ec2104 lecture 2

a function is _____ when small changes in independent variable x leads to _____ changes in the ____________

a function is continuous is small changes in independent variable x leads to small changes in the function value y

geometrically a function is continuous if it has no ______

geometrically a function is continuous if it has no breaks

in this definition, the 3 conditions are:

1. the function f must be defined at _______

2. the _______________ must exist

3. the limit must = ___

1. the function f must be defined at x = a

2. the limit of f(x) which is x → a must exist

3. the limit must = f(a)

if f and g are continuous at a then (f + g) and (f - g) must both be continuous at a

if f and g are continuous at a then fg and f/g must both be continuous at a

[f(x)^r] is continuous at a only if:

1. _____ is defined

2. r is a ___ _____

1. [f(a)]^r is defined

2. r is a real number

if f has an ______ on the interval l then its inverse f^-1 is ________ on ______

if f has an inverse on the interval l then its inverse f^-1 is continuous on f(l)

INTERMEDIATE VALUE THEOREM: let f be a function which is continuous upon (a,b)

• if f(a) and f(b) have ______ ____, then there is at least _ __ in (a, b) such that __ = 0

• if f(a) and f(b) have different signs, then there is at least one c in (a, b) such that f(c)= 0 (meaning c is a root)

INTERMEDIATE VALUE THEOREM: let f be a function which is continuous upon (a,b)

• if f(a) __ f(b), then for every intermediate value y in the open interval between __ and __ there is at least one c in (a,b) such that f(c) = _

• if f(a) =/= f(b), then for every intermediate value y in the open interval between f(a) and f(b) there is at least one c in (a,b) such that f(c) = y

strengths of IVT

1. tells you that at least 1 root exists

2. the range of a continuous function over an interval is itself an interval

limitations of

1. doesn’t tell you how many roots

2. doesn’t tell you how to get the roots

5 steps to using the IVT:

1. D_____ the function: Let f(x) be …

2. E________ f(x) c_________: The polynomial f(x) is continuous for all x

3. Choose an i_______: For x in [-2, 1]

4. Check for s____ r_______: f(-2) = -4.2 and f(1) = -0.6

5. Invoke IVT: So IVT implies that there exists at least 1 number c in [-2,1] such that f(c) = 0

5 steps to using the IVT:

1. Define the function: Let f(x) be …

2. Establish f(x) continuous: The polynomial f(x) is continuous for all x

3. Choose an interval: For x in [-2, 1]

4. Check for sign reversal: f(-2) = -4.2 and f(1) = -0.6

5. Invoke IVT: So IVT implies that there exists at least 1 number c in [-2,1] such that f(c) = 0

a function f(x) is differentiable at x=a as long as the limit in the definition of the derivative exists