2.2 Differentiation Techniques: The Power and Sum- Difference Rules 1.5
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
2.2 Differentiation Techniques: The Power
and Sum-Difference Rules
1.5
2
OBJECTIVES
• Find the derivative using the Constant Rule.
• Find the derivative using the Power Rule.
• Find the derivative using the Constant Multiple
• Rule and the Sum and Difference Rules.
• Find the derivative of sine and cosine.
• Use derivatives to find rates of change
Leibniz’s Notation:
• When y is a function of x, we will also designate the derivative, , as
• which is read “the derivative of y with respect to x.”
f x dy
dx,
• THEOREM 1: The Power Rule
• For any real number k,
dy
dxxk k xk 1
• Example 1: Differentiate each of the following:
• a) b) c)
• a) b) c)1 11
1
dx x
dx
5 5 1
4
5
5
dx x
dx
x
4 4 1
5
4
4
dx x
dx
x
y x 4y xy x5
• Example 2: Differentiate:
• a) b)
• a) b)1 1
12 2
1
2
1
2
1
2
1 ,
21
,
21
2
dy dyx x x
dx dx
x or
or
x
x
0.7 0.7 1
0.7 0.3
.3
0.7
0.7
7 =
10x
dx x
dxd
x xdx
y x0.7y x
• THEOREM 2:
• The derivative of a constant function is 0. That is,
d
dxc 0
• THEOREM 3:
• The derivative of a constant times a function is the constant times the derivative of the function. That is,
• d
dxcf (x) c
d
dxf (x)
• Example 3: Find each of the following derivatives:
•
• a) b) c)
• a) b)
d
dx7x4 d
dx( 9x)
d
dx
1
5x2
4 4
4 1
3
7 7
7 4
28
d dx x
dx dx
x
x
1 1
( 9 ) 9
9 1
9
d dx x
dx dx
x
• Example 3 (concluded):
• c)
• 2 2
2
2 1
33
1 1 1
5 5
1
51
25
2 2 , or
5 5
d d
dx x dx x
dx
dx
x
xx
• THEOREM 4: The Sum-Difference Rule
• Sum: The derivative of a sum is the sum of the derivatives.
• Difference: The derivative of a difference is the difference of the derivatives.
d
dxf (x) g(x)
d
dxf (x)
d
dxg(x)
d
dxf (x) g(x)
d
dxf (x)
d
dxg(x)
• Example 5: Find each of the following derivatives:
•
• a) b)
• a)
d
dx(5x3 7)
d
dx24x x
5
x
3 3
3 3 1
2
(5 7) (5 ) (7)
5 0 5 3
15
d d dx x
dx dx dxd
x xdx
x
• Example 5 (concluded):• b)
• d
dx24x x
5
x
d
dx(24x)
d
dxx
d
dx
5
x
24 d
dxx
d
dxx
1
2 5d
dxx 1
24 1x1 1 1
2x
1
2 1
5 1x 1 1
24 1
2x
1
2 5x 2 , or 24 1
2 x
5
x2
14
Finding the Equation of a Tangent Line
Example: Find the equation of the tangent line at x = 1 and take a “peek” at the graph. Then verify using your calculator.
43)( 25 xxxf
“peek” at the graph
15
Finding the Equation of a Tangent Line
5 2 ' 4
'
( ) 3 4 ( ) 5 6
( )
f x x x f x x x
f x
1x 5(1) 6(1) 1
Example: Find the equation of the tangent line at x = 1 and take a “peek” at the graph. Then verify using your calculator.
43)( 25 xxxf
So, the slope of the tangent line, is -1, at x = 1. And if x = 1, then y = 1- 3(1)+ 4 = 2. So, using the point (1,2)
2 1(1) 3,
3
b b
y x
• Example 6: Find the points on the graph of
• at which the tangent line is horizontal.
• Recall that the derivative is the slope of the tangent line, and the slope of a horizontal line is 0. Therefore, we wish to find all the points on the graph of f where
the derivative of f equals 0.
f (x) x3 6x2
• Example 6 (continued):• So, for
• Setting equal to 0:
f (x) 3x3 1 62x2 1
f (x) 3x2 12x
f (x) x3 6x2
f x
3x2 12x 0
3x(x 4) 0
3x 0 x 4 0
x 0 x 4
• Example 6 (continued):• To find the corresponding y-values for these x-
values, substitute back into
Thus, the tangent line to the graph
of is horizontal at the points (0, 0)
and (4, 32).
f (x) x3 6x2 .
f (0) 03 602
f (0) 0
f (4) 43 642
f (4) 32
f (x) x3 6x2
• Example 6 (concluded):
20
2
0
2
Consider the function siny
We could make a graph of the slope: slope
1
0
1
0
1Now we connect the dots!
The resulting curve is a cosine curve.
sin cosd
x xdx
21
2
0
2
We can do the same thing for cosy slope
0
1
0
1
0The resulting curve is a sine curve that has been reflected about the x-axis.
cos sind
x xdx
22
Derivatives Involving sin x, cos x
• Try the following
34 2cos sin2
dx x
dx
( ) cos( ) '( ) ?f x x f x
23
Consider a graph of displacement (distance traveled) vs. time.
time (hours)
distance(miles)
Average velocity can be found by taking:
change in position
change in time
s
t
t
sA
B
ave
f t t f tsV
t t
The speedometer in your car does not measure average velocity, but instantaneous velocity.
