Constructing the Antiderivative

Solving (Simple) Differential Equations

The Fundamental Theorem of Calculus (Part 2)

Chapter 6: Calculus~Hughes-Hallett

Review: The Definite Integral

Physically - is a summing up Geometrically - is an area under a

curve Algebraically - is the limit of the sum

of the rectangles as the number increases to infinity and the widths decrease to zero:

Review of The Fundamental Theorem of Calculus (Part 1)

If f is continuous on the interval [a,b] and f(t) = F’(t), then:

In words: the definite integral of a rate of change gives the total change.

)()()( aFbFdttfb

a

Differential and Integral Formulas

Properties of Antiderivative:

1. [f(x) g(x)]dx = f(x)dx g(x)dx (The antiderivative of a sum is the

sum of the antiderivatives.) 2. cf(x)dx = cf(x)dx

(The antiderivative of a constant times a function is the constant times the antiderivative of the function.)

The Definition of Differentials (given y = f(x))

1. The Independent Differential dx:If x is the independent variable, then the

change in x, x is dx; i.e. x = dx.

2. The Dependent Differential dy:If y is the dependent variable then:

i.) dy = f ‘(x) dx, if dx 0 (dy is the derivative of the function times dx.)ii.) dy = 0, if dx = 0.

Using the differential with the antiderivative.

2

2 2 12

2 12

212

212

12

12

12

, 2

2

[2 ], (multiplied by1)

[2 ] , (commutative law)

[2 ], (commutative law)

[2 ], (Formula 2 for integrals)

[ ], (substitute:u=2r,du=2dr)

r

r r

r

r

r

u

u

u

e dr Let u r

du dr

e dr e dr

e dr

e dr

e dr

e du

e du

e C

212

, (Formula 6 for integrals)

, (resubstitute)re C

Solving First Order Ordinary Linear Differential Equations

To solve a differential equation of the form dy/dx = f(x) write the equation in differential form: dy = f(x) dx and integrate: dy = f(x)dx

y = F(x) + C, given F’(x) = f(x) If initial conditions are given y(x1) = y1

substitute the values into the function and solve for c: y = F(x) + C y1 = F(x1) + C C = y1 - F(x1)

Example: Solve, dr/dp = 3 sin pwith r(0)= 6, i.e. r= 6 when p = 0

Solution:

3sin( ), . . : (0) 6, . . 6, 0

3sin( )

3sin( )

3 sin( )

3[ cos( )] , (0) 6 Theinitialcondition.

6 3cos(0)

6 3(1)

9

3cos( ) 9

drp I C r i e r when x

dp

dr p dp

dr p dp

dr p dp

r p C r

C

C

C

r p

The Fundamental Theorem of Calculus (Part 2)

If f is a continuous function on an interval, & if a is any number in that interval, then the function F, defined by F(x) = a

x f(t)dt

is an antiderivative of f, and equivalently:

)x(fdx

dt)t(fdx

a

Example:dx

1ttantd3

x

32

1)tan(2

]2[1)tan()(1tan

1tan1tan

2,

1tan1tan

26

2323

3

3

3

3

2

3

33 3

2

2

2

xxx

xxxdx

duuu

dx

du

du

ttd

dx

ttd

andxdx

duthenxuLet

dx

ttd

dx

ttd

ux

x

x

That’s all Folks!

Have a good Summer!

God Bless