Chapter 5

Question 1

Differentiability-

Let g: A to R be a function defined on an interval A. Given c in A, the derivative of g at c is…

Answer 1

g’(c) = lim as x goes to c of

[g(x) - g(c)] / (x - c)

provided the limit exists.

Question 2

g’(c) = lim as x goes to c of

[g(x) - g(c)] / (x - c)

provided the limit exists.

Answer 2

Differentiability-

Let g: A to R be a function defined on an interval A. Given c in A, the derivative of g at c is…

Question 3

Algebraic Differentiability Theorem- Let f and g be functions defined on an interval A, and assume both are differentiable at some c in A, then…

Answer 3

i. ) (f + g)’(c) = f’(c) + g’(c)
ii. ) (kf)’(c) = k*f’(c) for k in R
iii. ) (f*g)’(c) = f’(c)*g(c) + f(c)*g’(c)

Question 4

i. ) (f + g)’(c) = f’(c) + g’(c)
ii. ) (kf)’(c) = k*f’(c) for k in R
iii. ) (f*g)’(c) = f’(c)*g(c) + f(c)*g’(c)

Answer 4

Algebraic Differentiability Theorem- Let f and g be functions defined on an interval A, and assume both are differentiable at some c in A, then…

Question 5

Chain Rule-

Let f: A to R and g: B to R satisfy f(A) c B so that the composition g(f(x)) is defined. If f is differentiable at c in A, and if g is differentiable at f(c) in B, then…

Answer 5

g(f(x)) is differentiable at c with

[g(f(c))]’ = g’(f(c))*f’(c)

Question 6

g(f(x)) is differentiable at c with

[g(f(c))]’ = g’(f(c))*f’(c)

Answer 6

Chain Rule-

Let f: A to R and g: B to R satisfy f(A) c B so that the composition g(f(x)) is defined. If f is differentiable at c in A, and if g is differentiable at f(c) in B, then…

Question 7

Interior Extremum Theorem-

Let f be differentiable on an open interval (a,b). If f attains a maximum (or minimum) value at some point c in (a,b)…

Answer 7

then f’(c) = 0.

Question 8

then f’(c) = 0.

Answer 8

Interior Extremum Theorem-

Let f be differentiable on an open interval (a,b). If f attains a maximum (or minimum) value at some point c in (a,b)…

Question 9

Darboux’s Theorem-

If f is differentiable on an interval [a,b] and if a’ satifies

f’(a) < a’ < f’(b), then…

Answer 9

there exists a point c in (a,b) where f’(c) = a’.

Question 10

there exists a point c in (a,b) where f’(c) = a’.

Answer 10

Darboux’s Theorem-

If f is differentiable on an interval [a,b] and if a’ satifies

f’(a) < a’ < f’(b), then…

Question 11

Rolle’s Theorem-

Let f: [a,b] to R be continuous on [a,b] and differentiable on (a,b). If f(a) = f(b), then…

Answer 11

there exists a point c in (a,b) where f’(c) = 0.

Question 12

there exists a point c in (a,b) where f’(c) = 0.

Answer 12

Rolle’s Theorem-

Let f: [a,b] to R be continuous on [a,b] and differentiable on (a,b). If f(a) = f(b), then…

Question 13

Mean Value Theorem-

If f: [a,b] to R is continuous on [a,b] an differentiable on (a,b), then…

Answer 13

there exists a point c in (a,b) where

f’(c) = [f(b) - f(a)] / (b - a)

Question 14

there exists a point c in (a,b) where

f’(c) = [f(b) - f(a)] / (b - a)

Answer 14

Mean Value Theorem-

If f: [a,b] to R is continuous on [a,b] an differentiable on (a,b), then…

Question 15

If g: A to R is differentiable on an interval A and satisfies g’(x) = 0 for all x in A, then…

Answer 15

g(x) = k for some k in R.

Question 16

g(x) = k for some k in R.

Answer 16

If g: A to R is differentiable on an interval A and satisfies g’(x) = 0 for all x in A, then…

Question 17

If f and g are differentiable on an interval A and

f’(x) = g’(x)

for all x in A, then…

Answer 17

f(x) = g(x) + k for some k in R.

Question 18

f(x) = g(x) + k for some k in R.

Answer 18

If f and g are differentiable on an interval A and

f’(x) = g’(x)

for all x in A, then…

Question 19

Generalized Mean Value Theorem-

If f and g are continuous on the closed inteval [a,b] and differentiable on the open interval (a,b), then…

Answer 19

there exists a point c in (a,b) where

[f(b) - f(a)]*g’(c) = [g(b) - g(a)]*f’(c)

or, if g’ is not 0,

f’(c) / g’(c) = [f(b) - f(a)] / [g(b) - g(a)]

Question 20

there exists a point c in (a,b) where

[f(b) - f(a)]*g’(c) = [g(b) - g(a)]*f’(c)

or, if g’ is not 0,

f’(c) / g’(c) = [f(b) - f(a)] / [g(b) - g(a)]

Answer 20

Generalized Mean Value Theorem-

If f and g are continuous on the closed inteval [a,b] and differentiable on the open interval (a,b), then…

Question 21

L’Hospital’s Rule for 0/0 Case-

Let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of a. If f(a) = g(b) = 0 and g’(x) is not 0 for all x not a, then

Answer 21

lim as x goes to a of

f’(x) / g’(x) = L

implies lim as x goes to a of

f(x) / g(x) = L.

Question 22

lim as x goes to a of

f’(x) / g’(x) = L

implies lim as x goes to a of

f(x) / g(x) = L.

Answer 22

L’Hospital’s Rule for 0/0 Case-

Let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of a. If f(a) = g(b) = 0 and g’(x) is not 0 for all x not a, then

Question 23

Given g: A to R and a limit point c of A, we say that limit x to c g(x) = infinity if…

Answer 23

for every M > 0, there exists a δ > 0 such that whenever

0 < |x - c| < δ

it follows that g(x) > M.

Question 24

for every M > 0, there exists a δ > 0 such that whenever

0 < |x - c| < δ

it follows that g(x) > M.

Answer 24

Given g: A to R and a limit point c of A, we say that limit x to c g(x) = infinity if…

Question 25

L’Hopital’s Rule for inf/inf Case-

Assume f and g are differentiable on (a,b) and that g’(x) is not 0 for all x in (a,b). If lim x goes to a of g(x) = infinity, then…

Answer 25

lim as x goes to a of

f’(x) / g’(x) = L

implies that lim x goes to a of

f(x) / g(x) = L.

Question 26

lim as x goes to a of

f’(x) / g’(x) = L

implies that lim x goes to a of

f(x) / g(x) = L.

Answer 26

L’Hopital’s Rule for inf/inf Case-

Assume f and g are differentiable on (a,b) and that g’(x) is not 0 for all x in (a,b). If lim x goes to a of g(x) = infinity, then…