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Math 233 Calculus 3 - Fall 2017...12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS Question....
Transcript of Math 233 Calculus 3 - Fall 2017...12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS Question....
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Math 233 Calculus 3 - Fall 2017
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§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS
§12.1 - Three-Dimensional Coordinate Systems
After completing this section, you should be able to:
• Recognize right handed and left handed coordinate systems
• Determine whether an equation describes a plane. cylinder, spheres, or otherstandard shape in 3-dimensional space
• Describe the regions of 3-dimensional space defined by inequalities
• Complete the square to rewrite equations and inequalities involving 3 variables inmore standard form
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§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS
Definition. R3 means
By convention, we graph points in R3 using a right-handed coordinate system. Right-handed means:
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§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS
Identify the right-handed coordinate systems.
A B C
D E F
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§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS
Question. For two points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) inR3, what is the distancebetween P1 and P2?
Question. What is the equation of a sphere of radius r centered at the point (h, k, l)?
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§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS
Example. (# 10) Find the distance from (4,�2, 6) to each of the following:
a. The point (9,�1,�4)
b. The xy-plane (where z = 0)
c. The xz-plane
d. The x-axis
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§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS
Example. Describe the region (and draw a picture).
a. |z| 2
b. x2 + z2 9
c. x2 + y2 + z2 > 2z
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§12.6 QUADRIC SURFACES
§12.6 Quadric Surfaces
After completing this section, you should be able to:
• Match equations of quadric surfaces with their graphs
• Describe horizontal and vertical cross-sections of surfaces and predict their shapebased on the equation of the surface.
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§12.6 QUADRIC SURFACES
Match the equation to the graph using only your brain (no graphing software). Check your answer with graphing software.
A. B. C.
D. E. F.
G. H. I.
1. x2 + z2 = 4 2. 36x2 + 4y2 + 9z2 = 36 3. �x2� y2 + z = 0 4. �x2 + y2 + z = 0
5. x � y2� z2 = 0 6. �x2
� 4y2 + z2 = 0 7. x2 + y2� z2 = 1 8. �x2
� y2 + z2 = 19. �x2 + y2 + z2 = 1 10. �x2 + z = 0
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§12.2 VECTORS
§12.2 Vectors
After completing this section, you should be able to:
• Represent vectors with arrows, components and in terms of~i, ~j, ~k.
• Add and subtract vectors
• Find the length of a vector
• Rescale vectors to find other vectors in same direction
• Use vectors to solve problems from physics and geometry
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§12.2 VECTORS
Definition. A vector is a quantity with direction and magnitude (length).
Vectors are usually drawn with arrows.
Two vectors are considered to be the same if:
Example. Which pairs of vectors are equivalent?
Definition. A scalar is another word for .A scalar does not have a direction, in contrast to a vector.
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§12.2 VECTORS
Vector addition: Draw ~a +~b Vector subtraction: Draw ~a �~b
Multiplication of scalars and vectors: Draw 2~a and �3~a
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§12.2 VECTORS
Vector Components
If we place the initial point of a vector ~a at the origin, then the vector can be describedby
Definition. The components of a vector are
Note. Vectors in 3-dimensional space can be described in terms of three components.
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§12.2 VECTORS
Question. What are the components of the vector ~AB that starts at a point A = (3, 1)and ends at a point B = (�2, 5)?
Note. In general, the components of the vector that starts at a point A = (x1, y1) andends at a point B = (x2, y2) are:
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§12.2 VECTORS
Definition. Given two vectors ~a =< a1, a2 > and ~b =< b1, b2 >, and a scalar c
~a +~b =
~a �~b =
c~a =
Similarly, for three-dimensional vectors ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 >,
~a +~b =
~a �~b =
c~a =
Example. < 1, 2, 3 > + < 5, 7, 12 >=
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§12.2 VECTORS
Question. We have defined vector addition, subtraction, and scalar multiplicationtwice.
Why are these definitions equivalent?
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§12.2 VECTORS
1. Addition is commutative
2. Addition is associative
3. Additive identity (the zero vector)
4. Additive inverses
5. Distributive property
6. Distributive property
7. Associativity of scalar multiplication
8. Multiplicative identity
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§12.2 VECTORS
Definition. The length of a vector ~w =< w1,w2,w3 > is
||
~w|| =
Note. The length of a vector is also called:
Definition. A unit vector is a vector ...
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§12.2 VECTORS
Question. How can we rescale any vector to make it a unit vector? (Rescale meansmultiply by a scalar.)
Note. Rescaling a vector to make it a unit vector is also called ...
Example. Find a unit vector that has the same direction as < 5, 1, 3 >.
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§12.2 VECTORS
Definition. In 2 dimensions, the standard basis vectors are
Definition. In 3 dimensions, the standard basis vectors are
Note. Vectors can always be written in terms of
Example. Write < 3,�2, 7 > as a sum of multiples of standard basis vectors.
