1. consider f(x) = x 2 what is the slope of the tangent at a=0?

1. Consider f(x) = x1. Consider f(x) = x22

What is the slope of the What is the slope of the tangent at a=0?tangent at a=0?

7

6

5

4

3

2

1

-1

-4 -2 2 4

Consider f(x) = xConsider f(x) = x22

So slope at a local max or min of a smooth curve is

zero.

2xf '(x)f '(0) 2(0) 0

7

6

5

4

3

2

1

-1

-4 -2 2 4

2. f is 2. f is differentiable at differentiable at

a for which a for which graph?graph?

a

a a

a

a

A. B.

C. D.

E.

2. f is 2. f is differentiable at differentiable at

a for which a for which graph?graph?

ANSWER: BANSWER: B

a

a a

a

a

A. B.

C. D.

E.

3. Which 3. Which graph(s) have a graph(s) have a corner, but not a corner, but not a

cuspcusp at x = a? at x = a?

a

a a

a

a

A. B.

C. D.

E.

3. Which 3. Which graph(s) have a graph(s) have a corner, but not a corner, but not a

cuspcusp at x = a? at x = a?

ANSWER:ANSWER:A and EA and E

a

a a

a

a

A. B.

C. D.

E.

4. For graph A, 4. For graph A, what is the left-what is the left-hand derivativehand derivative

at x = a? at x = a?a

a a

a

a

A. B.

C. D.

E.

4. For graph A, 4. For graph A, what is the left-what is the left-hand derivativehand derivative

at x = a? at x = a?

ANSWER:ANSWER:-1-1

a

a a

a

a

A. B.

C. D.

E.

5. For graph E, 5. For graph E, what are the what are the left-hand and left-hand and

right-hand right-hand derivativesderivatives at x = a? at x = a?

a

a a

a

a

A. B.

C. D.

E.

5. For graph E, 5. For graph E, what are the left-what are the left-hand and right-hand and right-

hand derivativeshand derivatives at x = a? at x = a?

ANSWER: ANSWER:

0 and 0 and -∞-∞a

a a

a

a

A. B.

C. D.

E.

(At vertical tangents, as with limits, if we write f’ (a) = it means the slope “approaches ”, but that is not a real number slope.)

6. For graph D, 6. For graph D, what is the right-what is the right-hand derivativehand derivative

at x = a? at x = a?a

a a

a

a

A. B.

C. D.

E.

6. For graph D, 6. For graph D, what is the right-what is the right-hand derivativehand derivative

at x = a? at x = a?

ANSWER: dneANSWER: dnethere is no derivative at there is no derivative at

a discontinuitya discontinuity!!

a

a a

a

a

A. B.

C. D.

E.

When is f(x) NOT differentiable?When is f(x) NOT differentiable? Where it is not Where it is not

continuous continuous (but you can count a one-sided derivatives at closed endpts of closed intervals)

Where fWhere f’’ is not defined is not defined At a cornerAt a corner At a cuspAt a cusp At vertical tangents At vertical tangents

(As with limits, if we write f’ (a) = it means the slope “approaches ”, but that is not a real number slope.)

Corners

Cusp Vert. Tangents

7. Write f(x) = | x | 7. Write f(x) = | x | as a piece-wise as a piece-wise defined function.defined function.

7. Write f(x) = | x | 7. Write f(x) = | x | as a piece-wise as a piece-wise defined function.defined function.

ANSWER:ANSWER:

8. For f(x) = | x |, 8. For f(x) = | x |, find f ‘ (0).find f ‘ (0).

8. For f(x) = | x |, 8. For f(x) = | x |, find f ‘ (0).find f ‘ (0).

ANSWER:ANSWER:

m = -1

m = 1

asx 0 , f '(x) 1 andasx 0 , f '(x) 1sof '(0)d.n.e.

Consider f(x) = | x |Consider f(x) = | x |

Is f(x) differentiable at a = 0?

NO! It is continuous, but not NO! It is continuous, but not differentiable! This is a differentiable! This is a CORNERCORNER..

x 0

h 0

h 0

limf (x) 0 (it IS cont.)

f (0 h) f (0)but lim 1

hf (0 h) f (0)

but lim 1h

9. Must get all three correct 9. Must get all three correct to count the point.to count the point.

T F If a function f is differentiable at a, T F If a function f is differentiable at a, then it is continuous at a.then it is continuous at a.

T F If a function f is continuous at a, T F If a function f is continuous at a, then it is differentiable at a.then it is differentiable at a.

T F If a function f has a limit at a, then T F If a function f has a limit at a, then it is continuous at a.it is continuous at a.

9. Must get all three correct 9. Must get all three correct to count the point.to count the point.

TT If a function f is differentiable at a, If a function f is differentiable at a, then it is continuous at a.then it is continuous at a.

