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Derivatives, Epsilon-Delta Proof of Continuity and Integrals

A) If x, y > 0, then ln x - ln y = ln x
¯¯¯¯
ln y

b) If f'(a) = 0 and f"(a) = 0, then the function f does not have an extreme point at x = a.

c) For every real number x, we have ln(e^x²-¹) = (x - 1)(x + 1)

(e has the exponent x²-¹)

d) If there exists &#1028; > 0, such that for every &#948; > 0, we have &#9474;f(x) - 5&#9474; < &#1028; whenever
&#9474;x - 5&#9474;< &#948;, then lim f(x) = 5
x&#8594;3

e) If F(x) is an antiderivative of f(x), then G(x) = 3F + 5x +7 is an antiderivative of g(x) = 3f(x) + 5

Solution Preview

True or False. Please explain the answers:

a) If x, y > 0, then ln x - ln y = ln x
¯¯¯¯
ln y

It is False, as the correct equality is
lnx-lny=ln (x/y) for all x, y>0.

b) If f'(a) = 0 and f"(a) = 0, then the function f does ...

Solution Summary

Derivatives, Epsilon-Delta Proof of Continuity and Integrals are investigated.

$2.19