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§12.2 VECTORS
Review. Vectors can be represented with
•
•
•
Example. Which arrow(s) represent(s) the vector < 2, 1 >?
Example. Write < 2, 1 > in terms of the standard basis vectors~i and ~j.
Example. Find a unit vector in the direction of < 2, 1 >.
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§12.2 VECTORS
Example. Find the unit vectors that are parallel to the tangent line to y = x3 at (2, 8).
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§12.2 VECTORS
Example. The 2-d unit vector ~u makes an angle of ⇡3 with the positive x-axis. Find thecomponents of ~u and write ~u in terms of~i and ~j.
Example. The 2-d vector ~v has magnitude 9 and makes an angle of 5⇡6 with the positive
x-axis. Find the components of ~v and write ~v in terms of~i and ~j.
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§12.2 VECTORS
Example. Two forces act as shown. Find the magnitude of the resultant force and theangle it makes with the positive x-axis.
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§12.2 VECTORS
Example. Spiderman is suspended from two strands of spider silk as shown. Find thetension in each strand of spider silk. (You will need some additional information.)
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§12.3 DOT PRODUCT
§12.3 Dot Product
After completing this section, students should be able to:
• Compute the dot product of two vectors from the vector components or from themagnitudes and the angle between them
• Use the dot product to find the angle between two vectors
• Use the dot product to determine if two vectors are perpendicular
• Find the scalar and vector projections of one vector onto another
• Compute the work done by a constant force moving an object in a direction that isat an angle to the direction of the force
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§12.3 DOT PRODUCT
Definition. If ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 >, then the dot product of ~a and ~b isgiven by
~a ·~b =
Example. Find the dot product of < 4, 2,�1 > and < 7, 0, 5 >.
Question. Is ~a ·~b a vector or a scalar?
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§12.3 DOT PRODUCT
Note. Dot product satisfies the following properties:
1. Commutative Property:
2. Distributive Property:
3. Associative Property:
4. Multiplication by ~0:
Question. What do you get when you take the dot product of a vector with itself?
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§12.3 DOT PRODUCT
Definition. Dot product can also be defined in terms of magnitudes and angles:
Example. Suppose ~v and ~w meet at a 45� angle and ||~v|| = 4 and ||~w|| = 6. Find ~v � ~w.
Example. Suppose ~a � ~b = 20 and ||~a|| = 10 and ||~b|| = 4. What is the angle between ~aand ~b?
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§12.3 DOT PRODUCT
Note. If ✓ is the angle between ~a and ~b, then
cos(✓) =
Example. Find the angle between the vectors < 3, 1, 2 > and < 4, 6, 1 >.
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§12.3 DOT PRODUCT
Question. We have two definitions of dot product:
•
•
Are they equivalent?
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§12.3 DOT PRODUCT
Review. For ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 >, what are the two ways of defining~a �~b?
•
•
Example. If ~u is a unit vector, find ~u � ~v and ~u � ~w.
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§12.3 DOT PRODUCT
Question. What can be said about the angle between ~a and ~b, if
•
~a �~b = 0
•
~a �~b > 0
•
~a �~b < 0
Question. Is it possible to find the exact angle between two vectors if all we knowis the dot product and the magnitudes of the vectors?
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§12.3 DOT PRODUCT
Example. Find the acute angle between the lines. Round your answer to the nearestdegree.
5x � y = 5, 9x + y = 6
Example. Prove that for two lines that are not horizontal or vertical, the two lines areperpendicular if and only if their slopes are ...
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§12.3 DOT PRODUCT
Example. True or False:
(a) ~i · ~j = ~j ·~k = ~k ·~i = 0
(b) ~i ·~i = ~j · ~j = ~k ·~k = 1
Example. Find a unit vector that is orthogonal to both~i + ~j and~i +~k.
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§12.3 DOT PRODUCT
Scalar and vector projections
Definition. The scalar projection of ~bonto ~a is given by
comp~a~b =
Definition. The vector projection of ~bonto ~a is given by
proj~a~b =
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§12.3 DOT PRODUCT
Example. Find the scalar and vector projections of ~b onto ~a, where~b =< �2, 3,�6 >~a =< 5,�1, 4 >
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§12.3 DOT PRODUCT
Recall: The work done by a constant force, moving an object in the direction of theforce, is:
Definition. The work done by a constant force ~F moving an object along a vector ~D(not necessarily in the direction of the force) is:
Question. How can we write work in terms of the dot product?
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§12.3 DOT PRODUCT
Example. A tow truck drags a stalled car along a road. The chain makes an angle of30� with the road and the tension in the chain is 500N. How much work is done by thetruck in pulling the car 3 km?