FF If a function f is continuous at a, then If a function f is continuous at a, then it is differentiable at a.it is differentiable at a.

FF If a function f has a limit at a, then it If a function f has a limit at a, then it is continuous at a.is continuous at a.

DefinitionDefinition

A function f is differentiable on an A function f is differentiable on an open open intervalinterval if f if f’’(x) exists for every x in that (x) exists for every x in that

interval. interval. (i.e. left & right slopes are equal)(i.e. left & right slopes are equal)

A function is differentiable on a closed interval [a, b] if it is differentiable on the open interval

(a, b) and if the following (one-sided) limits exist:

h 0 h 0

f (a h) f (a) f (b h) f (b)lim and lim

h h

On the right side of the lower

bound.

On the left side of the upper

bound.

Can be called the right-hand derivative at a.

Can be called the left-hand

derivative at b.

At a closed endpoint of an interval: At a closed endpoint of an interval: (not elsewhere)(not elsewhere)

We accept the one-sided derivative as an overall derivative.

I.E: No derivative at an open endpt, but may be differentiable at the closed endpt

[a, b) [a, ) (a, b] (- , b]

If a is in an open interval (meaning in the middle of some interval), then f’(a) exists, IFF, both

right-hand and left-hand derivatives at a exist and are equal.

10. Must get all three 10. Must get all three correct to count the point.correct to count the point.

T F If the left hand deriv at x = a is ∞, T F If the left hand deriv at x = a is ∞, then (a, f(a) ) is a cusp.then (a, f(a) ) is a cusp.

T F If the left hand deriv at x = a is 5 T F If the left hand deriv at x = a is 5 and the right hand deriv at x = a and the right hand deriv at x = a

is - 5, then (a, f(a) ) is a cusp.is - 5, then (a, f(a) ) is a cusp.

T F A piece-wise defined function T F A piece-wise defined function contains a corner.contains a corner.

10. Must get all three 10. Must get all three correct to count the point.correct to count the point.

F F If the left hand deriv at x = a is ∞, If the left hand deriv at x = a is ∞, then (a, f(a) ) is a cusp.then (a, f(a) ) is a cusp.

FF If the left hand deriv at x = a is 5 If the left hand deriv at x = a is 5 and the right hand deriv at x = aand the right hand deriv at x = a

is - 5, then (a, f(a) ) is a cusp.is - 5, then (a, f(a) ) is a cusp.

FF A piece-wise defined function A piece-wise defined function contains a corner.contains a corner.

There is a There is a CORNERCORNER at P(a, f(a)) if at P(a, f(a)) if

f is continuous at a and:f is continuous at a and:

if right & left-hand derivatives exist at a

but are unequal

or

if ONE of those derivatives exists at a and

f’(x) ± as x a + or as x a –

CORNER CORNER at P(a, f(a)) if f is continuous at a and:at P(a, f(a)) if f is continuous at a and:if right & left-hand derivatives exist at a but are unequal

or

if ONE of those derivatives exists at a and

f’(x) ± as x a + or as x a –

Derivatives exist, but are unequal.

left-derivative exists, but right derivative goes to - .

11. Do this one in your 11. Do this one in your notebook to keep for notebook to keep for

reference.reference.Use the def of derivative to Use the def of derivative to

find f ‘ for find f ‘ for 12f (x) x x

11. Use the def of derivative 11. Use the def of derivative to find f ‘ for to find f ‘ for

12

12

x h x x h x

x h x

(x h) x 1 1or x

22 xh x h x

h 0

h 0

h

f (x) x x

lim

lim

12. Find f ‘ (9) and f ‘ (0) for 12. Find f ‘ (9) and f ‘ (0) for 12f (x) x x

12. Find f ‘ (9) and f ‘ (0) for 12. Find f ‘ (9) and f ‘ (0) for

1 1

2 21 1

or x22 x

f '(x)

AND asx 0 ,

f (x) x x

1 1f '(9)62 9

1f '(0)2 0

DefinitionDefinitionThe graph of a function has a vertical tangent line

x = a at the point P(a, f(a)) if f is continuous at a (use one-sided limit for cont if it is an endpt) and if

x alim f '(x)

1

2Consider f (x) x x f (x) isdefinedover[0, )

1f '(x) ; f 'is definedover (0, ) and right

2 xhand deriv.at x 0approaches

Some more vertical tangentsSome more vertical tangents

DefinitionDefinitionThe graph of a function has a CUSP at the point P(a, f(a)) if f is continuous at a

and if the following 2 conditions hold:i) f’(x) as x approaches a

from one sideii) f’(x) - as x approaches

a from the other side

Look for the “seagull”