aim: differentials course: calculus do now: aim: differential? isn’t that part of a car’s drive...
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Aim: Differentials Course: Calculus
Do Now:
Aim: Differential? Isn’t that part of a car’s drive transmission?
Find the equation of the tangent line for f(x) = 1 + sinx at (0, 1).
Aim: Differentials Course: Calculus
Linear Approximations
graph of function is
approximated by a straight
line.
y = x2
y = x2 = 2x - 1 y = x2 = 2x - 1
Aim: Differentials Course: Calculus
Linear Approximations
c
c, f(c)
y2 – y1 = m(x2 – x1) - point slope
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.
y – f(c) = f’(c) (x – c)
y = f(c) + f’(c)(x – c)
xx
(x, y)
Can the graph of a function be approximated by a straight line?
as x c, the limit of s(x) or y is f(c)
Equation of tangent line approximation
f
s
equation oftangent line
Aim: Differentials Course: Calculus
Model Problem
Find the tangent line approximation of
at the point (0, 1).
1 sinf x x
1st derivative of f ' cosf x x
y = f(c) + f’(c)(x – c)
y = 1 + cos 0 (x – 0)y = 1 + 1x
The closer x is to 0, the better the approximation.
1 sinf x x = 1 + x
Equation of tangent line approximation
Aim: Differentials Course: Calculus
Differential
dy
dx
derivative of y with respect to x
When we talk only of dy or dx we talk differentials
0
( )lim '( )x
f x x f xf x
x
As Δx gets smaller and smaller, before it reaches 0,
( )'( )
f x x f xf x
x
approximates
( ) '( )f x x f x x f x
Δyactual change
dyapproximation
of Δy
'( )x f x
y
x
also
the ratio dy dx is the slope of the tangent line
Aim: Differentials Course: Calculus
Differential Approximations
c
c, f(c)
c + Δx
Δx
f(c)
Δyf’(c)Δx
(c + Δx, f(c + Δx)
f(c + Δx)
When Δx is small, then Δy is also small and Δy = f(c + Δx) – f(c) and is
approximated by f’(c)Δx.
dy
'( )dy x f c y
= dx
the ratio dy dx is the slope of the
tangent line
Aim: 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
Δy dy Δy f’(c)Δx
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
Definition
Aim: Differentials Course: Calculus
Differential
Δx is an arbitrary increment of the independent variable x.
Definition
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.
y = f(x) = x2 f’(x) = 2x
2 2dy
x dy xdxdx
dy = f’(1)(0.01)
dy = 2(0.01) = 0.02
Δy = f(c + Δx) – f(c)
actual change in y
dy = f’(x)dx
approximate change in y
dy = 2(1)(0.01)
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.
dy = f’(x)dx
dy = 2(0.01) = 0.02
Δy = f(c + Δx) – f(c)
actual change in yapproximate change in y
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
dy
y
g x = 2x-1f x = x2
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.
1, 1
= 0.0201
Δx = 0.01
= 0.02
Aim: Differentials Course: Calculus
Error Propagation
estimations based on physical measurementsA(r) = πr2
7.19 cm7.21 cm7.18 cm
r = 7.2cm – exact measurement
A(7.2) = π(7.2)2 = 162.860
A = 163.313A = 162.408A = 161.957
difference is
propagated error
Aim: Differentials Course: Calculus
Error Propagation
x + Δx
Exact value
f( )
Measurement error
f(x)
Measurement value
= Δy
Propagated error
Propagation error – when a measured value that has an error in measurement is used to compute another value.
dy = f’(x)dx
approximate change in y
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.
34
3V r
r = 0.7 measured radius
-0.01 < Δr < 0.01 possible error
24dV
rdr
24dV r dr
V dV approximate ΔV by dV
30.06158 in
substitute r and dr
24 r dr24 (0.7) ( 0.01)
dy = f’(x)dx
approximate change in y
Aim: Differentials Course: Calculus
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.
2
3
4 Ra
4tio
3
of to dV r dr
VdV V
r
3
dr
r
Substit
3 0.01ute f or
0.a d
7 n dr r
0.0429 relative error 4.29%
Aim: Differentials Course: Calculus
Liebniz notation
Differential Formulas
Let u and v be differentiable functions of x.
2
Constant Multiple:
Sum or Difference:
Product:
Quotient:
d cu c du
d u v du dv
d uv u dv v du
u v du u dvd
v v
The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx.
Definition
dy = y’dx'dy
ydx
Aim: Differentials Course: Calculus
Differential Formulas
a. y = x2
b. y = 2sin x
c. y = xcosx
d. y = 1/x
2dy
xdx
2cosdy
xdx
sin cosdy
x x xdx
2
1dy
dx x
Function Derivative Differential
2dy x dx
2cosdy x dx
sin cosdy x x x dx
2
dxdy
x
Aim: Differentials Course: Calculus
Model Problem
Find the differential of composite functions
y = f(x) = sin 3x
y’ = f’(x) = 3cos 3x
dy = f’(x)dx = 3cos 3x dx
Original function
Apply Chain Rule
Differential Form
y = f(x) = (x2 + 1)1/2 Original function
Apply Chain Rule
Differential Form
1 22
2
1'( ) 1 2
2 1
xf x x x
x
2'( )
1
xdy f x dx dx
x
Aim: Differentials Course: Calculus
Approximating Function Values
Use differential to approximate 16.5
Let ( )f x x
then ( ) ( ) '( )f x x f x f x dx 1
2x dx
x
x = 16 and dx = 0.5
( ) 16.5f x x
116 0.5
2 16
14 0.5 4.0625
8
( ) '( )f x x f x x f x ( ) '( )f x x f x f x dx
Aim: Differentials 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.
x 1.9 1.99 2 2.01 2.1
f(x)
T(x)
5( )f x 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).