xxPropostes per Mates I (tardor 2021)

Propostes per Mates I (tardor 2021)

10xde juny dex2021

(Enllax a aquest document en html) (Enllax a la versix pdf d’aquest document) (Enllax a la font LaTeX) (Enllax als fitxers de les figures) (Enllax al fitxer de definicions latex)

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1 Introduccix x1.1 Els exercicis i problemes x1.2 Sobre els Clickers o ConcepTests 2 T0 El llenguatge de les Matemxtiques 3 T1 Arrel quadrada. Definicix de funcix Funcions elementals i valor absolut x3.1 Excercicis Pemb-Rau x3.2 ConcepTests 4 T2 Dominis, Recorregut. Circumferxncia x4.1 Excercicis Pemb-Rau x4.2 ConcepTests 5 T3 Funcions lineals i exponencials. Funcions quadrxtiques (repxs) x5.1 Excercicis Pemb-Rau x5.2 ConcepTests 6 T4 Desplaxaments de grxfiques. Desigualtats grxfiques x6.1 Excercicis Pemb-Rau x6.2 ConcepTests 7 T5 Funcions polinxmiques (repxs). Continuxtat. Teorema de Bolzano x7.1 Excercicis Pemb-Rau x7.2 ConcepTests 8 T6 Funcions potencials (repxs). Funcions exponencials i logaritmes x8.1 Excercicis Pemb-Rau x8.2 ConcepTests 9 T7 Derivada i recta tangent. Creixement x9.1 Excercicis Pemb-Rau x9.2 ConcepTests 10 T8 Derivada implxcita x10.1 Excercicis Pemb-Rau x10.2 ConcepTests 11 T9 Aproximacix lineal, diferencial. Aproximacions polinxmiques x11.1 Excercicis Pemb-Rau x11.2 ConcepTests 12 T10 xptims locals. Criteri de la primera i la segona derivada x12.1 Excercicis Pemb-Rau x12.2 ConcepTests 13 T11 Segona derivada. Concavitat i convexitat. Grxfiques x13.1 Excercicis Pemb-Rau x13.2 ConcepTests 14 T12 xptims globals i teorema dels valors extrems x14.1 Excercicis Pemb-Rau x14.2 ConcepTests 15 T13 Integral indefinida x15.1 Excercicis Pemb-Rau x15.2 ConcepTests 16 T14 Integral definida i cxlcul d’xrees x16.1 Excercicis Pemb-Rau x16.2 ConcepTests 17 NO T15 Aplicacions de la integral x17.1 Excercicis Pemb-Rau x17.2 ConcepTests 18 T15 Vectors a R^{n}. Representacix. Operacions. Combinacions lineals x18.1 Excercicis Pemb-Rau x18.2 ConcepTests 19 T16 Independxncia lineal. Producte escalar, ortogonalitat x19.1 Excercicis Pemb-Rau x19.2 ConcepTests 20 T17 Matrius. Transposada. Matrius simxtriques. Potxncia de matrius xx20.0.1 Cxlcul matricial amb R x20.1 Excercicis Pemb-Rau x20.2 ConcepTests 21 T18 Determinants. Determinants i dependxncia lineal. Producte vectorial xx21.0.1 Cxlcul de determinants amb R x21.1 Excercicis Pemb-Rau x21.2 ConcepTests 22 T19 Inversa. Matrius ortogonals xx22.0.1 Matrius inverses i ortogonals amb R x22.1 Excercicis Pemb-Rau x22.2 ConcepTests

1 Introduccix Aquest document xs una resposta a la peticix de la coordinacix de Matemxtiques d’orientar com el nou llibre de text pot cobrir el que es fa actualment a Matemxtiques I. Tambx es complementa amb la proposta de Clickers o qxestionaris de conceptes rxpids. S’inclou tanbx una proposta completa del que podria ser una nova primera setmana dedicada a una petita introduccix al llenguatge formal de les matemxtiques. Queda per tant per decidir la configuracix final del temari del curs de Matemxtiques I.

1.1 Els exercicis i problemes Per cada guia assenyalo els exercicis del llibre Pemb-RAU que pertoquen. Estan disponibles al web del llibre les solucions als exercicis, i als problemes.

1.2 Sobre els Clickers o ConcepTests Amb la doble intencix de fer les classes de teoria mxs interactives i d’aprofondir en els aspectes mxs conceptuals dels temes tractats es proposa que es destini part del temps de les classes a la formulacix i discusix de preguntes de resposta mxltiple en una seqxxncia com ara la segxent.

Es planteja la pregunta en pantalla

Amb un temps de reflexix individual de un o dos minuts es demana que els estudiants triin la resposta correcta via Aula Global o alguna alternativa usant els seus mxbils.

A la vista de les respostes obtingudes que es visualitzen en pantalla, s’obre un temps de discussix que pot ser a partir de preguntes directes del professor o en grups petits dels estudiants.

Es pot tornar a donar opcix als estudiants a respondre on-line, o es tanca el tema demanant a estudiants individuals que raonin la resposta finalment triada.

Pot ser que el tancament del procxs requereixi explicacions suplementxries per part del professor o per part d’estudiants avanxats.

