Transcript Aim: - MazesMath

Aim: Differential? Isn’t that part of a car’s
drive transmission?
Do Now:
Find the equation of the tangent
line for f(x) = 1 + sinx at (0, 1).
Aim: Differentials
Course: Calculus
Linear Approximations
y = x2
graph of
function is
approximated
by a straight
line.
y = x2 = 2x - 1
Aim: Differentials
y = x2 = 2x - 1
Course: Calculus
Linear Approximations
Can the graph of a
function be
approximated by a
straight line?
(x, y)
c, f(c)
f
By restricting
values of x to be
close to c, the values
of y of the tangent
line can be used as
approximations of
the values of f.
as x  c, the limit
of s(x) or y is f(c)
s
equation of
tangent line
x
c x
y2 – y1 = m(x2 – x1) - point slope
y – f(c) = f’(c) (x – c)
y = f(c) + f’(c)(x – c)
Aim: Differentials
Equation of
tangent line
approximation
Course: Calculus
Model Problem
Find the tangent line approximation of
f  x   1  sin x
at the point (0, 1).
f ' x   cos x
1st derivative of f
Equation of
tangent line
approximation
y = f(c) + f’(c)(x – c)
y = 1 + cos 0 (x – 0)
y = 1 + 1x
f  x   1  sin x
=1+x
Aim: Differentials
The closer x is to
0, the better the
approximation.
Course: Calculus
dy
dx
Differential
derivative of y with respect to x
also
the ratio dy  dx is the slope of the
tangent line
When we talk only of dy or dx we talk differentials
f  x  x   f ( x )
lim
 f '( x )
x 0
x
As Δx gets smaller and smaller, before it reaches 0,
 approximates
f  x  x   f ( x )
y
 f '( x )

x
x
f  x  x   f ( x )  x f '( x )
Δy
actual change
Aim: Differentials
 x f '( x )
approximation
dy
Course: Calculusof Δy
Differential Approximations
(c + Δx, f(c + Δx)
dy  x f '(c )  y
Δy
f’(c)Δx
dy
c, f(c)
f(c + Δx)
f(c)
c
c + Δx
Δx = dx
the ratio dy  dx is
the slope of the
tangent line
When Δx is small, then Δy is also small
and Δy = f(c + Δx) – f(c) and is
by f’(c)Δx.
Aim: approximated
Differentials
Course: Calculus
Differential
When Δx is small, then Δy is also small
and Δy = f(c + Δx) – f(c) and is
approximated by f’(c)Δx.
Δy = f(c + Δx) – f(c) actual change in y
 f’(c)Δx
approximate change in y
Definition
Let y = f(x) represent a function that is
differentiable in an open interval containing
x . The differential of x (denoted by dx) is
any nonzero real number. The differential of
y (denoted by dy) is
dy = f’(x)dx
Δy  dy
Δy  f’(c)Δx
Aim: Differentials
Course: Calculus
Differential
Definition
Δx is an arbitrary increment of the
independent variable x.
dx is called the differential of the independent
variable x, dx is equal to Δx.
Δy is the actual change in the variable y as x
changes from x to x + Δx; that is,
Δy = f(x + Δx) – f(x)
dy, called the differential of the dependent
variable y, is defined by dy = f’(x)dx
Aim: Differentials
Course: Calculus
Comparing Δy and dy
Let y = x2. Find dy when x = 1 and dx = 0.01.
Compare this value with Δy for x = 1 and
Δx = 0.01.
approximate change in y
actual change in y
dy = f’(x)dx
Δy = f(c + Δx) – f(c)
y = f(x) = x2  f’(x) = 2x
dy
 2 x  dy  2 xdx
dx
dy = f’(1)(0.01)
dy = 2(1)(0.01)
dy = 2(0.01) = 0.02
Aim: Differentials
Course: Calculus
Comparing Δy and dy
Let y = x2. Find dy when x = 1 and dx = 0.01.
Compare this value with Δy for x = 1 and
Δx = 0.01.
approximate change in y
actual change in y
dy = f’(x)dx
Δy = f(c + Δx) – f(c)
dy = 2(0.01) = 0.02
Δy = f(1 + 0.01) – f(1)
Δy = f(1.01) – f(1)
Δy = 1.012 – 12
Δy = 0.0201
values become closer to each other
when dx or Δx approaches 0
Aim: Differentials
Course: Calculus
Comparing Δy and dy
Let y = x2. Find dy when x = 1 and dx = 0.01.
Compare this value with Δy for x = 1 and
Δx = 0.01.
f x = x2
g  x  = 2 x-1
y
dy
= 0.0201
= 0.02
1, 1
Δx = 0.01
Aim: Differentials
Course: Calculus
Error Propagation
estimations based on physical measurements
A(r) = πr2
r = 7.2cm – exact measurement
A(7.2) = π(7.2)2 = 162.860
7.21
7.19
7.18 cm
cm
difference
is
propagated
error
A = 161.957
163.313
162.408
Aim: Differentials
Course: Calculus
Error Propagation
Propagation error – when a measured
value that has an error in measurement
is used to compute another value.
Measurement
error
 f(x)
f( x + Δx )
Exact
value
Propagated
error
= Δy
Measurement
value
approximate change in y
dy = f’(x)dx
Aim: Differentials
Course: Calculus
Model Problem
The radius of a ball bearing is measured to
be 0.7 inch. If the measurement is correct to
within 0.01 inch, estimate the propagated
error in the volume V of the ball bearing.
4 3
V  r
3
dV
 4 r 2  dV  4 r 2dr
dr
r = 0.7
measured radius
-0.01 < Δr < 0.01
possible error
V  dV
 4 r 2dr
approximate ΔV
by dV
 4 (0.7)2 (0.01) substitute r and dr
 0.06158 in
Aim: Differentials
3
approximate change in y
Course: Calculus
dy = f’(x)dx
Relative Error
The radius of a ball bearing is measured to
be 0.7 inch. If the measurement is correct to
within 0.01 inch, estimate the propagated
error in the volume V of the ball bearing.
dV 4 r 2dr