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§12.3 DOT PRODUCT
Extra Example. True or False and justify your answer.
|
~a �~b| ||~a|| · ||~b||
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§12.3 DOT PRODUCT
Extra Example. True or False and justify your answer.
||
~a +~b|| ||~a|| + ||~b||
Hint: ||~a +~b||2 = (~a +~b) � (~a +~b)
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§12.3 DOT PRODUCT
Extra Example. Use scalar projection to show that the distance from a point P1(x1, y1)to the line ax + by + c = 0 is
|ax1 + by1 + c|p
a2 + b2
Use this formula to find the distance from the point (�2, 3) and the line 3x� 4y+ 5 = 0.
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§12.4 CROSS PRODUCTS
§12.4 Cross Products
After completing this section, students will be able to:
• Compute the cross-product of two 3-dimensional vectors from their components.
• Compute the magnitude of the cross-product from the magnitude of the vectorsand the angle between them.
• Use the right hand rule to find the direction of the cross-product.
• Use cross product to find a vector perpendicular to two other vectors.
• Use cross product to determine if two vectors are parallel.
• Use cross product to find the area of a parallelogram or triangle.
• Use properties of cross product to determine if statements about vectors are trueor false.
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§12.4 CROSS PRODUCTS
Definition. The cross-product of two vectors ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 > isgiven by:
~a ⇥~b =
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§12.4 CROSS PRODUCTS
Example. For ~a =< 1, 2, 3 > and ~b =< 5,�1, 10 >, find ~a ⇥~b.
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§12.4 CROSS PRODUCTS
Properties of Cross Product:
• If ✓ is the angle between ~a and ~b, with 0 ✓ ⇡, then the length of ~a ⇥~b is givenby
||
~a ⇥~b|| =
• The vector ~a ⇥~b is perpendicular to ...
• The direction of ~a ⇥~b is given by ...
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§12.4 CROSS PRODUCTS
Example. For the two vectors shown, find ||~a ⇥ ~b|| and determine whether ~a ⇥ ~b isdirected into the page of out of the page.
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§12.4 CROSS PRODUCTS
Proposition. If ✓ is the angle between ~a and ~b, with 0 ✓ ⇡, then
||
~a ⇥~b|| = ||~a|| ||~b|| sin✓
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§12.4 CROSS PRODUCTS
Proposition. The vector ~a ⇥~b is perpendicular to both ~a and ~b.
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§12.4 CROSS PRODUCTS
Proposition. The direction of ~a ⇥~b is given by the right hand rule.
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§12.4 CROSS PRODUCTS
Review. The cross-product of two vectors ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 > isdefined in terms of components as:
or in terms of length and direction as:
Example. Find the cross product ~u ⇥ ~w where ~u, ~v, and ~w are vectors of length 5 thatlie in the x-y plane.
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§12.4 CROSS PRODUCTS
True or False: If two nonzero vectors ~a and ~b are parallel, then ~a ⇥~b = ~0.
True or False: If two nonzero vectors ~a and ~b have cross-product ~a ⇥~b = ~0, then ~a and~b are parallel.
Question. How can you use cross product to:
• find a vector perpendicular to two vectors?
• determine if two vectors are parallel?
Question. Is it possible to find the exact angle between two vectors if all we know isthe cross product and the magnitudes of the vectors?
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§12.4 CROSS PRODUCTS
Example. Find a unit vector orthogonal to both~i + ~j and~i +~k.
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§12.4 CROSS PRODUCTS
Example. Use the cross product to write the area of the parallelogram in terms of ~aand ~b.
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§12.4 CROSS PRODUCTS
Example. Find the area of the triangle with vertices P(0, 0,�3), Q(4, 2, 0), and R(3, 3, 1).
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§12.4 CROSS PRODUCTS
True or False
1. ~a ⇥~b is a scalar.
2. ~a ⇥ ~a = ~0.
3. For ~a,~b , ~0, if ~a ⇥~b = ~0 then ~a = ~b.
4. ~i ⇥ ~j = ~k
5. ~a ⇥~b = ~b ⇥ ~a
6. (~a ⇥~b) ⇥ ~c = ~a ⇥ (~b ⇥ ~c)
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§12.4 CROSS PRODUCTS
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§12.4 CROSS PRODUCTS
Extra Example. Are these three vectors coplanar?
~u = 2~i + 3~j +~k
~v =~i � ~j
~w = 7~i + 3~j + 2~k
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§12.4 CROSS PRODUCTS
Extra Example. 1. Find all vectors ~v such that < 1, 2, 1 > ⇥ ~v =< 3, 1,�5 >.
2. Explain why there is no vector ~v such that < 1, 2, 1 > ⇥ ~v =< 3, 1, 5 >.
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§12.5 LINES AND PLANES
§12.5 Lines and Planes
After completing this section, students should be able to:
• Recognize equations of lines in parametric form, symmetric form, and vector form.