Hi ha un llibre que pot ser interessant: Teaching Mathematics with Classroom Voting With and Without Clickers. Aqux tenim l’index.

Moltes de les preguntes proposades a continuacix provenen de la coleccix de la Cornell U. Aquests fan servir un codi:

Q “Quick Check”Designed to quickly check students’ basic understanding of the material

P “Probing” Usually requires some thought and extension beyond basic concepts.

D “Deep” Difficult questions that will usually require instructor intervention to help guide students in the right direction.

Un altre recurs interessant: http://mathquest.carroll.edu/ La seva coleccix completa. Mxs avall fem referxncia a algunes parts.

Moltes altres sxn possibles, perx la idea xs que no siguin preguntes que requereixin gaire cxlculs sino mxs aviat la comprensix i reflexix sobre els temes que s’han explicat a classe o que se suposa que els estudiants han revistat pre-classe.

2 T0 El llenguatge de les Matemxtiques

Es tracta de fer una lleu introduccix al llenguatge formal, limitat a l’xlgebra de conjunts i la de Boole elemental. Posem davant els conjunts perqux probablement els sxn mxs familiars i els poden ajudar a entrar en el mxs abstracta de la lxgica formal, que tractem nomxs somerament, sense entrar en taules de veritat, demostracions, etc. que poden ser objecte de tractament a mates II o III.

El material per aquest tema es troba a

(en html) (versix pdf ) (font LaTeX) (Enllax als fitxers de les figures) (Enllax al fitxer de definicions latex)

3 T1 Arrel quadrada. Definicix de funcix Funcions elementals i valor absolut

Convindria aprofitar aquest tema per esmentar que les funcions poden ser de una o de mxs vairables, simplement algun exemple. Tambx la idea de funcix composta es pot introduir ja.

Vegeu els capxtols 3.3 i 3.4 del Pem-Rau:

3.3xxFunctionsx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.46
x3.4xxMappingsx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.54

3.1 Excercicis Pemb-Rau

Exercicis 3.3.1, 3.3.2, 3.3.3, 3.3.5 PIC

Exercicis 3.4.1, 3.4.2 i problemes 3-1, 3-2, 3-3, 3-4 PIC PIC

3.2 ConcepTests

Questions senzilles sobre el concepte de funcix, domini, recorregut, es poden trobar a Classroom Voting Questions: Precalculus. Functions and Change

Per exemple

Which of the following functions has its domain identical with its range?

(a) f(x) = x^{2}

(b) g(x) = \sqrt{x}

(c) h(x) = x^{4}

(d) i(x) = |x|

4 T2 Dominis, Recorregut. Circumferxncia

Potser aquest ema i l’anterior es podrien fondre, no?

4.1 Excercicis Pemb-Rau

4.2 ConcepTests

Questions senzilles sobre el concepte de funcix, domini, recorregut, es poden trobar a Classroom Voting Questions: Precalculus. Functions and Change

Per exemple Which of the following functions has its domain identical with its range?

(a) f(x) = x^{2}

(b) g(x) = \sqrt{x}

(c) h(x) = x^{4}

(d) i(x) = |x|

O de Cornell:

[Q] True or False. If f(x) = \frac{x^{2} − 4}{x − 2} and g(x) = x + 2, then we can say the functions f and g are equal.

Answer: False. Note that even if the two functions have the same rule, they are defined on different domains, i.e., f is not defined at 2.

5 T3 Funcions lineals i exponencials. Funcions quadrxtiques (repxs)

No entenc molt bx perqux es tracten aqux les funcions exponencials. Em semblaria mxs coherent tractar lineals/potencials.

El tractament de les quadrxtiques jo el faria basat en “quadrar l’expressix”, xs a dir transformar-la en la forma y = a(x − p)^{2} + q

1.1xxStraightxlinexgraphsx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.1
x1.2xxAnxeconomicxapplication:xsupplyxandxdemandx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.8
x2xxLINEARxINEQUALITIES
x2.1xxInequalities
x
x4.1xxQuadraticxfunctionsxandxequationsxx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.58
x4.2xxMaximisingxandxminimisingxquadraticxfunctionsx.x.x.x.x.x.x.x.x.x.x.x.x.x.65
x4.3xxIndicesx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.67

El 4.3 va sobre potxncies, exponents i funcions potencials.

El llibre tracta les funcions exponencials desprxs de la derivacix i juntament amb els logaritmes. Llavors entra en el tema del creixement amb totes les eines a punt. Tx sentit.

