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Chapter 13 – Vector Functions13.2 Derivatives and Integrals of Vector Functions

13.2 Derivatives and Integrals of Vector Functions

Objectives: Develop Calculus of

vector functions.

Find vector, parametric, and general forms of equations of lines and planes.

Find distances and angles between lines and planes

13.2 Derivatives and Integrals of Vector Functions

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Definition – Derivatives of Vector FunctionsThe derivative r’ of a vector

function is defined in much the same way as for real-valued functions:

if the limit exists.

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Definition – Tangent VectorThe vector r’(t) is called the

tangent vector to the curve defined by r at the point P, provided:◦r’(t) exists◦r’(t) ≠ 0

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VisualizationSecant and Tangent Vectors

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Definition – Unit Tangent Vector

We will also have occasion to consider the unit tangent vector which is defined as:

'( )( )

| '( ) |

tt

tr

Tr

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TheoremThe following theorem gives us

a convenient method for computing the derivative of a vector function r: ◦Just differentiate each component of r.

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Second DerivativeJust as for real-valued functions,

the second derivative of a vector function r is the derivative of r’, that is, r” = (r’)’.

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Differentiation Rules

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IntegralsThe definite integral of a

continuous vector function r(t) can be defined in much the same way as for real-valued functions—except that the integral is a vector.

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Integral NotationHowever, then, we can express

the integral of r in terms of the integrals of its component functions f, g, and h as follows using the notation of Chapter 5.

*

1

* * *

1 1 1

( ) lim ( )

lim ( ) ( ) ( )

nb

ia ni

n n n

i i in

i i i

t dt t t

f t t g t t h t t

r r

i j k

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Integral Notation - Continued

Thus,

◦ This means that we can evaluate an integral of a vector function by integrating each component function.

( ) ( ) ( ) ( )b b b b

a a a at dt f t dt g t dt h t dtr i j k

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Fundamental Theorem of CalculusWe can extend the Fundamental

Theorem of Calculus to continuous vector functions:

◦ Here, R is an antiderivative of r, that is, R’(t) = r(t).

◦ We use the notation ∫ r(t) dt for indefinite integrals (antiderivatives).

b

a(t) ( ) ( ) ( )

b

adt t b a r R R R

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Example 1- pg. 852 #8Sketch the plane curve with the

given vector equation.Find r’(t).Sketch the position vector r(t)

and the tangent vector r’(t) for the given value of t.

( ) (1 cos ) (2 sin ) , / 6t t t t r i j

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Example 2- pg. 852 #9Find the derivative of the vector

function.

2( ) sin , , cos 2t t t t t tr

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Example 3Find the unit tangent vector T(t)

at the point with the given value of the parameter t.

2( ) 4 , 1t t t t t r i j k

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Example 4- pg. 852 #24Find the parametric equations for

the tangent line to the curve with the given parametric equations at the specified point.

2

, , ; (1,0,0)t t tx e y te z te

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Example 5- pg. 852 #31Find the parametric equations for

the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen.cos , , sin ; ( , ,0)x t t y t z t t

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Example 6- pg. 856 #36Evaluate the integral.

1

2 20

4 2

1 1

tdt

t t j k

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Example 7Evaluate the integral.

cos sint t t dt i j k

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More Examples

The video examples below are from section 13.2 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 3◦Example 4