0
limt
f t t f tdsV t
dt t
(The velocity at one moment in time.)
24
Rates of ChangeSome Examples: population growth rates, production rates, velocity and acceleration
*Common use for rate of change is to describe motion of an object along a straight line.
time
distancerate
saverage velocity slope of the secant line
t
0
( ) ( )( ) lim '( )
t
instantaneous
slope of the tangent li
ve
n
loci
e
tyt
s t t s tv t s t
Without Calculus
WITH Calculus
25
Motion Along a Straight Line
s(t)= position function
t
s(t)
(t+t)
s(t+t)
t (sec)
s (ft)
( ) ( )( )
s t t s tv t
t
Ave. velocity during the time interval from t to (t+t)
0
( ) ( )( ) lim
t
s t t s tv t
t
'( )s t
= Instantaneous velocity at time t
Slope of secant line
26
Motion Along a Vertical Line
v(t) > 0,
v(t) < 0,
v(t) = 0, stopped instantaneously
s(t) = position (ft)
v(t) = s´(t) = velocity (ft/sec)
rate of change of position
a(t) =v´(t) = s´´(t) = accel (ft/s2)
rate of change of velocity
speed = |v(t)| Speed is never negative
Note: Velocity is speed with a direction.
27
Velocity is the first derivative of position.
28
Example: Free Fall Equation
21
2s g t
GravitationalConstants:
2
ft32
secg
2
m9.8
secg
2
cm980
secg
2132
2s t
216 s t
32 ds
V tdt
Speed is the absolute value of velocity.
29
Acceleration is the derivative of velocity.
dva
dt
2
2
d s
dt example: 32v t
32a
If distance is in: feet
Velocity would be in:feet
sec
Acceleration would be in:ft
sec sec
2
ft
sec
30time
distance
acc posvel pos &increasing
acc zerovel pos &constant
acc negvel pos &decreasing
velocityzero
acc negvel neg &decreasing acc zero
vel neg &constant
acc posvel neg &increasing
acc zero,velocity zero
It is important to understand the relationship between a position graph, velocity and acceleration
31
1 •• Which of these position versus time curves best shows the motion of an object with constant positive
acceleration?
Determine the Concept The slope of an x (t) curve at any point in time represents the speed at that instant. The way the slope changes as time increases gives the sign of the acceleration. If the slope becomes less negative or more positive as time increases (as you move to the right on the time axis) then the acceleration is positive. ie: the graph is concave up!!!! If the slope becomes less positive or more negative then the acceleration is negative. ie: the graph is concave down!!! So the slope of an x (t) curve at any point in time represents the acceleration at that instant.
32
tion (m)
a) The slope is both negative and increasing, therefore the velocity is negative and the acceleration is positve
b) The slope is positive but decreasing, therefore, therefore, the velocity is positive and decreasing, and the acceleration is negative.
c) The slope is positive and constant, therefore, the velocity is positive and the acceleration is 0
d) The slope is positive and increasing, therefore, the velocity is positive and increasing and the acceleration is positive.
e) The slope of the curve is 0, therefore, the velocity and acceleration are both 0.
33
Rates of Change:
Average rate of change = f x h f x
h
Instantaneous rate of change = 0
limh
f x h f xf x
h
These definitions are true for any function.
( x does not have to represent time. )
Suppose that in June a chain of stores had combined daily sales of
ice cream cones given by
where s is the number (in hundreds) of cones sold and x is the day of the
month.
5048.01.0 2 xxs
(a) how many cones were sold on June 3rd?
(b) at what rate were sales changing on June 10th?
(c) at what rate were sales changing on June 28th?
(d) on what day was the rate of change of sales equal to 10 cones per day?
36
Example of Free Falling Object
Example: A ball is thrown straight down from the top of a 220-foot building with an initial velocity of –22 feet/second. 1) What is its velocity after 3 seconds? 2) What is its velocity after falling 108 feet?
220
0
00216)( stvtts
37
3y xExample: Find at x = 2.dy
dx
nDeriv ( x ^ 3, x,2) ENTER returns 12
MATH: 8: (F(X),X,VALUE)
38
Warning:
The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.
Examples: 1/ , ,0nderiv x x returns
, ,0d abs x x returns 1
39
YOU SHOULD NOW BE ABLE TO:
• Find the derivative using the Constant Rule.
• Find the derivative using the Power Rule.
• Find the derivative using the Constant Multiple Rule and the Sum and Difference Rules.
• Find the derivative of sine and cosine.
• Use derivatives to find rates of change.
40
Constant & Power Rules
0)(
dx
cd
Ex:(6)d
dx
1)(
dx
xd 1)( nn
nxdx
xd
Power Rule
Ex: 3( )d x
dx
Constant Rule
41
Constant Multiple, Sum & Difference Rules
( ) ( )dcf x cdf x
dx dx
Ex: 6(3 )d x
dx
dx
xdg
dx
xdf
dx
xgxfd )()())()((
Ex: 835)( 34 xxxxf
42
Sine and Cosine Rules
xdx
xdcos
sin x
dx
xdsin
cos
Example: xxy cos3sin2
Related Documents
