Theoretical Test 2

Question 0

Lipschitz Continous

Answer 0

Let f : S->R be a function such that there exists a number K such that for all x and y in S we have |f(x)

Question 1

Uniform Continuity

Answer 1

Let S ⇢ R. Let f : S -> R be a function. Suppose that for any e > 0 there exists a delta>0 such that whenever x,c in S and |x-c|<delta then |f(x)-f(c)|<e

Question 2

Derivative Definition

Answer 2

Let I be an interval, let f : I->R be a function, and let c be in I. Suppose that the limit L := lim [f(x)-f(c)]/(x-c) exists. Then we say that f is differentiable at c and we say that L is the derivative of f at c and we write f’(c) := L.

Question 3

Relative Max and Min

Answer 3

Let S ⇢ R be a set and let f : S->R be a function. The function f is said to have a relative maximum at c in S if there exists a delta>0 such that for all x in S such that |x-

Question 4

Partition Definition

Answer 4

A partition P of the interval [a,b] is a finite set of numbers {x0,x1,x2,…,xn} such that a=xo<xn=b. We write delta(xi)=xi-x(i-1)

Question 5

Darboux Sums

Answer 5

(lower integral from b to a)f(x) dx := sup{L(P, f) : P a partition of [a,b]}
(upper integral from b to a) f(x) dx := inf{U(P, f) : P a partition of [a,b]}.

Question 6

Continuity and Integrable functions

Answer 6

Let f : [a,b] -> R be a continuous function. Then f in R[a,b]

Question 7

Linearity of Integrable Functions

Answer 7

Let f and g be integrable and a be in R.

Then a*f(x) is integrable and you can pull the constant out of the integral.

f+g is integrable and you can split the integrals

Question 8

Taylor’s Theorem

Answer 8

Suppose f : [a, b] -> R is a function with n continuous derivatives on [a, b] such that f(n+1) exists on (a, b). Given distinct points x0 and x in [a, b], we can find a point c between x0 and x such that f(x)=Pn(x)+f^(n+1)(c)/(n+1)!^n+1

Question 9

Mean Value Theorem

Answer 9

Let f : [a, b]->R be a continuous function differentiable on (a, b). Then there exists a point c in (a, b) such that
f(b)

Question 10

Rolle’s Theorem

Answer 10

Let f : [a, b]->R be continuous function differentiable on (a, b) such that f(a) = f(b) = 0. Then there exists a c in (a,b) such that f’(c) = 0

Question 11

Inverse Function

Answer 11

Let f be a strictly monotone continuous function on an interval I which is differentiable at x0 in I with f’(x0) not = 0. Then the inverse function g is differentiable at y0 = f(x0) and
g’(y0)= 1/f’(x0)

Question 12

Chain Rule

Answer 12

Let I1,I2 be intervals, let g: I1->I2 be differentiable at c in I2, and
f : I2->R be differentiable at g(c). If h: I1->R is defined by h(x) := (f of g)(x) = f(�g(x))�,
then h is differentiable at c and h’(c)=f’(g(c))g’(c)

Question 13

cauchy criterion

Answer 13

let f:[a,b]->R be bounded. It is integrable if and only if for all e>0, there exists a partition such that U(P,f)-L(P,f)<e

• choose subinterval size by (b-a)/n
• to solve, take sum from k=1 to n of infimum-supremums

Question 15

Integrable Definition

Answer 15

f:[a,b]->R be bounded such that lower integral = upper integral. Then it’s Riemann integrable

Question 16

proof that if a derivative is 0, the function is a constant

Answer 16

choose x,y in I such that x(c)-0 and results that f(x)=f(y)

Question 17

proof of quotient rule

Answer 17

use definition of derivate [f(+h)-f(x)/h], expand, then add and subtract f(x)g(x), simplify.

Question 18

proof “if f is differentiable at c, it is continuous at c”

Answer 18

we know f’(c)=lim[f(x)-f(c)]/x-c and lim(x-c)=0 exist. then f(x)-f(c)=[[f(x)-f(c)]/x-c](x-c). then limf(x)-f(c) exists and lim(f(x)-f(c))=[[f(x)-f(c)]/x-c](x-c)=f’(c)*0=0, thus limf(x)=f(c), so f is continuous at c

Question 19

proof that a lipschitz continuous function is uniform continuous

Answer 19

Let f:s->R be a function and let K be a constant such that for all x,y in S we have |f(x)-f(y)|0. Take delta=e/K. For all x,y in S such that |x-y|<Kdelta=Ke/K=e. Thus f is uniformly continuous