Chapter 9 and 11

Question 1

the derivative of a function f : R → R

Answer 1

f’: S→R

f’= lim y→x f(y)-f(x)/y-x

Question 2

K is a D-Domain and f: K→R is differentiable at x.
x is differentiable if there exists an f(x)= r:K→R and f’(x)=R s.t.
2 conditions (Theorem 9.2.2)

Answer 2

1. lim x→0 r(y)/y-x =0

2. for all y in K, f(y)=f’(x)(y-x)+f(x)+r(y)

Question 3

Rolle’s Theorem. Usual setup. Assume a<b></b>

Answer 3

1. f(a)=f(b)
   2.f is continuous on [a,b]
   3.f is differentiable on (a,b) then
   There exists a c in (a,b) s.t. f’(c)=0

Question 4

Mean Value Theorem

Answer 4

Usual setup. Let a,b be elements of R s.t. [a,b] is contained in K. Sps f is continuous on [a,b] and differentiable on (a,b). Then, there exists a c in [a,b] s.t. f’(c)=f(b)-f(a)/b-a

Question 5

Darboux’s Theorem

Answer 5

Let K be a D-Domain and f be differentiable on [a,b] contained in K. Then, f’ satisfies the IVP on [a,b]

Question 6

a Riemann sum for a function f corresponding to a partition P

Answer 6

For a function f defined on [a, b], a partition P of [a, b] into a
collection of subintervals [x0, x1], [x1, x2], · · · , [xn−1, xn], and for each i = 1, 2, · · · , n, a point x∗i in [xi−1, xi] the sum of f(x∗i)(xi − xi−1) from i=1 to n is call a ________ _____ for f determine by the partition P. Let |P| = max {xi − xi−1 for all
i = 1, 2, · · · , n} denote the longest length of all the subintervals.

Question 7

the Riemann integral of a function f over [a, b]

Answer 7

For all epsilon greater than 0 there exists a delta s.t. if |||P||-0|

Question 8

D-Domain

Answer 8

a subset K contained in R is a _________ if K is the union of countably many “non-degenerate” disjoint intervals

Question 9

differentiable at a point

Answer 9

Let K be a D-Domain and f: K → R, x is an element of K. f is ________ at x if the limit as y→x of f(y)-f(x)/y-x exists

Question 10

differentiable on S

Answer 10

If f is _________ at every point in a set S contained in K, then f is _________ on S

Question 11

lim x→0 r(y)/y-x=0 implies that

Lemma 9.2.4 Usual setup

Answer 11

r(y)→0 as y approaches x where y-x is the distance from the tangent line to the curve

Question 12

If f is differentiable at x, then

Answer 12

f is continuous at x.

Question 13

usual setup. g: K→R. f and g are differentiable at x s.t. x=k then, 3 things constant, add/subtract, product rule (Theorem 9.2.6)

Answer 13

1. for all k in R, (kf)’(x)= k*f’(x)
2. (f+ or -g)’(x)= f’(x) + or - g’(x)
3. (f*g)’(x)= f’(x)g(x)+f(x)g’(x)

Question 14

Chain Rule Theorem

Answer 14

Let K and K’ be D-Domains. Sps f: K→R and g: K’→R. f[K] is contained in K’

Question 15

Chain Rule. Sps f is differentiable at x in K and g is differentiable at f(x) then

Answer 15

the derivative of the composition of f in g(x) =g’(f(x))*f’(x)

Question 16

Let (a,b) be elements of R. Let f: (a,b)→R and a constant function. Then,

Answer 16

the derivative at x is 0 for all x in (a,b)

Question 17

Global maximum

Answer 17

Let X be a metric space, f:X→R, z be an element of X. Then, f(z) is a __________ for a function f if f(z) is greater than or equal to f(x) for all x in X

Question 18

Global minimum

Answer 18

Let X be a metric space, f:X→R, z be an element of X. Then, f(z) is a __________ for a function f if f(z) is less than or equal to f(x) for all x in X

Question 19

Local maximum

Answer 19

for f if there exists an r>0 s.t. f(z) is greater than or equal to f(x) for all x in an open ball radius r about z

Question 20

Local minimum

Answer 20

for f if there exists an r>0 s.t. f(z) is less than or equal to f(x) for all x in an open ball radius r about z

Question 21

Local Extreme Theorem (9.4.2)

Answer 21

Usual setup. If f is differentiable at some c in intK where c is a local extremum, then f’(c)=0

Question 22

Interior

Answer 22

open ball around x fully contained in the set

Question 23

Corollary of the MVT. Usual setup. Let I be contained in K be an interval. Then

Answer 23

f is constant iff its derivative is 0 for all x in I

Question 24

Usual setup. Let I be an interval in K. g:K→R. Then, (Corollary 9.4.8)

Answer 24

f’(x)=g’(x) for all x in I iff f(x)=g(x)+c where c is constant

Question 25

Antiderivative

Answer 25

Let K be a D-Domain. F: K→R is an _______ for f:K→R if F’(x)=f(x) for all x in K

Question 26

Corollary 9.4.8 implies that if f has an antiderivative then

Answer 26

it has infinitely many

Question 27

Theorem 9.5.2 Usual setup interval I is contained K. Sps f is differentiable on I. Then, 2 things

Answer 27

1. f is increasing on I iff f’(x) is greater than or equal to 0
2. f is decreasing on I iff f’(x) is less than or equal to 0

Question 28

Theorem 9.5.3 Usual setup. Let K be a D-Domain. f: K→R is differentiable on [a,b] contained in K . If f’(x) is not 0 for all x in [a,b], then

Answer 28

f is injective on the closed interval

Question 29

Lemma 9.5.4 Let K be a D-domain, f is differentiable on [a,b] contained in K. If f’(a) and f’(b) are opposite signs, then

Answer 29

there exists a c in (a,b) s.t. f’(c)=0

Question 30

Intermediate Value Property

Answer 30

Let I be an interval on R. f: I→R has the _____ ______ _____ if for all a and b in I and N is b/w f(a) and f(b) then there exists a c in I s.t. f’(c)=N

Question 31

Uniqueness of the Integral

Answer 31

Let a<b></b>

Question 32

||P|| Mesh

Answer 32

maximum size of subintervals because the widths of the subintervals can be different

Question 33

Let a<b></b>

Answer 33

a) the integration of f from c to c is defined to be 0

b) the negative integration of f from b to a

Question 34

Let a<b></b>

Answer 34

f is bounded because the range is a bounded set

Question 35

Translation Invariance Theorem

Answer 35

Let a<b></b>