• Recognize equations of planes.
• Write the equation of a line given its direction and a point on the line, or given twopoints, or similar information.
• Write the equation of a plane given three points, or one point and a normal vector,or through a line and a point, or through two intersecting or parallel lines, or fromsimilar information.
• Determine if planes intersect and find lines of intersection and angles of intersec-tion.
• Determine if lines intersect, are parallel, or are skew.
• Use scalar projection to find the distance from a point to a plane, or a point toa line, or between two skew or parallel lines, or between two planes, or similarconfigurations.
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§12.5 LINES AND PLANES
Example. Is the line through (�4,�6, 1) and (�2, 0,�3) parallel to the line through(10, 18, 4) and (5, 3, 14)?
What is the equation of the line through the origin, that is parallel to the line through(�4,�6, 1) and (�2, 0,�3)?
What is the equation of the line through (�4,�6, 1) and (�2, 0,�3)?
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§12.5 LINES AND PLANES
How else could you write an equation of the line through (�4,�6, 1) and (�2, 0,�3)?
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§12.5 LINES AND PLANES
Note. The equation of a line through the point (x0, y0, z0) in the direction of the vector< a, b, c > can be described:
with the parametric equations:
or, with ”symmetric equations”:
or, with the vector equation:
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§12.5 LINES AND PLANES
Example. Find the equation of the plane through the point (�3, 2, 0) and perpendicularto the vector < 1,�2, 5 >
Note. The plane through the point (x0, y0, z0) and perpendicular to the vector < a, b, c >is given by the equation
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§12.5 LINES AND PLANES
Review. Which of these equations represents a line in 3-dimensional space?
1. 3x + 5y = 2
2. 3(x � 1) + 5(y � 3) � 4(z � 2) = 0
3. x = 7 + 4t, y = 5 � 3t, z = 7t
4. x�74 =
�y+53 = z
7
5. ~r(t) =< 7 + 4t, 5 � 3t, 7t >
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§12.5 LINES AND PLANES
Review. The equation of a line in 3-dimensional space can be described
with parametric equations:
with ”symmetric equations”:
with the vector equation:
where < a, b, c > represents ...
and < x0, y0, z0 > represents ...
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§12.5 LINES AND PLANES
Review. The equation of a plane in 3-dimensional space can be described by:
where < a, b, c > represents ...
and < x0, y0, z0 > represents ...
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§12.5 LINES AND PLANES
Example. Find an equation of the line though the point (3,�1, 2) and perpendicular tothe plane 4x � 6y + z = 13.
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§12.5 LINES AND PLANES
Example. Find an equation for the plane through the points (3,�1, 2), (8, 2, 4), and(�1,�2,�3).
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§12.5 LINES AND PLANES
Question. What information is enough to determine the equation for a line in the plane?
Question. What information is enough to determine the equation for a line in 3-dimensional space?
Question. What information is enough to determine the equation for a plane in 3-dimensional space?
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§12.5 LINES AND PLANES
Example. Find the line of intersection of the planes 2x � y + z = 5 and x + y + z = 1.
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§12.5 LINES AND PLANES
Example. Determine whether the lines L1 and L2 are parallel, skew, or intersecting.
L1 : x = 3 + 2t, y = 4 � t, z = 1 + 3t
L2 : x = 1 + 4s, y = 3 � 2s, z = 4 + 5s
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§12.5 LINES AND PLANES
Example. Determine whether the planes are parallel, perpendicular or neither. Ifneither, find the angle between them.
x + 2y + 2z = 1, 2x � y + 2z = 1
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§12.5 LINES AND PLANES
Extra Example. Find the distance between the point (1, 2, 3) and the plane 5x+4y�3z =10.
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§12.5 LINES AND PLANES
Extra Example. Find the distance between the skew lines
L1 : x = 3 + 2t, y = 4 � t, z = 1 + 3t
L2 : x = 1 + 4s, y = 3 � 2s, z = 4 + 5s
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§13.1 VECTOR FUNCTIONS
§13.1 Vector Functions
After completing this section, students should be able to:
• Define a vector valued function.
• Find the domain of a vector valued function.
• Find the limit of a vector valued function.
• Match equations of vector valued functions with their graphs by considering theprojections of the graphs onto the xy, yz, and xz planes.
• Give a vector valued equation for the intersection of two surfaces.
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§13.1 VECTOR FUNCTIONS
Definition. A vector function or vector-valued function is:
If we think of the vectors as position vectors with their initial points at the origin, thenthe endpoints of ~v(t) trace out a in R3 (or in R2).
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§13.1 VECTOR FUNCTIONS
Example. Sketch the curve defined by the vector function ~r(t) =< t, sin(5t), cos(5t) >.
