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Course: MATH 367, Spring 2011School: Texas A&M
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7 Homework Identify the congruent triangles and show they are congruent. Note a picture have can more than one pair of triangles. 1. 2. 3. 4. 5. 6. 7. 8. 12
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7 Homework Identify the congruent triangles and show they are congruent. Note a picture have can more than one pair of triangles.
1.
2.
3.
4.
5.
6.
7.
8.
12
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Texas A&M - MATH - 367
An axiomatic system example. Assume that a club of two or more students is organized into committees in such a way that each of the following conditions are satised. a) Every committee is a set of one or more students. b) For each pair of students, there
Texas A&M - MATH - 367
Denition 28 Parallel Two lines are parallel in a plane if they dont intersect. Postulate 16 Parallel Postulate Through a point not on a given line there is exactly one parallel to the line. Postulate 17 Lobachevsky and Bolya Postulate Through a point not
Texas A&M - MATH - 367
Theorem 29 If A and B are equidistant from P and Q then every point between A and B has the same property. Theorem 30 If a line L contains the midpoint of P Q and contains another point which is equidistant from P and Q, then L P Q.Theorem 40 Through a g
Texas A&M - MATH - 367
Denition 33 polygon A polygon is the union of n segments in a plane, intersecting at and only at their endpoints, such that exactly two segments contain each endpoint and no two consecutive segments are on the same line. We are going to assume that any po
Texas A&M - MATH - 367
Denition 12 Segment If A and B are two points then the segment between them is the set of points A X B , for all X in S , along with A and B . It is written AB Denition 13 Ray Is the set of points C on AB with A not between B and C . It is written AB . De
Texas A&M - MATH - 367
Theorem 14 Segment construction [3.6.C-2] Given a segment AB and a ray CD, there is exactly one point E on CD such that AB CE . = Theorem 15 Segment Addition [3.6.C-3] If A B C and D E F and AB DE and = BC EF then AC DF . = = Denition 22 Convex A set G is
Texas A&M - MATH - 367
Postulate 12 Space separation postulate [4.5.ss-1] Given a plane in space. The set of all points that do not lie in the plane is the union of two sets S1 , S2 (the book uses H1 and H2 we will reserve those for half-planes) such that each of the sets is co
Texas A&M - MATH - 367
Denition 7 Logical System consists of undened terms, denitions, assumptions and theorems. The undened terms for geometry are set, point, line, plane. Denition 8 One-to-one correspondence is if two sets have the same number of elements. If two sets have a
Texas A&M - MATH - 367
Denition 1 Negation If p is a statement, the statement p is the negation of p. Example 1 Form the negation of the following statements. a The moon is rising. b ABC is a remote interior angle. c Point C is between points A and B . d m 3 = 25 Solution a The
Texas A&M - MATH - 251
Answers to exam 2, version A1. In general, the directional derivative of f at (a, b) in the direction of a unitvector u is Du f (a, b) = f (a, b) u. So,=exy + xyexy , x2 exy=f2e, eat (1, 1). The vector going from (1, 1) to (2, 3) is 1, 2 , but thi
Texas A&M - MATH - 251
Answers to exam 3, version A 1. If D is the region bounded by x and x2 , put in = ky to getx = =which integrates out to 5 . 8Dx dA dA D1 x xky dy dx 0 x2 , 1 x 2 ky dy dx 0 x2. The solid that we're integrating over has its top as part of a spherical
Texas A&M - MATH - 251
Exam 1, version A 9/19/08 1. For the function f (x, y ) = 16 4x2 + 9 y 2 , (a) (6 pts.) sketch the domain. (b) (6 pts.) nd the range. 2. (8 pts.) Find two unit vectors perpendicular to both 2, 1, 1 and 1, 2, 1 . 3. (10 pts.) Find the equation of the plane
Texas A&M - MATH - 251
Math 251.512Exam 2, version A10/15/081. (11 pts.) If f (x, y ) = xexy , nd the directional derivative of f at (1, 1) in thedirection from (1, 1) to (2, 3).2. (11 pts.) FindRxdA, where R is the rectangle 1 x 2, 0 y 4.1 + 2yx, use dierentials to
Texas A&M - MATH - 251
