Quiz 4 Oct. 31, 2007

Calculus I Name:

TA/classroom: Student ID:

Quiz 4

Oct. 31, 2007

1. (10 pts) Given function f (x) = ¹_x (for x 6= 0), compute the f⁰(2) by definition (f⁰(a) = lim

h→0

f (a + h)− f(a)

h ).

f⁰(2) = lim

h→0

1 h( 1

2 + h− 1 2)

= lim

h→0

1

h(2− (2 + h) (2 + h)2 )

= lim

h→0

1

h( −h (2 + h)2)

=− 1 4 2. (10 pts) Given function f (x) =√

x + 1 (for x≥ −1), compute the f⁰(x) by definition (f⁰(x) = lim

h→0

f (x + h)− f(x)

h ).

f⁰(x) = lim

h→0

√x + h + 1−√ x + 1 h

= lim

h→0

√x + h + 1−√ x + 1

h ·

√x + h + 1 +√ x + 1

√x + h + 1 +√ x + 1

= lim

h→0

(x + h + 1)− (x + 1) h(√

x + h + 1 +√ x + 1)

= lim

h→0

√ 1

x + h + 1 +√ x + 1

= 1

2√ x + 1