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§13.1 VECTOR FUNCTIONS
Example. Consider the vector function ~r(t) =t2� t
t � 1~i +p
t + 8~j +sin(⇡t)
ln t~k
1. What is the domain of ~r(t)?
2. Find limt!1~r(t)
3. Is ~r(t) continuous on (0,1)? Why or why not?
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§13.1 VECTOR FUNCTIONS
Review. Match the vector equations with the curves.
1. ~r(t) =< t2, t4, t6 >
2. ~r(t) =< t + 2, 3 � t, 2t � 1 >
3. ~s(t) =< cos(t),� cos(t), sin(t) >
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§13.1 VECTOR FUNCTIONS
Example. At what points does the helix ~r(t) =< sin(t), cos(t), t > intersect the spherex2 + y2 + z2 = 5?
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§13.1 VECTOR FUNCTIONS
Example. Find parametric equations for the curve of intersection of the paraboliccylinder y = x2 and the top half of the ellipsoid x2 + 4y2 + 4z2 = 16.
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§13.1 VECTOR FUNCTIONS
Example. Find a vector function that represents the curve of intersection of the hyper-boloid z = x2
� y2 and the cylinder x2 + y2 = 1.
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§13.1 VECTOR FUNCTIONS
Extra Example. Consider the vector function ~r(t) = te�t~i +t3 + t
2t3� 1~j +
1p
t~k
1. What is the domain of ~r(t)?
2. Find limt!1~r(t)
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
§13.2 Derivatives and Integrals of Vector Functions
By the end of this section, students should be able to:
• Compute the derivative of a vector function.
• Compute the integral of a vector function.
• When ~r(t) represents the position of a particle at time t, explain the meaning of~r0(t), its direction, and its magnitude.
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Suppose a particle is moving according to the vector equation ~r(t). How can we find atangent vector that gives the direction and speed that the particle is traveling?
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Definition. The derivative of the vector function ~r(t) is the same thing as the tangentvector, defined as
d~rdt= ~r 0(t) =
If ~r(t) =< r1(t), r2(t), r3(t) >, then
~r 0(t) =
The derivative of a vector function is a (circle one) vector / scalar.
The unit tangent vector is:~T(t) =
The tangent line at t = a is:
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Example. For the vector function ~r(t) =< t2, t3 >
1. Find ~r 0(1).
2. Sketch ~r(t) and ~r 0(1).
3. Find ~T(1).
4. Find the equation for the tangent line at t = 1.
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Definition. If ~r(t) =< f (t), g(t), h(t) >, thenZ~r(t) dt =
and
Z b
a~r(t) dt =
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Example. ComputeZ 2
1
1t~i + et~j + tet~k.
END OF VIDEO
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Review. If ~r(t) = 5t2~i + sin(t)~j � 3~k, how do we compute ~r 0(t)?
Question. For a vector function ~r(t) =< r1(t), r2(t), r3(t) >, what is the limit definition of~r 0(t)?
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Question. If we think of ~r(t) as a space curve, what does ~r 0(t) represent geometrically?
Question. If ~r(t) represents the position of a particle at time t,
(a) what does the direction of ~r 0(t) signify?
(b) what does the magnitude of ~r 0(t) signify?
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Example. Find the tangent vector, the unit tangent vector, and the tangent line for thefollowing curves at the point given
1. ~r(t) =< t, t2, t3 > at t = 1
2. ~r(t) =< t2, t4, t6 > at t = 1
3. ~p(t) =< t + 2, 3 � t, 2t � 1 > at t = 0
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Example. At what point do the curves ~r1(t) =< t, 1�t, 3+t2 > and ~r2(t) =< 3�t, t�2, t2 >intersect? Find their angle of intersection correct to the nearest degree.
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Derivative rules - see textbook
• Is there a product rule for derivatives of vector functions?
• Is there a quotient rule for derivatives of vector functions?
• Is there a chain rule for derivatives of vector functions?
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Example. Show that if ||~r(t)|| = c (a constant), then ~r 0(t) is orthogonal to ~r(t) for all t.
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Review. If ~r(t) =< f (t), g(t), h(t) >, thenZ~r(t) dt =
and
Z b
a~r(t) dt =
Example. Find ~p(t) if ~p 0(t) = cos(⇡t)~i + sin(⇡t)~j + t~k and ~p(1) = 6~i + 6~j + 6~k.
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§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
Extra Example. (# 53) Show that if ~r is a vector function such that ~r 00 exists, then
ddt
[~r(t) ⇥ ~r 0(t)] = ~r(t) ⇥ ~r 00(t)
Extra Example. (#57) If ~u(t) = ~r(t) � [~r 0(t) ⇥ ~r 00(t)], show that
~u 0(t) = ~r(t) · [~r 0(t) ⇥ ~r 000(t)]
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§14.1 FUNCTIONS OF TWO OR MORE VARIABLES
§14.1 Functions of Two or More Variables
After completing this section, students should be able to:
• Match equations of the form z = f (x, y) to graphs of surfaces and graphs of levelcurves.