9xxEXPONENTIALxANDxLOGARITHMICxFUNCTIONS163
x9.1xxThexexponentialxfunctionx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.163
x9.2xxNaturalxlogarithmsx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.168
x9.3xxTimexinxeconomicsxx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.174

5.1 Excercicis Pemb-Rau Exercicis 4.4.1 a 4.4.5: PIC Exercicis 9.1.1 a 9.1.3 PIC Problema 9-2 PIC Si es tractxs amb potxncies i funcions potencials, exercicis 4.3.1 a 4.3.9: PIC

5.2 ConcepTests Moltes questions a: Classroom Voting Questions: Precalculus. Lines and Parabolas I tambx a Classroom Voting Questions: Precalculus. Exponential Functions

6 T4 Desplaxaments de grxfiques. Desigualtats grxfiques El tema del desplaxament de grxfiques crec que s’hauria de contextualitzar mxs, no hi veig sentit si es planteja com unes regles que s’aprenen de memxria per respondre preguntes tipus. Si de cas, s’ha de lligar amb exemples on hi hagi canvis d’origen, canvis d’escala, etc. Del llibre:

2xxLINEARxINEQUALITIES
x2.1xxInequalitiesx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.21
x2.2xxEconomicxapplicationsx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.25
x2.3xxLinearxprogrammingx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.29

Tindria sentit imtropduir aqui la programacix lineal amb resolucix grxfica?

6.1 Excercicis Pemb-Rau Sobre desiguatats lineals, 2.2.1 a 2.2.3: PIC En el problema PR 3-2 es tracta el tema de canvi de variable per tractar el desplaxament.

6.2 ConcepTests Classroom Voting Questions: Precalculus. New Functions From Old: Compositions, Inverses, and Transforms

7 T5 Funcions polinxmiques (repxs). Continuxtat. Teorema de Bolzano Aquest tema el trobo desproporcionadament complex i difxcil, sobretot si el comparo amb els primers del curs. Crec que tractar la continuxtat i els lxmits cap a ± sense haver parlat abans de limits... Potser algunes coses de polinomis s’haurien de moure als T2 o T3. En el llibre Pem-Rau:

5.4xxLimitsxandxcontinuityx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.92

En el llibre, el capxtol comenxa amb succession, series i progressions geomxtriques en economia, temes que aqux no tractem.

Pel que fa a lxmits, fa una definicix “verbal” i desprxs tambx entra en ϵ/δ. Dona les regles aritmxtiques dels lxmits, i posposa les demostracions aun capxtol 31 on fa l’anxlisi amb rigor. Tracta de passada els polinomis com a exemple de funcions continues, tracta breument la indeterminacix 0/0 i els lxmits laterals. Desprxs tracta la continuxtat. I el T. Valor Intermedi, el de Bolzano no l’esmenta.

7.1 Excercicis Pemb-Rau

5.4.1 a 5.4.4: PIC PIC

7.2 ConcepTests

En aquest primer test, si es creu que els estudiants no tenen clara la distribucix dels irracionals a la recta real, es podria canviar la definicix: x^{2} o −x^{2} segons si la tercera xifra de l’expressix decimal de x xs o no zero.

[D] Consider the function

. f(x) = \{ \begin{matrix}x^{2} & x is rational, x ≠ 0 \\ −x^{2} & x is irrational \\ undefined & x = 0 \end{matrix}

Then

there is no a for which lim_{x)a} f(x) exists

there may be some a for which lim_{x)a} f(x) exists, but it is impossible to say without more information

lim_{x)a} f(x) exists only when a = 0

lim_{x)a} f(x) exists for infinitely many a

Answer: (c). Students should be encouraged to draw the graph and discuss.

[P] True or False. At some time since you were born your weight in pounds equaled your height in inches.

Answer: True. Students must consider the difference of two functions: f(t) = height(t) − weight(t), functions of time. At birth, f(birth) > 0 and right now, f(now) < 0, hence by IVT f(T) = 0, where T is some time in the past. It is important to stress that this technique of looking at the difference of two functions is recurrent in calculus.

[Q] True or False. As x increases to 100, f(x) = 1/x gets closer and closer to 0, so the limit as x goes to 100 of f(x) is 0. Be prepared to justify your answer.

Answer: False. As x increases to 100, f(x) = 1/x gets closer and closer to 0, gets closer and closer to 1/1000, but not as close as to 1/100. The question points out the weakness of the statement ”f(x) gets closer to L as x ) a, and therefore lim_{x)a} f(x) = L”.

[Q] You know the following statement is true:

If f(x) is a polynomial, then f(x) is continuous.

Which of the following is also true?

If f(x) is not continuous, then it is not a polynomial.

If f(x) is continuous, then it is a polynomial.

If f(x) is not a polynomial, then it is not continuous.

Answer: (a). This may seem like an easy logic question, but students tend to have difficulties; it might be a good time to review some logic. This question prepares students for reasoning that even if differentiability implies continuity, continuity does not imply differentiability. Ask for examples of functions that are continuous, but not polynomials.

[P] True or False. At some time since you were born your weight in pounds equaled your height in inches.

Answer: True. Students must consider the difference of two functions: f(t) = height(t) − weight(t), functions of time. At birth, f(birth) > 0 and right now, f(now) < 0, hence by IVT f(T) = 0, where T is some time in the past. It is important to stress that this technique of looking at the difference of two functions is recurrent in calculus.

[D] True or False. x^{100} − 9x^{2} + 1 has a root in [0, 2].

Answer: True. This problem is not a direct application of IVT, plugging in 0 and 2, we get positive numbers, so the student must choose some other number in [0, 2] to test. Choosing 0 and 1, or 1 and 2, IVT immediately applies.