Exam 3, version A 11/12/08 Conversion from spherical to cartesian coordinates:Math 251.512x = sin cos y = sin sin z = cos dV =2 sin d d d1. (10 pts.) Determine x of the center of mass of a plate bounded by y = x and y = x2 , if density is proportional
Texas A&M - MATH - 251
Math 251.504Exam 1, version ASolutions1. (11 pts.) Find the equation of the plane containing the points (0, 1, 1), (2, 1, 2), and (3, 0, 1).Answer: To nd the equation of a plane, we need a point in the plane and a vector normal tothe plane. Weve alre
Texas A&M - MATH - 251
Math 251.504Exam 1, version BSolutions1. (11 pts.) Find the equation of the plane containing the points (1, 1, 1), (1, 1, 0), and (3, 0, 1).Answer: To nd the equation of a plane, we need a point in the plane and a vector normal tothe plane. Weve alre
Texas A&M - MATH - 251
Math 251.504Exam 2, version ASolutions1. (10 pts.) Evaluate D x dA, where D is the triangular region withvertices (0, 0), (1, 1), and (1, 4). Solution: In the order dy dx, thedouble integral becomes1 1 4xxy |y=4x dxx dy dx =y =x0x0 1=3x2 dx
Texas A&M - MATH - 251
Math 251.504Exam 2, version BSolutions1. (10 pts.) Evaluate D x dA, where D is the triangular region withvertices (0, 0), (1, 1), and (1, 3). Solution: In the order dy dx, thedouble integral becomes1 1 3xxy |y=3x dxx dy dx =y =x0x0 1=2x2 dx
Texas A&M - MATH - 151
11.1: VectorsDenitions:vector:additionscalar multiplicationsubtractionmagnitude:unit vector:i and j:1Examples:Given the vectors a =< 3, 5 > and b starts at the point (1, 1) and ends at the point (1, 3),write i + j in terms of a and b.Given a
Texas A&M - MATH - 151
11.2: Dot ProductDenitions:The dot product of the vectors a and b is given byDot Product computation formulaFrom the denition, it follows that the angle between two vectors is given bya and b are orthogonal if and only ifOrthogonal complementsScal
Texas A&M - MATH - 151
11.3: Vector Functions and Parametrized CurvesDenitions:(Recall) function:Vector Valued function:Parametrized Curve:Eliminating the ParameterVector and Parametric Equations of a Line1Examples:Given the curve parametrized by r(t) = (t2 + 1)i + (t
Texas A&M - MATH - 151
12.1/2.2: Intro to Calculus and LimitsGoal #1: To nd the slope of a line tangent to a curve at a given point.Concept of a Limit (Maplet):Innite Limits and Vertical Asymptotes:1Examples:x2 + 1x 1 x 1limOn Beyond Average: Find the vertical asympto
Texas A&M - MATH - 151
12.3: Analytic Computation of LimitsProperties of Limits: (pp 91-93. Basis for the techniques used in the following examples.)Examples:lim x3 3x2 + 1x 1limx42x + 8x2 + x 12lim r(t) where r(t) =t25t3 + 4t3i+t2 4t2j1Squeeze Theorem: If g
Texas A&M - MATH - 151
12.5: ContinuityDenitions:f is continuous at x = aRemovable DiscontinuitiesSource for understanding: Maplet Left and Right Hand Limits and Continuities, located athttp:/calclab.math.tamu.edu/maple/maplets/ (NetID login)1Theorems:Limits inside Con
Texas A&M - MATH - 151
12.6: Limits at InnityIn 2.2, we learned that if y as x a, then the graph of f has a vertical asymptote atx = a. Similarly, if y L as x , then the graph of the function has a horizontal asymptoteat y = L.Key Limit:limx1=xComputing Limits at Inn
Texas A&M - MATH - 151
12.7: Tangents, Velocities, and Rates of ChangeWe are now ready to nd a formal way of computing the slope of the line tangent to the curve. Re-viewthe animation from 2.1 posted on my webpage. What happens as the second xcoordinate moves closerto the g
Texas A&M - MATH - 151
13.1: The DerivativeNow that we can nd the slope of the line tangent to a curve at any point (provided the limit of theslope exists), we can talk about a new function based on this calculation.Denition: The derivative function of a function f (or the
Texas A&M - MATH - 151
13.2: Derivative RulesDerivative Rules:If f and g are dierentiable functions, then.dn(x ) =dxd(cf (x) =dxd(f (x) g (x) =dxd(f (x) g (x) =dxddxf (x)g (x)=Examples:2 xx.Compute the derivative of f (x) = 3x+31Find the derivative o
Texas A&M - MATH - 151
3.3-Rates of ChangeRecall: The derivative of a function can measure:Position/Velocity/AccelerationExamples:A particle moves in a line according to the function s = f (t) = t3 6t2 + 9t, where t is in secondsand s is in feet.a) Find the velocity at ti
Texas A&M - MATH - 151
