Jan 21 - Jan 25

Question 1

Continuity Definition

Answer 1

A function f is continous at a number a if

limf(x) = f(a)
x->a

this means

1. f(a) is defined
2. limf(x) = f(a)
   x->a
3. lim f(x) = f(a)
   x->a
   `

Question 2

When is a function continuous from the left and when is it continuous from the right?

Answer 2

A function f is continuous from the right at a number a if

lim f(x) = f(a)
x->a+

and f is continuous from the left at a if

lim f(x) = f(a)
x->a

Question 3

Define When A Function Is Continuous on an Interval

Answer 3

If it is continuous at every number in the interval,

(If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left)

Question 4

Continuity Theorem

Answer 4

If f and g are continuous at a and c is a constant, then the following functions are also continuous at a

1. f + g
2. f - g
3. cf
4. fg
5. f/g if g(a) DOES NOT = 0

A) Any polynomial is continuous everywhere; that is, it is continuous on R = (-infinity, infinity)

b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Question 5

What type of Functions are continuous at every number in their domains?

Answer 5

Polynomials, rational functions, root functions, trig functions, inverse trig functions, exponential functions, logarithmic functions.

Question 6

If f is continuous at b and lim g(x) = b while x approaches a, in other words this means….

Answer 6

lim f(g(x)) = f (lim g(x))
x->a               x->a

Question 7

Composite Function Theorem

Answer 7

If g is continuous at a and f is continuous at g(a), then the composite function f o g given by ( f o g )(x) = f(g(x)) is continuous at a.

This theorem is often expressed informally by saying a continuous function of a continuous function is a continuous function.,

Question 8

The Intermediate Value Theorem

Answer 8

Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) is not equal to f(b). Then there exists a number in c in (a, b) such that f(c) = N.

This theorem states that a continuous function takes on every intermediate value between the function values of f(a) and f(b).

Question 9

Formula TO Calculate Average Velocity

Answer 9

s (t) = 4.9t^ 2

avg velocity = change in position / time elapsed

Question 10

Intuitive Definition of A Limit

Answer 10

Suppose f(x) is defined when x is near the number a.

lim f(x) = L
x-> a

Question 11

Heaviside Function H is defined by

Answer 11

H(t) = {0 if t < 0

{1 if t >/ 0

Question 12

Definition of One-Sided Limits

Answer 12

lim f(x) = L
x->a -

“the left-hand limit of f(x) as x approaches a is equal to L if we can make the values of f(x) arbitrarily close to L “

Question 13

Intuitive Definition of an Infinite Limit

Answer 13

Let f be a function defined on both sides of a, except possibly at a itself. Then

lim f(x) = infinity 
x-> a

Let f be a function defined on both sides of a, except possibly a itself. Then

lim f(x) = -infinity 
x-> a

Question 14

Vertical Asymptote

Answer 14

The vertical line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:

lim f(x) = infinity 
x->a

lim f(x) = - infinity 
x->a

lim f(x) = infinity 
x->a-

lim f(x) = - infinity 
x->a- 

lim f(x) = infinity 
x-> a+

lim f(x) = - infinity
x->a+