus why the functions in parts (a) and (b) have relative minima at
x
=
0
.
Definition Definition Lowest point, either on the entire domain or on the given range of a function is called minimum. The plural form of 'minimum' is 'minima'.
A driver is traveling along a straight road when a buffalo runs into the street. This driver has a reaction time of 0.75 seconds. When the driver sees the buffalo he is traveling at 44 ft/s, his car can decelerate at 2 ft/s^2 when the brakes are applied. What is the stopping distance between when the driver first saw the buffalo, to when the car stops.
Topic 2
Evaluate S
x
dx, using u-substitution. Then find the integral using
1-x2
trigonometric substitution. Discuss the results!
Topic 3
Explain what an elementary anti-derivative is. Then consider the following
ex
integrals: fed dx
x
1
Sdx
In x
Joseph Liouville proved that the first integral does not have an elementary anti-
derivative Use this fact to prove that the second integral does not have an
elementary anti-derivative. (hint: use an appropriate u-substitution!)
1. Given the vector field F(x, y, z) = -xi, verify the relation
1
V.F(0,0,0) = lim
0+ volume inside Se
ff F• Nds
SE
where SE is the surface enclosing a cube centred at the origin and having edges of length 2€. Then,
determine if the origin is sink or source.
Elementary and Intermediate Algebra: Concepts and Applications (7th Edition)
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