• Describe the graphs of functions of three variables w = f (x, y, z) in terms of thelevel curves f (x, y, z) = k
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§14.1 FUNCTIONS OF TWO OR MORE VARIABLES
Example. Consider the function of two variables f (x, y) = pxy
1. What is its domain?
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§14.1 FUNCTIONS OF TWO OR MORE VARIABLES
For the function f (x, y) = pxy ...
2. What are its level curves?
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§14.1 FUNCTIONS OF TWO OR MORE VARIABLES
For the function f (x, y) = pxy ...
3. What does its graph look like?
END OF VIDEO
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§14.1 FUNCTIONS OF TWO OR MORE VARIABLES
1. z = sinp
x2 + y2 2. z = 1x2+4y2 3. z = sin(x) sin(y)
4. z = x2y2e�(x2+y2) 5. z = x3� 3xy2 6. z = sin2(x) + y2
4
I. II. III.
IV. V. VI.
A. B. C.
D. E. F.
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§14.1 FUNCTIONS OF TWO OR MORE VARIABLES
Functions of 3 or more variables
To visualize functions f (x, y, z) of three variables, it is handy to look at level surfaces.Example. f (x, y, z) = x2 + y2 + z2
(a) Guess what the level surfaces should look like.
(b) Graph a few level surfaces (e.g. x2 + y2 + z2 = 10, x2� y2 + z2 = 20, x2
� y2 + z2 = 30)on a 3-d plot.
Example. f (x, y, z) = x2� y2 + z2
1. Guess what the level surfaces should look like.
2. Graph a few level surfaces (e.g. x2� y2 + z2 = 0, x2
� y2 + z2 = 10, x2� y2 + z2 = 20)
on a 3-d plot.
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§14.2 LIMITS AND CONTINUITY
§14.2 Limits and Continuity
After completing this section, students should be able to:
1. Use a graph to build intuition for whether or not a limit of a function of twovariables exists.
2. Prove that a limit of a function of two variables does not exist by finding two pathsalong which the function approaches di↵erent values.
3. Prove that a limit of a function of two variables does exist by converting to polarcoordinates and using the squeeze theorem.
4. State the definition of continuity for functions of two variables in terms of limits.
5. Determine where a function is continuous.
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§14.2 LIMITS AND CONTINUITY
Recall LIMITS from Calculus 1:
Informally, limx!a
f (x) = L if the y-values f (x) get closer and closer to the same number Lwhen x approaches a from either the left of the right.
Does limx!0
f (x) exist for these functions?
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§14.2 LIMITS AND CONTINUITY
LIMITS for functions of two variables:
Informally, lim(x,y)!(a,b)
f (x, y) = L if the z-values f (x, y) ...
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§14.2 LIMITS AND CONTINUITY
For each function, decide if lim(x,y)!(0,0)
f (x, y) exists. The color is based on height: low is
blue and high is red.
A) lim(x,y)!(0,0)
x2� y2
x2 + y2 B) lim(x,y)!(0,0)
1 � x2
x2 + y2 + 1
C) lim(x,y)!(0,0)
xx2 + y2 D) lim
(x,y)!(0,0)
x3
x2 + y2
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§14.2 LIMITS AND CONTINUITY
Example. lim(x,y)!(0,0)
x2� y2
x2 + y2
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§14.2 LIMITS AND CONTINUITY
Example. lim(x,y)!(0,0)
1 � x2
x2 + y2 + 1
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§14.2 LIMITS AND CONTINUITY
Example. lim(x,y)!(0,0)
xx2 + y2
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§14.2 LIMITS AND CONTINUITY
Example. lim(x,y)!(0,0)
x3
x2 + y2
END OF VIDEO
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§14.2 LIMITS AND CONTINUITY
Review. Informally, lim(x,y)!(a,b)
f (x, y) = L if the z-values f (x, y) ...
lim(x,y)!(0,0)
x2� y2
x2 + y2
lim(x,y)!(0,0)
x3
x2 + y2
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§14.2 LIMITS AND CONTINUITY
Does lim(x,y)!(0,0)
f (x, y) exist? The color is based on height: low is blue and high is red.
A) lim(x,y)!(0,0)
2xyx2 + y2 B) lim
(x,y)!(0,0)
xy2
x2 + y4
C) lim(x,y)!(0,0)
10 sin(xy)x + y + 20
D) lim(x,y)!(0,0)
3x2y � 3x2� 3y2
x2 + y2
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§14.2 LIMITS AND CONTINUITY
Example. Show that lim(x,y)!(0,0)
2xyx2 + y2 does not exist.