8 T6 Funcions potencials (repxs). Funcions exponencials i logaritmes

El Perm-Rau tracta les exponencials despr´s d’haver vist la derivada. Tx sentit perqux el mxs interessant de la funcix exponencial xs el creixement. Es pot donar per suposat que ja saben alguna cosa de derivades i fer-ho aixx?

4.3xxIndicesx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.67
x4.4xxLogarithmsxxxx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.73
x
x9.1xxThexexponentialxfunctionxxx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.163
x9.2xxNaturalxlogarithmsxxx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.168

8.1 Excercicis Pemb-Rau Exercicis 4.3: PIC Problemes cap. 4: PIC Exercicis 9.1: PIC Exercicis 9.2: PIC Problemes cap 9: PIC PIC

8.2 ConcepTests Classroom Voting Questions: Precalculus: Powers, Polynomials, and Rational Functions Sobre logaritmes Sobre exponencials

9 T7 Derivada i recta tangent. Creixement

6xxINTRODUCTIONxTOxDIFFERENTIATIONxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx100
x6.1xxThexderivativex.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.101
x6.2xxLinearxapproximationsxandxdifferentiabilityx.x.x.x.x.x.x.x.x.x.x.x.x.x108
x6.3xxTwoxusefulxrulesxx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x112
x6.4xxDerivativesxinxeconomicsx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.115
x7xxMETHODSxOFxDIFFERENTIATIONxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx119
x7.1xxThexproductxandxquotientxrulesx.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.119
x7.2xxThexcompositexfunctionxrulex.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x.x121

9.1 Excercicis Pemb-Rau

Exercicis 6.1: PIC

Exercicis 6.2: PIC

Exercicis 6.3: PIC

Exercicis 6.4 i Problemes capxtol 6: PIC PIC

Exercicis 7.1: PIC

Exercicis 7.2: PIC

9.2 ConcepTests

Digues quines de les segxents sxn certes o falses per una funcix f(x) definida i diferenciable a tot R. Prepara’t per raonar si xs certa perqux, i si xs falsa a mostrar un contraexemple.

f′(x) > 0 per tot x xs condicix necessxria per f xs creixent a tot R.

f′(x) > 0 per tot x xs condicix suficient per f xs creixent a tot R.

f′(x) <= 0 per tot x xs condicix necessxria per f xs decreixent a tot R.

f′(x) <= 0 per tot x xs condicix suficient per f xs decreixent a tot R.

Si la funcix xs creixent a tot R, no pot tenir cap punt estacionari.

Si x_{0} xs un punt estacionari de la funcix, la funcix no pot ser decreixent a x_{0}.

N’hi ha una xmplia col·leccix a el projecte Carrol U., perx aqux en recollim alguns de la Cornell:

[Q] If the radius of a circle increases from r_{1} to r_{2}, then the average rate of change of the area of the circle is

less than 2πr_{2} greater than 2πr_{1} equal to 2π\frac{r_{1} +r_{2}}{2} all of the above

Answer: (d). This is a straightforward application of the definition of average rate of change. Once we get (c) as an answer, (a) and (b) follow:

. \frac{πr_{2}^{2} − πr_{1}^{2}}{r_{2} − r_{1}} = π(r_{2} + r_{1}) = 2π\frac{r_{2} + r_{1}}{2}

[P] The derivative of f(x) = x|x| at x = 0 is 0. does not exist, because |x| is not differentiable at x = 0 does not exist, because f is defined piecewise does not exist, because the left and right hand limits do not agree.

Answer: (a). Instructors should encourage the use of the limit definition of derivative; f′(0) = 0

[P] If f′(a) exists, lim_{x)a} f(x)

it must exist, but there is not enough information to determine it exactly equals f(a) equals f′(a) it may not exist

Answer: (b). If f is differentiable at a, it must be continuous at a, and therefore the limit equals f(a). Many students will like answers (a) and (d).

[Q] A slow freight train chugs along a straight track. The distance it has traveled after x hours is given by a function f(x). An engineer is walking along the top of the box cars at the rate of 3 mi/hr in the same direction as the train is moving. The speed of the man relative to the ground is

f(x) + 3 f′(x) + 3 f(x) − 3 f′(x) − 3

Answer (b). Once the students realize that f′(x) is the speed of the train after x hours, most of them will give the right answer.

[Q] Suppose f′(x) exists for all x in (a, b). Read the following four statements:

f(x) is continuous on (a, b) f(x) is continuous at x = a f(x) is defined for all x in (a, b) f′(x) is differentiable on (a, b)

I and III I, II and III All of the above None of the above Only I

Answer: (a). Students will know that I holds. They might have questions about II and not be quite sure about III.

[Q] The Constant Multiple Rule tells us

. \frac{d}{dx}cf(x) = c\frac{d}{dx}f(x)

and the Product Rule says

. \frac{d}{dx}cf(x) = c\frac{d}{dx}f(x) + f(x)\frac{d}{dx}c

True or False. The two rules agree. Be prepared to justify your answer.

Answer: True. Some students will be unable to recognize that we get the same result. Remind them that the derivative of a constant function is zero.

