Blog Post 4

Continuity

A function is continuous if…

1. The limit exists at x=a

2. f(a) exists. In other words, if there is no hole/asymptote

3. The limit at x=a is equivalent to f(a)

Example where function is not continuous:

x+3, x

f(x)= 7, x=2

x^2+3x +1, x>4

This function is not continuous because limit as x approaches 4 from the left does not equal the limit as x approaches 4 from the right.

Intermediate Value Theorem

Working example:

F(x)=2x^2-16 on interval [1,3]

F(1)=2(1)^2-16 →2-16 → f(x)=-14

F(3)= 2(3)^2-16 = 18-16 = 2

Since f is continuous on [1,3] and f(1)< 0 < f(3), then there exists c in [1,3] such that f(c) = 0

failing example:

f(x)=x2+ x -9 on interval [1,2]

f(1)=(1)2 + 1 -9 = -7

f(2) = (2)^2 + 2 -9 = -3

Since f is continuous on [1,2] and f(1)< 0 >f(4), then it cannot be concluded that there exists c in [1,2] such that f(c) = 0

Derivatives

• Type 1: The derivative that is found using the limit as h approaches 0 of the difference quotient. f(x+h) - f(x)/h.

1. f(x) = 3x + 2
2. 3(x+h) +2 - (3x + 2)/h
3. 3x+3h + 2 - 3x - 2/h
4. 3h/h
5. 3

• Type 2: The derivative that is found using the limit as x approaches a of the slope formula. f(x) - f(a)/ x-a

1. f(x)= 4x- as x approaches 1
2. 4x-4 - (4(1) -4)/x-1
3. (4)(x-1)/x-1
4. 4

A.) A derivative is a function that finds the slope of a curve of a polynomial.

B.) for me, the hardest part of finding a derivative is remembering to distribute the negative.

Instantaneous Velocity vs. Average Velocity

Instantaneous velocity is the tangent slope and one point while the average velocity is the slope of a function over an interval.