Blog Post 4: Derivatives, etc.

1. A function is continuous at x= a if

1. the limit exists at x = a
2. f(a) exists
3. limit at x=a equals f(a)

This function is not continuous at x=1

The limit of x=1 exists

The function exists at f(1)

The limit of 1 is 2, which does not equal f(1) which equals 0

Since the limit and the output of the function are not the same at x=1, the function is not continuous at x=1.

2. Intermediate Value Theorem:

1. solution of f(x)=3x-1 on interval [0,4]

f(0) = -1

f(4) = 11

     Since f is continuous on [0, 4] and f(0) = -1 < 0 < 11 = f(4), then there exists c in [0, 4] such that f(c) = 0.

     2.  solution of f(x)=|4x-7|+12 on interval [7,10]

f(7) = 33

f(10) = 45

     Since f is continuous on [7, 10] and f(7) = 33 > 0 < 45 = f(10), then it cannot be concluded that there exists c in [7, 10] such that f(c) = 0. 

3. The derivative of a function is the slope of the tangent line at x = a. Derivatives can be found using the limit as h approaches 0 of the difference quotient or the limit as x approaches a of the slope formula. 

derivative of f(x) = 2x-6 at x = 4

Difference quotient:

Slope formula:

The hardest part is remembering to do all of the steps.

4. Instantaneous velocity is the slope of the tangent line at one point while the average velocity is over an interval.