Ratio of dV to V
4 3
V
r
3
3dr

r
3  0.01

Substitute for dr and r
0.7
 0.0429  4.29% relative error
Aim: Differentials
Course: Calculus
Differential Formulas
Definition
The differential of x (denoted by dx) is any
nonzero real number. The differential of y
(denoted by dy) is dy = f’(x)dx.
dy
 y '  dy = y’dx
dx
Liebniz notation
Let u and v be differentiable functions of x.
Constant Multiple:
d  cu  c du
Sum or Difference:
d  u  v   du  dv
Product:
d  uv   u dv  v du
Quotient:
 u  v du  u dv
d 
2
v
v
 Course: Calculus
Aim: Differentials
Differential Formulas
Function
Derivative
Differential
dy
a. y = x2
dy  2 x dx
 2x
dx
b. y = 2sin x dy  2cos x
dy  2cos x dx
dx
dy    x sin x  cos x  dx
dy
c. y = xcosx
  x sin x  cos x
dx
dy
1
dx
d. y = 1/x
 2
dy   2
dx
x
x
Aim: Differentials
Course: Calculus
Model Problem
Find the differential of composite functions
y = f(x) = sin 3x
Original function
y’ = f’(x) = 3cos 3x
Apply Chain Rule
dy = f’(x)dx = 3cos 3x dx Differential Form
y = f(x) = (x2 + 1)1/2
Original function
Apply Chain Rule

1 2
f '( x ) 
x 1
2
 2x 
12
x
x2  1
Differential Form
x
dy  f '( x )dx 
dx
x2  1
Aim: Differentials
Course: Calculus
Approximating Function Values
Use differential to approximate 16.5
xx  ff ((xx))  fx'(fx'()dx
ff  xx  
x)
Let f ( x ) 
x
then f ( x  x )  f ( x )  f '( x ) dx
1
 x
dx
2 x
x = 16 and dx = 0.5
f ( x  x )  16.5
1
 16 
 0.5 
2 16
1
 4   0.5   4.0625
8
Aim: Differentials
Course: Calculus
Model Problem
Use differential to approximate
Aim: Differentials
99.4
Course: Calculus
Model Problem
Find the equation of the tangent line T to
the function f at the indicated point. Use
this linear approximation to complete the
table.
f ( x)  x5
x
1.9
1.99
2
2.01
2.1
f(x)
T(x)
Aim: Differentials
Course: Calculus
Model Problem
The measurement of the side of a square is
found to be 12 inches, with a possible error of
1/64 inch. Use differentials to approximate
the possible propagated error in computing
the area of the square.
Aim: Differentials
Course: Calculus
Model Problem
The measurement of the radius of the end of
a log is found to be 14 inches, with a possible
error of ¼ inch. Use differentials to
approximate the possible propagated error in
computing the area of the end of the log.
Aim: Differentials
Course: Calculus
Model Problem
The radius of a sphere is claimed to be 6
inches, with a possible error of 0.02 inch. Use
differentials to approximate the maximum
possible error in calculating (a) the volume of
the sphere, (b) the surface area of the sphere,
and (c) the relative errors in parts (a) and (b).
Aim: Differentials
Course: Calculus