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§14.2 LIMITS AND CONTINUITY
Example. Show that lim(x,y)!(0,0)
xy2
x2 + y4 does not exist.
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§14.2 LIMITS AND CONTINUITY
Example. Show that lim(x,y)!(0,0)
f (x, y) =10 sin(xy)x + y + 20
does exist.
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§14.2 LIMITS AND CONTINUITY
Example. Show that lim(x,y)!(0,0)
3x2y � 3x2� 3y2
x2 + y2 does exist.
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§14.2 LIMITS AND CONTINUITY
Summary: for practical purposes, the best way to show that a limit does not exist is to:
For practical purposes, the best ways to show that a limit exists is to:
•
•
•
•
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§14.2 LIMITS AND CONTINUITY
Recall CONTINUITY from Calculus 1: A function f is continuous at the point x = a if:
1.
2.
3.
Recall from Calculus 1: common functions that are continuous include:
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§14.2 LIMITS AND CONTINUITY
CONTINUITY for functions of of two (or more) variables:
A function f (x, y) is continuous at the point (x, y) = (a, b) if:
1.
2.
3.
Common functions that are continuous include:
•
•
•
•
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§14.2 LIMITS AND CONTINUITY
Example. Where is f (x, y) =
p
4 � x2 +p
4 � y2
1 � x2� y2 continuous?
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§14.3 PARTIAL DERIVATIVES
§14.3 Partial Derivatives
After completing this section, students should be able to:
• Compute partial derivatives.
• Use average rates of change to approximate partial derivatives.
• For functions of two variables, explain the geometric meaning of a partial deriva-tive as the slope of a tangent line to a curve in the intersection of a surface and aplane.
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§14.3 PARTIAL DERIVATIVES
Example. (from book) The wave heights h in the open sea depend on the speed ⌫ ofthe wind and the length of time t that the wind has been blowing at that speed. So wewrite h = f (⌫, t).
1. What is f (40, 20)?
2. If we fix duration at t = 20 hours andthink of g(⌫) = f (⌫, 20) as a functionof ⌫, what is the approximate valueof the derivative dg
d⌫
����⌫=40
?
3. If we fix wind speed at 40 knots, andthink of k(t) = f (40, t) as a function ofduration t, what is the approximatevalue of the derivative dk
dt
���t=20?
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§14.3 PARTIAL DERIVATIVES
Definition. For a function f (x, y) defined near (a, b), the partial derivatives of f at (a, b)are:
fx(a, b) = the derivative of f (x, b) with respect to x when x = a, and
fy(a, b) = the derivative of f (a, y) with respect to y when y = b.
In terms of the limit definition of derivatives, we have:
fx(a, b) = limh!0
fy(a, b) = limh!0
Geometrically, f (x, b) can be thought of as
So fx(a, b) = ddx f (x, b)|x=a can be thought of as
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§14.3 PARTIAL DERIVATIVES
Note. To compute fx, we just take the derivative with x as our variable, holding allother variables constant. Similarly for the partial derivative with respect to any othervariable.
Example. f (x, y) = xy. Find fx(1, 2) and fy(1, 2).
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§14.3 PARTIAL DERIVATIVES
Notation. There are many notations for partial derivatives, including the following:
fx@ f@x
@z@x f1 D1 f Dx f
Note. Partial derivatives can also be taken for functions of three or more variables. Forexample, if f (x, y, z,w) is a function of 4 variables, then fz(3, 4, 2, 7) means:
END OF VIDEO
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§14.3 PARTIAL DERIVATIVES
Review. Which of the following is the limit definition of@ f@x
(1, 2)?
A. limh!0
f (1 + h, 2 + h) � f (1, 2)h
B. limh!0
f (1 + h, 2) � f (1, 2 + h)h
C. limh!0
f (1 + h, 2) � f (1, 2)h
D. limh!0
f (1, 2 + h) � f (1, 2)h
Review. Based on the graph of z = f (x, y) shown
• is@ f@x
(1, 2) positive, negative, or 0?
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§14.3 PARTIAL DERIVATIVES
• is@ f@y
(1, 2) positive, negative, or 0?
Question. For f (x, y) = 3x2y + 5y2 + cos(xy), find@ f@x
.
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§14.3 PARTIAL DERIVATIVES
Example. For f (x, y, z) = ex sin(y�z), find Dx f and Dz f .
Example. For z = f (x)g(y), find an expression for@z@x
.
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§14.3 PARTIAL DERIVATIVES
Example. For x2� y2 + z2
� 2z = 12, find@z@y
.
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§14.3 PARTIAL DERIVATIVES
Notation. Second derivatives are written using any of the following notations:
Example. For f (x, y) = x2 + x2y2� 2y2, calculate fxx, fxy, fyx, and fyy.
