The Derivative + velocity

What is the definition of a derivative? (given its condition?)

What is an alternative definition?

What is the formula for finding the derivative of f at any a?

Definition:

IF:

lim [f(x)-f(a)]/x-a exists,

x→a

THEN:

f ’(a) = lim [f(x)-f(a)]/x-a

x→a

• f’(a) = derivative of a function f at a number a (“f prime of a”)

• If the limit does NOT exist, then we say - “the function is not differentiable at the number a”

Alternative definition:

f’(a) = lim [f(a+h)-f(a)]/h

h→0

• (let h=x-a)

Formula for finding the derivative of f at any a:

f’(x) = lim [f(x+h)-f(x)]/h

h→0

Conceptually, what is a derivative?

The slope at any point of your curve

What does it mean when f is differentiable at a?

f’(a) exists

(derivative of function f exists at a)

When is f differentiable on an open interval? Which ones?

f is differentiable on an open interval IF it is differentiable at every number in the interval

• (a,b)

• (a,∞)

• (-∞,a)

• (-∞,∞)

If f(x)=x²-2x, find a formula for f’(x)

answer

What is the Leibnitz notation for f’(x) and f’(a)?

Leibnitz notation for f’(x) and f’(a):

d/dx (f(x)) = f’(x)

df/dx = f’(x)

df/dx | x=a = f’(a)

How would you know if f is continuous at a from the function’s differentiability?

What would you write to prove that f is continuous at a based on the function’s differentiability?

image

How can functions FAIL to be differentiable?

1. If lim x→a [f(x)-f(a)]/x-a is infinite (can also be on one side only too)

2. graph is not smooth (sharp lines)

3. Vertical tangent line

How would you prove that f(x) = cube root of x is not differentiable at x=0?

answer

True or false: If a function f(x) is continuous at x=a, then f(x) is differentiable at x=a

FALSE
ex) Just because a function is continuous and a limit exists doesn’t mean its differentiable

ex) f(x)=|x| has sharp lines (no smooth curves), so f(x) is not differentiable even though it is continuous at a point a

Graphically, how can you visually see where a point is not differentiable?

Image

How do you interpret these derivatives?

f’(2)=0

f’(2)>0

f’(2)<0

f’(2)=0 → slope of tangent is 0

f’(2)>0 → slope of tangent is >0 (f is INCREASING near 2)

f’(2)<0 → slope of tangent is <0 (f is DECREASING near 2)