[Q] If f and g are both differentiable and h = f ∘ g, h′(2) equals

f′(2) ∘ g′(2)

f′(2)g′(2)

f′(g(2))g′(2)

f′(g(x))g′(2)

Answer: (c). Even though students may have memorized the Chain Rule formula, some may not be able to apply it to this type of problem.

[Q] True or False. \frac{d}{dx} ln(π) = \frac{1}{π} .

Answer: False. Students must observe that ln(π) is a constant, and thus \frac{d}{dx} ln(π) = 0.

10 T8 Derivada implxcita

La regla de la cadena la he inclxs en l’anterior seccix, la derivada implxcita el llibre la tracta molt mxs endavant, amb el teormea de la funcix implxcita. Perx estx bx tractar-ho aqux, de fet s’hauria d’aprofitar per enllaxcar amb el que veuen a Inroduccix a Micro, isoquantes i demxs. Aixx ho tracta bx el llibre

7.2xxThexcompositexfunctionxrulex.x.x.x.x.x.121
x15xIMPLICITxRELATIONSxxxxxxxxxxxxxxxxxxxxxxx293
x15.1xImplicitxdifferentiationx.x.x.x.x.x.x.x293

10.1 Excercicis Pemb-Rau Exercicis 15.1: PIC Problemes cap. 15: PIC

10.2 ConcepTests De tipus mxs calculets, poc interessants: Classroom Voting Questions: Calculus I: 3.7 Implicit Differentiation

Considera una funcix d’utilitat del tipus U(K, L) = f(K) + g(L). Podem derivar l’expressix de iso-utilitat U(K, L) = u_{0} (on u_{0} xs constant) implxcitament respecte K, considerant que L depxn de K per obtenir \frac{dK}{dL} o bx respecte L, considerant que K depxn de L i obtindrem \frac{dL}{dK} . Aleshores es compleix:

\frac{dK}{dL} = 1/\frac{dL}{dK}

\frac{dK}{dL} = −1/\frac{dL}{dK}

Cap de les dues

11 T9 Aproximacix lineal, diferencial. Aproximacions polinxmiques

11.2 ConcepTests

11.1 Excercicis Pemb-Rau 4,5 i 6, 9, 10 del capxtol [P] If e^{.5} is approximated by using the tangent line to the graph of f(x) = e^{x} at (0,1), and we know f′(0) = 1, the approximation is

.5
1 + e^{.5}
1 + .5

Answer: (c). Most students will be able to get a correct formula for the linearization of e^{x} at 0, L(x) = 1 + x, but a good number will have difficulties knowing how to use this information to get an approximation for e^{.5}.

Considerem f(x) = x^{2}. Utiiltzant la aproximacix lineal de la derivada, quina xs la millor aproximacix per a 2.01^{2}?

2 + 0.01
4 + 0.04
4 + 0.02
2 + 0.02

12 T10 xptims locals. Criteri de la primera i la segona derivada

12.1 Excercicis Pemb-Rau

12.2 ConcepTests

[P] An article in the Wall Street Journal’s “Heard on the Street” column (Money and Investment August 1, 2001) reported that investors often look at the “change in the rate of change” to help them “get into the market before any big rallies.” Your stock broker alerts you that the rate of change in a stock’s price is increasing. As a result you can conclude the stock’s price is decreasing can conclude the stock’s price is increasing cannot determine whether the stock’s price is increasing or decreasing.

Answer: (c). If f is the stock price, and we found out that f′ is increasing, this does not lead to any conclusion about the monotonicity of f. Per una funcix diferenciable a tot R , i un punt x_{0} ∈ R, digues quines sxn certes o falses

f′(x_{0}) = 0 xs condicix necessxria per tal que x_{o} sigui un xptim local de f. f′(x_{0}) = 0 xs condicix suficient per tal que x_{o} sigui un xptim local de f. f′(x_{0})x0 xs condicix suficient per tal que x_{o} no sigui un xptim local de f. x_{0} xs un xptim global xs condicix suficient per f′(x_{0}) = 0.

Per una funcix diferenciable dues vegades a tot R , i un punt x_{0} ∈ R, digues quines sxn certes o falses Si f′′(x) > 0 per tot x, f no pot tenir un mxnim global. Si f′′(x) > 0 per tot x, f pot no tenir un mxnim global. Si x_{0} xs un mxxim global de f en R , f′′(x_{0}) no pot ser zero.

13 T11 Segona derivada. Concavitat i convexitat. Grxfiques

13.1 Excercicis Pemb-Rau

13.2 ConcepTests

14 T12 xptims globals i teorema dels valors extrems 14.1 Excercicis Pemb-Rau

14.2 ConcepTests

[Q] You are given a continuous function, for which f′′(x) > 0 for all reals, except at x = a.

True or False. f might have an absolute maximum at x = a.

Be prepared to give a counterexample or justify your answer.

Answer: True. Consider a function with a cusp at x = a and f(a) = 1, which remains positive, and decreases to a horizontal asymptote at y = 0 .

[Q] True or False. If f(x) is continuous on a closed interval, then it is enough to look at the points where f′(x) = 0 in order to find its absolute maxima and minima. Be prepared to justify your answer.

