Theorems and Defs

Question 1

lim x→a [f(x)]^n =

Answer 1

= [lim x→a f(x)]^n

Where n is a positive integer

Question 2

lim x→a c =

Answer 2

= c

Question 3

lim x→a x =

Answer 3

= a

Question 4

lim x→a x^n =

Answer 4

= a^n

Where n is a positive integer

Question 5

lim x→a n√x =

Answer 5

= n√a

Where n is a positive integer

Question 6

lim x→a n√f(x) =

Answer 6

= n√ [lim x→a f(x)]

If n is positive, we assume that the lim x→a f(x) > 0.

Question 7

What is the (ε, δ)-definition of a limit?

Answer 7

(∀ε > 0) ∃ (δ > 0) ∋ (∀x ∈ D) (0 < |x-a| < δ ⇒ | f(x) - L | < ε)

Question 8

When is a function continuous?

Answer 8

We say that f(x) is continuous on an interval i if f(x) is continuous at each x ∈ i, though you might need to consider right/left continuity at the end points.

This is essentially a domain question in disguise.

Question 9

What is continuity?

Answer 9

If lim x→a f(x) = f(a)

Question 10

What are the three things implied by the definition of a limit being continuous at a number?

i.e. lim x→a f(x) = f(a)

Answer 10

1. f(a) is defined; i.e. a is in the domain of f
2. lim x→a f(x) actually exists
3. lim x→a f(x) = f(a)

Question 11

When can we say that a function is discontinuous or has a discontinuity at a?

Answer 11

When the function is defined near a (defined on an open interval containing a, though maybe not at a) but is not continuous at a.

Question 12

What can we say about continuity and the first five limit laws?

Answer 12

That if f and g are continuous at a and c is constant, then the limit laws are all continuous as well.

Question 13

Which types of functions are continuous on their domains?

Answer 13

1. Polynomials and rational functions.
2. Root functions
3. Trig and inverse trig functions
4. Exponential and logarithmic functions

Question 14

What is Theorem 8?

Answer 14

If f is continuous at b and lim x→a g(x) = b, then
lim x→a f [g(x)] = f(b)

lim x→a f [g(x)] = f [lim x→a g(x)]

Question 15

What is Theorem 9?

Answer 15

It applies continuity to Theorem 8;

i.e. a continuous function of a continuous function is a continuous function

Question 16

What is the Intermediate Value Theorem (IVT)?

Answer 16

Suppose that f is continuous on I = [a,b] and f(a) ≠ f(b);

then for any number N between f(a) and f(b) ∃ c ∈ [a,b] ∋ f(c) = N

Question 17

What are the three kinds of discontinuities?

Answer 17

Jump, removable, and infinite

Question 18

What is left continuous?

Answer 18

lim x→a- f(x) = f(a)

Question 19

What is right continuous?

Answer 19

lim x→a+ f(x) = f(a)

Question 20

d/dx arcsin(x) = ?

Answer 20

= 1 / sqrt(1 - x^2)

Question 21

d/dx arccos(x) = ?

Answer 21

= -1 / sqrt(1 - x^2)

Question 22

d/dx arctan(x) = ?

Answer 22

= 1 / (1 + x^2)

Question 23

d/dx b^x = ?

Answer 23

= ln(b) * b^x

Question 24

d/dx log-b (x) = ?

Answer 24

= 1/ [x * ln(b)]

Question 25

d/dx tan(x) = ?

Answer 25

= sec^2(x)

Question 26

d/dx cot(x) = ?

Answer 26

= -csc^2(x)

Question 27

d/dx csc(x) = ?

Answer 27

= -csc(x) * cot(x)

Question 28

d/dx sec(x) = ?

Answer 28

= sec(x) * tan(x)

Question 29

How does tan related to cos and sin?

Answer 29

tan(x) = sin(x) / cos(x)

Question 30

What is the MVT?

Answer 30

Suppose that f(x) is continuous on [a,b] and differentiable on (a,b), then ∃ at least one c ∈ (a,b) ∋ f'(c) = f(b) - f(a) / b-a

Question 31

What is the definition of a derivative?

Answer 31

f'(x) = lim h→0 [ f(x+h) - f(x) / h ] | ...as long as the limit exists!

Question 32

How do we calculate a linear approximation?

Answer 32

L(x) = f'(a) * (x-a) + f(a)

Question 33

What is the distance formula?

Answer 33

D(x,y) = sqrt [ (x1-x2)^2 + (y1-y2)^2 ]