Definitions

Question 1

What is the definition of continuity?

Answer 1

Question 2

What is the statement of sequential continuity?

Answer 2

Question 3

What is the Intermediate Value Theorem?

Answer 3

Question 4

What is the statement of existence and continuity for the inverse of a continuous and strictly increasing function f?

Answer 4

Question 5

What is the statement of boundedness of continuous functions?

Answer 5

Let f: [a, b] -> ℝ be continuous, then f is bounded.

Question 6

What is the statement of the radius of convergence?

Answer 6

If Σa_nxⁿ is a power series one of the following three properties hold:

(i) The series only converges if x = 0
(ii) The series converges for all real numbers x
(iii) There is a positive number R with the property that the series converges if |x| < R and diverges if |x| > R

Question 7

What is the statement of continuity of power series?

Answer 7

Question 8

Define the exponential function exp(x)

Answer 8

Question 9

Recall the two inequalities for the exponential referenced in lectures.

Answer 9

1. 1 + x =< e^x for all real x
2. e^x =< 1/(1-x) if x < 1

Question 10

What is the statement of the existence of the logarithm?

Answer 10

There is a continuous strictly increasing function defined on the interval (0, ∞) such that:

(i) e^log(x) = x
(ii) log(e^y) = y

We have that log(uv) = log(u) + log(v)

Question 11

What is the definition of powers?

Answer 11

Question 12

What is the definiton of limits?

Answer 12

Question 13

If f: I -> R is defined on the open interval I and c ∈ I then f is continuous at c is an equivalent statement to what?

Answer 13

Question 14

What is the definition of the derivative from first principles?

Answer 14

Question 15

What are the sum and product rules for derivatives?

Answer 15

Question 16

What is the chain rule for derivatives?

Answer 16

Question 17

What is the derivative g’ of a function f: (a, b) -> R with a positive derivative? A function being differentiable with a positive derivative implies its inverse is differentiable.

Answer 17

g = f^-1

g’(x) = 1/f’(g(x))

Question 18

What is the statement of Rolle’s theorem?

Answer 18

Suppose f: [a, b] -> R is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and that f(a) = f(b), Then there is a point c in the open interval where f’(c) = 0.

Question 19

What is the statement of the Mean Value Theorem.

Answer 19

Suppose f: [a, b] -> R is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there is a point c ∈ (a, b) where

f’(c) = (f(b) - f(a))/(b - a)

Question 20

What is the statement of the radius of convergence of Σna_nx^n-1 ?

Answer 20

Let Σa_nxⁿ be a power series of radius of convergence R. Then the series Σna_nx^n-1 has the same radius of convergence.

Question 21

What is the characteristic property of the exponential?

Answer 21

If x and y are real numbers then exp(x + y) = exp(x) exp(y).

Question 22

What is the definition of cos(x)?

Answer 22

For x ∈ R:

cos(x) = 1 - x²/2 + x⁴/24 - … + (-1)^k(x^2k)/(2k)! + …

Question 23

What is the definition of sin(x)?

Answer 23

For x ∈ R:

sin(x) = x - x³/6 + x⁵/120 - … + (-1)^k(x^2k+1)/(2k+1)! + …

Question 24

What are the trigonometric addition formulae?

Answer 24

cos(x + y) = cos(x)cos(y) − sin(x)sin(y)

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

Question 25

What is L'Hôpital's rule?

Answer 25

If f, g : I -\> **R** are differentiable on the open interval I containing c and f(c) = g(c) = 0: lim_x->cf(x)/g(x) = lim_x->cf'(x)/g'(x)

Question 26

What is the statement of Taylor's theorem with the Lagrange remainder?

Answer 26

If f: I -\> **R** is n times differentiable on the open interval I containing a and b then: f(b) = f(a) + f'(a)(b - a) + (f''(a)/2)(b - a)² + ... + ((f^(n-1)(a)/(n-1)!)/(n-1)!)(b-a)^n-1 + (f⁽ⁿ⁾(t)/n!)(b-a)ⁿ for some point t between a and b.