Answer: False. Encourage the students to think of the different ways in which this would fail -i.e. the endpoints, points where the derivative does not exist.

[P] Let f(x) be a differentiable function on a closed interval with x = a being one of the endpoints of the interval. If f′(a) > 0 then,

f could have either an absolute maximum or an absolute minimum at x = a.

f cannot have an absolute maximum at x = a.

f must have an absolute minimum at x = a.

Answer: (a). Students should think about the differences between local and absolute extrema. Ask students to draw pictures of functions satisfying f′(a) > 0, with a being the left endpoint, and then a being the right endpoint.

[Q] If f is continuous on [a, b], then

there must be numbers m and M such that m ≤ f(x) ≤ M, for x ∈ [a, b]

there must be local extreme values, but there may or may not be an absolute maximum or absolute minimum value for the function.

any absolute max or min would be at either the endpoints of the interval, or at places in the domain where f′(x) = 0

Answer: (a). (c) omits the possibility that the max or min could occur at a point on the graph where f′(x) does not exist (such as a cusp), and (b) is false by the Extreme Value Theorem.

[P] Let f be a continuous function on the closed interval 0 ≤ x ≤ 1. There exists a positive number A so that the graph of f can be drawn inside the rectangle 0 ≤ x ≤ 1, −A ≤ y ≤ A.

The above statement is:

Always true.

Sometimes true.

Not enough information.

Answer: (a). This problem emphasizes one important application of the EVT which is at the same time a very geometric result; that we can put a continuous function on a closed interval inside a box! On a closed interval, m ≤ f(x) ≤ M, so if we take A = max{|m|, |M|}, then on this closed interval, f fits in this sort of box. This idea will ultimately show up in finding bounds for integrals, so it would be great to introduce this idea at this point.

15 T13 Integral indefinida

15.1 Excercicis Pemb-Rau

15.2 ConcepTests

[P] Suppose you are told that the acceleration function of an object is a continuous function a(t). Let’s say you are given that v(0) = 1.

True or False: You can find the position of the object at any time t.

Answer: False. The goal is to test whether students understand that they need one initial condition for each antiderivative they have to find.

[P] Let f(x) = \frac{1}{x^{2}} , and F(x) be an antiderivative of f with the property F(1) = 1. True or False. F(−1) = 3.

Answer: False. f(x) is not continuous at 0 ! This problem may help them remember the continuity condition in finding antiderivatives. Students are almost surely going to answer true, and the answer will surprise them.

[Q] If f is an antiderivative of g, and g is an antiderivative of h, then

h is an antiderivative of f

h is the second derivative of f

h is the derivative of f′′ Answer: (b). This follows from the definition of antiderivative. This kind of problem makes the connection between antiderivatives and derivatives.

[Q] True or False: An antiderivative of a sum of functions, f + g, is an antiderivative of f plus an antiderivative of g.

Answer: True. This is an easy check based on the text reading.

[P] True or False: An antiderivative of a product of functions, fg, is an antiderivative of f times an antiderivative of g.

Answer: False. Students should differentiate an antiderivative of f times an antiderivative of g, to see that they will not get fg back.

16 T14 Integral definida i cxlcul d’xrees

16.1 Excercicis Pemb-Rau

16.2 ConcepTests

[Q] True or False. If a piece of string has been chopped into n small pieces and the i^{th} piece is Δx_{i} inches long, then the total length of the string is exactly ∑ _{i=1} ^{n} Δx_{i} .

Answer: True. This problem helps students to use the summation notation to represent physical quantities. It also gets the students to distinguish between the estimating procedure of a Riemann Sum versus cases in which we can have an exact value.

[P] You want to estimate the area underneath the graph of a positive function by using four rectangles of equal width. The rectangles that must give the best estimate of this area are those with height obtained from the:

Left endpoints

Midpoints

Right endpoints

Not enough information

Answer: (d). Students often hold onto the idea that the midpoint estimate is the best. With this example the instructor can point out that there are cases in which it does not work as well as some of the other height choices. A specific function which has a long thin spike at the midpoint is a good counterexample.

[P] Suppose you are slicing an 11 inch long carrot REALLY thin from the greens end to the tip of the root. If each slice has a circular cross section f(x) = π(r(x))^{2} for each x between 0 and 11, and we make our cuts at x_{1}, x_{2}, x_{3}, ... , x_{n}, then a good approximation for the volume of the carrot is

∑ _{i=1} ^{n} f(x_{i})x_{i}
∑ _{i=1} ^{n} [f(x_{i+1} ) − f(x_{i})]x_{i}
∑ _{i=1} ^{n} f(x_{i})[x_{i+1} − x_{i}]

Answer: (c). This is a quick application of the area approximations in this section. Once students understand the set up of the problem, the right answer should follow immediately from the text reading.

[P] Read the following four statements and choose the correct answer below.