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§14.3 PARTIAL DERIVATIVES
Theorem. (Clairout’s Thm) Let f (x, y) be defined on a disk D containing (a, b). If fxy
and fyx are both defined and on D, then fxy(a, b) = fyx(a, b).
Example. Consider the function
f (x, y) =
8>><>>:xyx2�y2
x2+y2 if (x, y) , (0, 0)
0 if (x, y) = (0, 0)
It turns out that
• for all x D2 f (x, 0) =
• for all y D1 f (0, y) =
Therefore,
• D2,1 f (0, 0) =
• D1,2 f (0, 0) =
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§14.3 PARTIAL DERIVATIVES
Question. True or False: There is a function f with continuous second partial deriva-tives such that fx = x and fy = x.
Extra Example. The wave equation is given by
@2u@t2 = a2@
2u@x2
where u(x, t) represents displacement, t represents time, x represents the distance fromone end of the wave, and a is a constant.
Verify that the function u(x, t) = sin(x � at) satisfies the wave equation.
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
§14.4 Tangent Planes and Linear Approximations
After completing this section, students should be able to:
• Find the equation for the tangent plane of a surface z = f (x, y) at a point.
• Use the tangent plane to approximate a function.
• Compute the di↵erential and use it to estimate errors.
• Explain the relationship between the tangent plane, the linearization equation, andthe di↵erential.
• Compute the tangent plane from information about the function’s value and partialderivative values.
• Use linearization to interpolate between numerical values in a 2 x 2 talbe.
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Tangent line:
Tangent plane:
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Example. Find the tangent plane to the surface z = y2� x2 at the point (1, 2, 3).
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Find a general formula for the tangent plane of z = f (x, y) at the point (x0, y0, z0).
Notice that this formula is analogous to the formula for a tangent line for a function ofone variable.
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Usually, the tangent plane at a point (x0, y0, z0) is a good approximation of the surfacenear that point.
Example. Find the tangent plane for the surface z = f (x, y) when (x, y) = (1, 1), wheref (x, y) = 1 � xy cos(⇡y). Use it to approximate f (1.02, 0.97).
END OF VIDEO
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Example. Suppose we have a surface z = f (x, y) and we know that f (1, 2) = 6, fx(1, 2) =�7 and fy(1, 2) = 4. What is the equation for the tangent plane to the surface at thepoint (1, 2, 6)?
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Example. Find the tangent plane for the surface z = sin(x)+ y2 at (x, y) = (0, 2) and useit to approximate sin(0.2) + (1.9)2
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
This method of approximating a function’s value with the height of the tangent planecan be written in terms of a linearization equation or in the language of di↵erentials.
Definition. The linearization of f (x, y) is written as:
Recall: In Calc 1, for a function y = f (x), the di↵erential was defined as d f = f 0(x)dx.
Definition. The di↵erential for the function y = f (x, y) is:
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
� f represents the actual change in a function f (x, y) � f (a, b). d f represents the corre-sponding change in the tangent plane between (a, b) and (x, y).
Di↵erentials are useful for estimating errors and making approximations, because thedi↵erential is linear in all its variables, and linear functions are easier to calculate.
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Example. Use di↵erentials to estimate the amount of metal in a closed cylindrical canthat is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cmthick and the metal in the sides is 0.05 cm thick.
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Example. Four positive numbers, each less than 50, are rounded to the first decimalplace and multiplied together. Use di↵erentials to estimate the maximum possibleerror in the computed product that might result from the rounding.
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
When the tangent plane approximation works and when it doesn’t.
Note. Anytime fx and fy exist at a point (a, b), it is possible to write down an equationfor a tangent plane at the point where (x, y) = (a, b).
• Usually ...
• Sometimes ...
Definition. (Informal definition) A function is f is called di↵erentiable at (a, b) if ...
Theorem. If fx and fy exist in near (a, b) and are at (a, b), then f isdi↵erentiable at (a, b).
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Example. Here is an example of a function that is not di↵erentiable at (x, y) = (0, 0).
f (x, y) =
8>><>>:xy
x2+y2 if (x, y) , (0, 0)
0 if (x, y) = (0, 0)
Question. In what sense is the tangent plane not a good approximation to the function?
Question. Is this function continuous at (0, 0)?
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Question. What was the relationship between di↵erentiability and continuity in Cal-culus 1?
Question. What is the relationship between di↵erentiability and continuity for func-tions of several variables?
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§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
Recap: The tangent plane, the linearization, and the di↵erential are related as follows:
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§14.5 THE CHAIN RULE
§14.5 The Chain Rule
After completing the section, students should be able to:
• Compute partial derivatives using the Chain Rule, using formulas for functions,or using numerical information such as tables of values or contour graphs.
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