13.4: Derivatives of Trig Functionssin x=x 0xKey Limit: limProof : (zoom in on graph of y = sin x at x = 0)Key Limit: limx 0cos x 1=xProof :We can use these limits to nd the derivative of f (x) = sin x using the denition:1Similarly, we can
Texas A&M - MATH - 151
13.7: Derivatives of Vector FunctionsRecall denition:What the derivative of a vector function tells us:Examples:Find the velocity and speed if the position function is given by r(t) = (5 cos t)i + (5 sin t)j at thepoint (3, 4).1Find a unit tangent
Texas A&M - MATH - 151
13.8: Higher DerivativesSecond derivative: derivative of the rst derivativeWhat the second derivative tells us:Examples:Label each of the graphs below as the original function, rst derivative, or second derivative.1Find and simplify the rst and sec
Texas A&M - MATH - 151
13.9: Tangents of Parametrized CurvesTo nd the slope of the tangent line for a parametrized curve, use the fact thatdy=dxExamples:Find an equation of the line tangent to the curve given by x = 3 cos t, y = 4 sin t at the pointwhere t = .61The c
Texas A&M - MATH - 151
13.10: Related RatesIdea: As certain quantities change over time, quantities which are related to them (usually via aformula) also change over time.Problem Solving Strategies:Examples:A spherical snowball is melting at a rate of 5 cubic centimeters
Texas A&M - MATH - 151
13.5: Chain RuleFrom 3.4: we knowdd(sin x) = cos x. Does(sin 2x) = cos 2x?dxdxRecall: The composition of 2 functions f and g is dened byDene f and g for the above function.The Chain Rule: If f and g are dierentiable functions, y = f (u) and u =
Texas A&M - MATH - 151
13.6: Implicit DierentiationThe equation F (x, y ) = 0 implicitly denes a relation (not necessarily a function) between y and x.The graph of F (x, y ) = 0 is the set of all points (x, y ) such that the equation holds ( cfw_(x, y )|F (x, y ) =0 ) Given
Texas A&M - MATH - 151
13.11: Linear and Quadratic ApproximationPurpose: To understand Linear (Dierential) and Quadratic Approximation to a function near acertain point.Recall: Given y = f (x), the tangent line at x = a is the best approximation to the graph of f nearx = a
Texas A&M - MATH - 151
14.1: Exponential FunctionsAn exponential function is a function of the form f (x) = ax , a > 0.Graph and Graphical Properties of f (x) = ax :Properties of Exponential Functions:Using the limit denition of the derivative, we see that, if f (x) = ax ,
Texas A&M - MATH - 151
14.2: Inverse Functionsfunctions vs. one-to-one functions:If f is one-to-one, the inverse of f is a function f 1 such thatIf (a, b) is on the graph of y = f (x), thenIf f is one-to-one and dierentiable at x = g (a), where g = f 1 , thenExamples:Sho
Texas A&M - MATH - 151
14.3: Review of Logarithmic FunctionsDenition and Properties of LogarithmsGraphs of Logarithmic Functions:Change of Base FormulaExamples:Compute log5 10 + log5 20 3 log5 2.1The formula to compute the amount of money A in an account earning 100r% i
Texas A&M - MATH - 151
14.4: Derivatives of Logarithmic FunctionsWhy do we know the function g (x) = ln x is dierentiable?Other Bases:Logarithmic Dierentiation1. .2. .3. .1Examples:Compute and simplifyd(ln(x).dxGiven f (x) = x ln(x2 + 1), nd f (x)Find the derivat
Texas A&M - MATH - 151
14.5: Exponential Growth and DecayA solution to the dierential equation y = ky is:Exponential Growth and Decay:Goal: Use given information to nd C and k .Examples:Aggigium is a radioactive substance with a half-life of 80 days. If there are 2015g of
Texas A&M - MATH - 151
14.6: Inverse Trig Functions and Their Derivativessin, cos, tan and one-to-one functions:y = sin1 x (or arcsin x) if and only ify = cos1 x if and only ify = tan1 x if and only ifDerivative of y = sin1 x:dcos1 x =dxdtan1 x =dxdsec1 x =dxExa
Texas A&M - MATH - 151
14.8: LHospitals RuleGoal: Given a limit of indeterminate form (0/0, /, etc.) with dierentiable functions, nd thelimit.LHospitals Rule: If f and g are dierentiable and g (x) = 0 for all x near a, and lim f (x) = lim g (x) = 0xaor lim f (x) = and lim
Texas A&M - MATH - 151
1Ch 0, App D: ReviewFunctions:Graph of a Function:Combining Functions:Trignometric Functions (VERY IMPORTANT FOR ENGINEERS!) See Appendix D and the Formula Page (before the Title Page in your text).Examples: Find the domain of f (x) =x|x 2| 111
Texas A&M - MATH - 166