If f is continuous on the interval [a, b], then:

∫_{a} ^{b}f(x)dx is the area bounded by the graph of f, the x-axis and the lines x = a and x = b

∫_{a} ^{b}f(x)dx is a number

∫_{a} ^{b}f(x)dx is an antiderivative of f(x)

∫_{a} ^{b}f(x)dx may not exist

(ii) only

(i) and (ii) only

(i) and (iii) only

(iv) only

Answer: (a). This problem attempts to clarify most of the misconceptions that students have about definite integrals, and to help them move away from the idea that they always represent the area under a curve.

[P] Water is pouring out of a pipe at the rate of f(t) gallons/minute. You collect the water that flows from the pipe between t = 2 and t = 4. The amount of water you collect can be represented by:

∫_{2}^{4}f(x)dx

f(4) − f(2)

(4 − 2)f(4)

the average of f(4) and f(2) times the amount of time that elapsed

Answer: (a). This question might also help students see the definite integral as total change rather than the standard area interpretation. At the same time it differentiates it from averages. The following 2 problems to be used in a sequence:

True or False. If ∫ f(x) dx = ∫ g(x) dx, then f(x) = g(x). Answer: True. See below

True or False. If f′(x) = g′(x), then f(x) = g(x). Answer: False. As students often get confused in the mechanics of the process of going back and forth between functions, their derivatives and antiderivatives, a discussion using the above problem can help them clarify their misunderstandings.

[P] Suppose the function f(t) is continuous and always positive. If G is an antiderivative of f, then we know that G:

is always positive.

is sometimes positive and sometimes negative.

is always increasing.

There is not enough information to conclude any of the above.

Answer: (c). f is the derivative of G, thus f > 0 implies G′ > 0, and therefore G is increasing. This is to demonstrate to students that they can apply the Increasing/Decreasing Test outside the context of problems like those in Chapter 4. Note that here we are just referring to the antiderivative of f. (a) may be a popular answer since we think of an integral of a positive function as ”adding”positive small pieces. But the choice between (a) or (b) for a particular antiderivative G, depends on the constant.

[Q] True or False. If f is continuous on the interval [a, b], \frac{d}{dx} (∫_{a}^{b} f(x)dx) = f(x).

Answer: False. Students often do not realize that definite integrals evaluated at constant endpoints a and b are constant, and in order to apply the FTC one must have one at least one of the endpoints as a variable. Note that if a and/or b were defined as functions of x, then the answer would be True.

[P] Below is the graph of a function f.

PIC Let g(x) = ∫_{0}^{x}f(t) dt. Then for 0 < x < 2, g(x) is

increasing and concave up. increasing and concave down. decreasing and concave up. decreasing and concave down.

Answer: (b). This problem can help students realize that an integral of this form is a specific function (not a general antiderivative), and the integrand is its derivative. Students can induce information about the concavity of g just like they normally would with any other function, only that this time they have to look at f.

[P] Below is the graph of a function f.

PIC

Let g(x) = ∫_{0}^{x}f(t) dt. Then g(0) = 0, g′(0) = 0 and g′(2) = 0

g(0) = 0, g′(0) = 4 and g′(2) = 0
g(0) = 1, g′(0) = 0 and g′(2) = 1
g(0) = 0, g′(0) = 0 and g′(2) = 1

Answer: (b). g(0) = 0, g′(0) = f(0) = 4, and g′(2) = f(2) = 0.

17 NO T15 Aplicacions de la integral

17.1 Excercicis Pemb-Rau 18 T15 Vectors a R^{n}. Representacix. Operacions. Combinacions lineals

18.1 Excercicis Pemb-Rau

18.2 ConcepTests

Si tenim u_{1}, u_{2}, u_{3} vectors no nuls de R^{3} i sabem que u_{1} xs combinacix lineal dels altres dos amb u_{1} = λu_{2} + μu_{3}, llavors

Podem assegurar que u_{2} xs CL dels altres dos. Nomxs podem assegurar que u_{2} xs CL dels altres dos si λx0 Nomxs podem assegurar que u_{2} xs CL dels altres dos si λ i μ sxn diferents de zero.

Si tenim u_{1}, u_{2}, u_{3}, u_{4} vectors no nuls de R^{3} i sabem que u_{1} xs combinacix lineal dels altres tres amb u_{1} = λ_{2}u_{2} + λ_{3}u_{3} + λ_{4}u_{4}, llavors

Podem assegurar que u_{2} xs CL dels altres tres Nomxs podem assegurar que u_{2} xs CL dels altres tres si λ_{2}x0 Nomxs podem assegurar que u_{2} xs CL dels altres tres si totes les λ sxn diferents de zero. Pot ser que u_{2} no sigui CL dels altres tres.

Segons les propietats de les operacions amb vectors, (λ + μ)(u + v) xs igual a λ(u + v) + μ)(u + v) (λ + μ)u + (λ + μ)v λu + λv + μu + μv Totes les anteriors sxn certes.

Cap de les anteriors xs certa.

19 T16 Independxncia lineal. Producte escalar, ortogonalitat 19.1 Excercicis Pemb-Rau 19.2 ConcepTests

Quina dels segxents trios de vectors sxn una base ortogonal de R^{3}?

