1. Equal derivatives Verify that the functions ƒ 1 x 2 = tan 2 x and g 1 x 2 = sec 2 x have the same

1. Equal derivatives Verify
that the functions ƒ1×2
=
tan2 x and g1x2
=
sec2 x have
the same derivative. What can you say about the difference ƒ –
g?
Explain. 50.
100-m speed The Jamaican sprinter Usain Bolt set a world record
of 9.58 s in the 100-meter dash in the summer of 2009. Did his speed ever
exceed 37 kmhr
during the race? Explain.

2. Condition for nondifferentiability Suppose
ƒ_1x2
6
0
6
ƒ_1x2, for
x
6
a, and
ƒ_1x2
7
0
7
ƒ_1x2, for
x
7
a.
Prove that ƒ
is
not differentiable at a. (Hint:
Assume
ƒ
is
differentiable at a, and apply the Mean Value
Theorem to ƒ_.)
More generally,
View complete question »

1. Equal derivatives Verify
that the functions ƒ1×2
=
tan2 x and g1x2
=
sec2 x have
the same derivative. What can you say about the difference ƒ –
g?
Explain. 50.
100-m speed The Jamaican sprinter Usain Bolt set a world record
of 9.58 s in the 100-meter dash in the summer of 2009. Did his speed ever
exceed 37 kmhr
during the race? Explain.

2. Condition for nondifferentiability Suppose
ƒ_1x2
6
0
6
ƒ_1x2, for
x
6
a, and
ƒ_1x2
7
0
7
ƒ_1x2, for
x
7
a.
Prove that ƒ
is
not differentiable at a. (Hint:
Assume
ƒ
is
differentiable at a, and apply the Mean Value
Theorem to ƒ_.)
More generally, show that if ƒ_ and ƒ_ change
sign at the same point, then ƒ is not
differentiable at that point.

 

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