Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600...
Transcript of Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600...
![Page 1: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/1.jpg)
Limit Definition of the
Derivative
![Page 2: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/2.jpg)
Objective
To use the limit definition to find the
derivative of a function.
.
![Page 3: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/3.jpg)
What is a derivative?
A function
the rate of change of a
function
the slope of the line tangent to
the curve
![Page 4: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/4.jpg)
The tangent line
single point
of intersection
![Page 5: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/5.jpg)
slope of a secant line
ax
f(x)
f(a)
f(a) - f(x)
a - x
![Page 6: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/6.jpg)
slope of a (closer) secant line
ax
f(x)
f(a)
f(a) - f(x)
a - x
x
![Page 7: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/7.jpg)
closer and closer…
a
![Page 8: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/8.jpg)
watch the slope...
![Page 9: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/9.jpg)
watch what x does...
ax
![Page 10: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/10.jpg)
The slope of the secant line gets closer and closer
to the slope of the tangent line...
![Page 11: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/11.jpg)
As the values of x get closer and closer to a!
ax
![Page 12: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/12.jpg)
The slope of the secant lines
gets closer
to the slope of the tangent line...
...as the values of x
get closer to a
Translates to….
![Page 13: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/13.jpg)
limax
f(x) - f(a)
x - a
Equation for the slope
Which gives us the the exact slope
of the line tangent to the curve at a!
as x goes to a
![Page 15: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/15.jpg)
The Difference Quotient
The derivative is the slope of the tangent
line to a graph f(x), and is usually denoted
f’(x).
To calculate the slope of the tangent line
we will use the difference quotient.
![Page 16: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/16.jpg)
The Difference Quotient
![Page 17: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/17.jpg)
Limit Definition of the Derivative
The derivative is the formula which gives
the slope of the tangent line at any point x
for f (x), and is denoted
provided this limit exists.
0
( ) ( )'( ) lim
x
f x x f xf x
x
![Page 18: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/18.jpg)
Derivatives
The derivative of the function y = f (x) may be
expressed as …
'( )f x
'y
dy
dx
Prime notation
Leibniz notation
“f prime of x”
“y prime”
“the derivative of y with respect to x”
![Page 19: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/19.jpg)
Derivatives
The process of finding derivatives is called
differentiation.
A function is differentiable at a point if its
derivative exists at that point.
![Page 20: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/20.jpg)
Limit Definition of the Derivative
Use the limit definition to find the
derivative of:2( ) 3 5f x x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
![Page 21: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/21.jpg)
Limit Definition of the Derivative
2( ) 3 5f x x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
2 2
0
( ) 3( ) 5 ( 3 5)'( ) lim
x
x x x x x xf x
x
2 2 2
0
2 ( ) 3 3 5 3 5'( ) lim
x
x x x x x x x xf x
x
![Page 22: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/22.jpg)
Limit Definition of the Derivative
0
(2 3)'( ) lim
x
x x xf x
x
2
0
2 ( ) 3'( ) lim
x
x x x xf x
x
'( ) 2 3f x x
A formula for finding the
slope of the tangent line
of f (x) at a given point.
![Page 23: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/23.jpg)
Limit Definition of the Derivative
Use the limit definition to find the
derivative of:2( ) 8 1f x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
![Page 24: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/24.jpg)
Limit Definition of the Derivative
2( ) 8 1f x x
0
( ) ( )'( ) lim
x
f x x f xf x
x
2 2
0
8( ) 1 (8 1)'( ) lim
x
x x xf x
x
2 2 2
0
8( 2 ( ) ) 1 8 1'( ) lim
x
x x x x xf x
x
![Page 25: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/25.jpg)
Limit Definition of the Derivative
0
(16 8 )'( ) lim
x
x x xf x
x
'( ) 16f x x
A formula for finding the
slope of the tangent line
of f (x) at a given point.
2 2 2
0
8 16 8( ) 1 8 1'( ) lim
x
x x x x xf x
x
![Page 26: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/26.jpg)
similarly...
aa+h
f(a+h)
f(a)
f(x+h) - f(x)
(x+h) - x
= f(x+h) - f(x)
h
(For this particular curve, h is a negative value)
h
![Page 27: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/27.jpg)
thus...
lim f(a+h) - f(a)
hh 0
AND
lim f(x) - f(a)x a x - a
Give us a way to calculate the slope of the line tangent at a!
![Page 28: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/28.jpg)
Which one should I use?
(doesn’t really matter)
![Page 29: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/29.jpg)
A VERY simple example...
want the slope
where a=2
xy 2xy
2xy
![Page 30: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/30.jpg)
ax
axax
ax
ax
ax
afxf
))((limlim
)()(lim
22
4)2lim()lim( xax
as x a=2
![Page 31: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/31.jpg)
h
xhx
h
xfhxf 22)(lim
)()(lim
h
hxh
h
xhxhx )2(lim
2lim
222
4)2lim( hx
As h 0
![Page 32: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/32.jpg)
back to our example...
When a=2,
the slope is 4
xy 2xy
2xy
![Page 33: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/33.jpg)
Differentiability
Not every function is differentiable at all points.
Some common situations in which a function will not be differentiable at a point include:
1. Vertical tangent lines
2. Discontinuities (like a hole, break, or vertical
asymptote)
3. Sharp turns (called cusps & nodes)
![Page 34: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/34.jpg)
Differentiability
![Page 35: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/35.jpg)
Differentiability
![Page 36: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/36.jpg)
Differentiability
![Page 37: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/37.jpg)
Differentiability
![Page 38: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/38.jpg)
CALCULUS JEOPARDY!
$200
Answer:
It’s computed by finding the limit of the
difference quotient as ∆x approaches 0.
Question:
What is the derivative?
![Page 39: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/39.jpg)
CALCULUS JEOPARDY!
$400
Answer:
It’s used to find the slope of a function
at a point.
Question:
What is the derivative?
![Page 40: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/40.jpg)
CALCULUS JEOPARDY!
$600
Answer:
It’s used to find the slope of the tangent
line to a graph f (x), and is usually denoted
f’(x).
Question:
What is the derivative?
![Page 41: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/41.jpg)
CALCULUS JEOPARDY!
$800
Answer:
It’s used to find the instantaneous rate of
change of a function.
Question:
What is the derivative?
![Page 42: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/42.jpg)
The Derivative is…
computed by finding the limit of the difference
quotient as ∆x approaches 0.
the slope of a function at a point.
the slope of the tangent line to a graph f (x), and
is usually denoted f’(x).
the instantaneous rate of change of a function.
![Page 43: Limit Definition of the Derivative · Derivatives The derivative of ... CALCULUS JEOPARDY! $600 Answer: It’s used to find the slope of the tangent line to a graph f (x), and is](https://reader034.fdocuments.net/reader034/viewer/2022042623/5fb175987df78d376e0d0f4c/html5/thumbnails/43.jpg)
in conclusion...
The derivative is the the slope of the line
tangent to the curve (evaluated at a point)
it is a limit (3 ways to define it)
cool site to go to for additional
explanations:http://archives.math.utk.edu/visual.calculus/2/