Homework #1Due: 3:00pm on January 28, 2010Name:Math 166 Section:Row:This assignment is due by 3:00pm on January 28, 2010 You can turn it in to me in classor drop it by the oce, Blocker 640D. Be sure that you follow the homework rules, they can befo
Texas A&M - MATH - 166
Homework #2Due: 3:00pm on February 4, 2010Name:Math 166 Section:Row:This assignment is due by 3:00pm on February 4, 2010 You can turn it in to me in classor drop it by the oce, Blocker 640D. Be sure that you follow the homework rules, they can befo
Texas A&M - MATH - 166
Homework #4Due: 3:00pm on February 18, 2009Math 166 Section:Name:Row:This assignment is due by 3:00pmon February 18, 2009 You can turn it in to me in classor drop it by the oce, Blocker 640D. Be sure that you follow the homework rules, they can bef
Texas A&M - MATH - 166
Homework #5Due: 3:00pm on February 25, 2010Math 166 Section:Name:Row:This assignment is due by 3:00pmon February 25, 2010 You can turn it in to me in classor drop it by the oce, Blocker 640D. Be sure that you follow the homework rules, they can bef
Texas A&M - MATH - 166
Homework #6Due: 3:00pm on March 4, 2010Math 166 Section:Name:Row:This assignment is due by 3:00pmon March 4, 2010 You can turn it in to me in class ordrop it by the oce, Blocker 640D. Be sure that you follow the homework rules, they can befound on
Texas A&M - MATH - 166
Homework #8Due: 3:00pm on March 25, 2010Math 166 Section:Name:Row:This assignment is due by 3:00pmon March 25, 2010 You can turn it in to me in class ordrop it by the oce, Blocker 640D. Be sure that you follow the homework rules, they can befound o
Texas A&M - MATH - 166
Homework #9Due: 3:00pm on April 1, 2009Math 166 Section:Name:Row:This assignment is due by 3:00pm on April 1, 2009 You can turn it in to me in class or drop it bythe oce, Blocker 640D. Be sure that you follow the homework rules, they can be found on
Texas A&M - MATH - 166
Homework #10Due: *12:30pm on April 9, 2010*Math 166 Section:Name:Row:This assignment is due by *12:30pmon April 9, 2010* You can turn it in to me in class or drop itby the oce, Blocker 640D. Be sure that you follow the homework rules, they can be fo
Texas A&M - MATH - 166
Homework #12Due: 3:00pm on April 22, 2010Name:Math 166 Section:Row:This assignment is due by 3:00pm on April 22, 2010 You can turn it in to me in class or drop it bythe oce, Blocker 640D. Be sure that you follow the homework rules, they can be found
Texas A&M - MATH - 166
Homework #13Due: 3:00pm on April 29, 2010Name:Row:Math 166 Section:This assignment is due by 3:00pm on April 29, 2010 You can turn it in to me in class or drop it bythe oce, Blocker 640D. Be sure that you follow the homework rules, they can be found
Texas A&M - MATH - 166
Quiz #2AnswersJune 2, 2011Math 1661. Determine the truth value of the following statements if you know that p and r are bothTRUE and s and q are both FALSE.(a) r p r is false and p is true. Since the exclusive or is true if exactly one of the state
Texas A&M - MATH - 166
Quiz #3AnswersJune 6, 2011Math 1661. If n(U ) = 100, n(A B ) = 8, n(A B ) = 19 and n(A) = 14, nd n(B ).use the formula: n(A B ) = n(A) + n(B ) n(A B ) (or draw an venn diagram)19 = 14 + n(B ) 8Answer: n(B ) = 132. An experiment is to select a lett
Texas A&M - MATH - 166
Quiz #4AnswersJune 10, 2011Math 1661. An exam contains ve multiple choice questions each with 6 answers. How many dierentways can a student answer the exam if they are allowed to leave questions blank.7 7 7 7 7 = 75 = 168072. In how many ways can 7
Texas A&M - MATH - 166
Quiz #5AnswersJune 13, 2011Math 1661. A club consists of 15 freshmen, 20 sophmores, 10 juniors, and 25 seniors. Seven members ofthe club are selected to win a prize of $100. How many ways can this group be selected if itwill contain exactly one fres
Texas A&M - MATH - 166
Quiz #6aAnswersJune 15, 2011Math 1661. A class contains the following students as listed in the table. Let the random variable Xdenote the number of freshmen students selected in a sample of 6.class7 freshmen5 sophomores12 JuniorsCompute P (X =
Texas A&M - MATH - 166
Quiz #7aAnswersJune 21, 2011Math 1661. Bob put $700 into an account that has a simple interest rate of 12.3% per year. At the endof 9 years, how much interest will have earned?Solve by either method:I=PrtI = 700 .123 9I = 774.9A = P(1 + rt)A =