chapter 13 – vector functions
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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.
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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VisualizationSecant and Tangent Vectors
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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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.
13.2 Derivatives and Integrals of Vector Functions
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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’)’.
13.2 Derivatives and Integrals of Vector Functions
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Differentiation Rules
13.2 Derivatives and Integrals of Vector Functions
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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.
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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Example 2- pg. 852 #9Find the derivative of the vector
function.
2( ) sin , , cos 2t t t t t tr
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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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
13.2 Derivatives and Integrals of Vector Functions
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Example 6- pg. 856 #36Evaluate the integral.
1
2 20
4 2
1 1
tdt
t t j k
13.2 Derivatives and Integrals of Vector Functions
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Example 7Evaluate the integral.
cos sint t t dt i j k
13.2 Derivatives and Integrals of Vector Functions
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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
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