( \begin{matrix}1 \\ −1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )

( \begin{matrix}1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )

( \begin{matrix}1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{matrix} ) , ( \begin{matrix}1/\sqrt{2} \\ −1/\sqrt{2} \\ 0 \end{matrix} )

Tots tres Cap dels tres

Siguin u, v vector ortonormals de R^{3}, llavors, ||u + v|| xs 2

\sqrt{2}

1 no es pot saber del cert 20 T17 Matrius. Transposada. Matrius simxtriques. Potxncia de matrius

20.0.1 Cxlcul matricial amb R Unes notes breus sobre l’xs en R en versix html o en versix pdf

20.1 Excercicis Pemb-Rau

20.2 ConcepTests

Donada A (matriu 3x2) i B (matriu 2x3)

Mai pot ser AB = BA AB = BA si sxn son commutatives AB = BA en alguns casos A^{2} = B^{2}

La matriu A tal que A(a, b, c)′ = (c, b, a)′ per tots a, b, c ∈ R xs

( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} )
( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )
( \begin{matrix}−1 & 0 & 1 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} )

Cap de les anteriors Suposa que A, B sxn matrius quadrades invertibles. Quina de les segxents igualtats es compleix sempre?

((AB)^{−1})′ = (A^{−1})′(B^{−1})′ ((AB)^{−1})′ = (B^{−1})′(A^{−1})′ ((AB)^{−1})′ = B^{−1}A^{−1}B′A′

((AB)^{−1})′ = B′A′B^{−1}A^{−1}

Cap de les altres

Suposa que A, B sxn matrius quadrades ortogonals. Quina de les segxents xs tambx ortogonal?

( \begin{matrix}A & 0 \\ 0 & B \end{matrix} )

( \begin{matrix}A & I \\ I & B \end{matrix} )

A^{2} − B^{2}

Cap de les altres

21 T18 Determinants. Determinants i dependxncia lineal. Producte vectorial

21.0.1 Cxlcul de determinants amb R

Unes notes breus sobre l’xs en R en versix html o en versix pdf

21.1 Excercicis Pemb-Rau

21.2 ConcepTests

Formem una matriu n × n amb els nombres 1xn^{2} consecutius disposats en files. El determinant de la matriu: == xs zero = xs una potxncia de n = depxn de n = xs n!/2

Resposta: xs zero: si restem la fila 2 menys la fila 1 tenim tots els elements iguals a n, la posem enlloc de la primera. Fem el mateix a la segona: fila 3- fila 2, tambx tots iguals a n. Queden dues files iguals, el determinant xs zero.

Si tots els nombres d’una matriu quadrada sxn enters positius el determinant pot ser un enter positiu un enter no negatiu qualsevol enter qualsevol racional qualsevol nombre real

Sigui A una matriu quadrada d’ordre n i B = kA per alguna constant k. Llavors el determinant de B, |B|, xs igual a:

k|A| K^{n} |A| \frac{1}{k} |A| \frac{1}{k^{n}} |A| k + |A| k^{n} + |A| no es pot saber amb aquesta informacix

Sigui A una matriu quadrada d’ordre |A^{2}| = 1. Llavors, el determinant de A xs 1 0 o 1 -1, 0, o 1 -1 o 1 pot ser qualsevol nombre positiu. pot ser qualsevol nombre real.

Podem que A_{ij} denoti l’adjunt del terme a_{ij} d’una matriu quadrada d’ordre n. Quina xs la relacix entre |A_{ij} | i |A_{ji} | ?

Sxn sempre iguals |A_{ij} | = −|A_{ji} | Sxn iguals si la matriu xs simxtrica. Si |A_{ij} | = 0, aleshores |A_{ji} | = 0. Els seu producte xs igual a 1. Cap de les altres xs necessxriament certa.

22 T19 Inversa. Matrius ortogonals

22.0.1 Matrius inverses i ortogonals amb R Unes notes breus sobre l’xs en R en versix html o en versix pdf

22.1 Excercicis Pemb-Rau

22.2 ConcepTests

El determinant d’una matriu ortogonal xs: ( i perqux?) = 1 = 0 == 1 x -1 = pot ser qualsevol valor diferent de 0 Resposta: |A| = |A^{T}| i |A · A^{T}| = |I| = 1, per tant |A|^{2} = 1. Tria la corecta. Siguin A, b matrius quadrades invertibles. Aleshores la inversa de ABA^{−1} xs

A^{−1}B^{−1}A x AB^{−1}A^{−1} ABA^{−1} A^{−1}BA la matriu pot no ser invertible

Considera les tres proposicions segxents que tracten d’una matriu quadrada invertible A.

A = A^{−1} AA = I A^{−1}A^{−1} = I

Tria la correcta:

(1) i (2) sxn equivalents pero (3) no.

(1) i (3) sxn equivalents pero (2) no.

(2) i (3) sxn equivalents pero (1) no. Les tres sxn equivalents No tenen res a veure entre sx

Suposa que A xs una matiur quadrada d’ordre n amb elements entres i |A| = 1. Qux pots dir dels elements de A^{−1}?

Han de ser tots enters Poden ser racionals i algun no enter Pode ser qualsevol nombre real A^{−1} pot no existir La resposta depxn de si n xs parell o senar