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

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).


Linear Approximations

graph of function is

approximated by a straight

line.

y = x2

y = x2 = 2x - 1 y = x2 = 2x - 1


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


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


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


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


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


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


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)




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


dy

y

g x = 2x-1f x = x2



1, 1

= 0.0201

Δx = 0.01

= 0.02


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


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



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



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%


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


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


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


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


Model Problem

Use differential to approximate 99.4


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


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.


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.


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).

Documents

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