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Math 54 Fifth Exam ReviewMathematics 54 - Elementary Analysis 2

Institute of Mathematics

University of the Philippines-Diliman

28 November 2014

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Fifth Long Exam Schedule

FIFTH LONG EXAMINATION

2 December 2014, Tuesday

Discussion Class Time and Room

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Final Exam Schedule

FINAL EXAMINATION

8 December 2014, Monday

1:45 pm - 3:45 pm

WFQ2THQ9 (Esguerra) - MB 314

WFQ2THR9 (Esguerra) - MB 318

WFQ2THQ10 (A. Velasco) - MB 319

WFQ2THR10 (A. Velasco) - MB 320

WFQ2THQ11 (Arias) - MB 305

WFQ2THR11 (Arias) - MB 306

WFQ2THQ10 (Bargo) - AV 1

WFQ2THR10 (Bargo) - AV 1

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Final Exam Schedule

FINAL EXAMINATION

8 December 2014, Monday

1:45 pm - 3:45 pm

WFV2THU13 (Damasco) - MBAN 403

WFV2THV13 (Bargo) - MBAN 404

WFV2THU14 (Boydon) - MB 301

WFV2THV14 (Ramos) - MB 321

WFV2THU15 (Ong) - MB 302

WFV2THV15 (Ong) - MB 303

WFV2THU16 (Santos) - MBAN 307

WFV2THV16 (Wong) - MBAN 101/102

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Fifth Long Exam Coverage

Functions of Several Variables

Limits and Continuity of Functions of Several Variables

Partial Derivatives

Differentiability, Differentials, and Local Linear Approximation

Chain Rule

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Sample Questions

I. Write TRUE if the statement is always true, and FALSE if otherwise.

1. The level curve to the graph of f (x,y)= 16x2 y2 at k= 7 is a circle of

radius 3.

2. If f is a polynomial or a rational function of two variables and

(a,b) dom f , then lim(x,y)(a,b)

f (x,y)= f (a,b).

3. The slope of the tangent line to the curve of intersection of the graph of

z= g(x,y) and the plane x= a at the point (a,b,g(a,b)) is gy(a,b).

4. There is a function f of two variables such that fx(x,y)= x2+8xy and

fy(x,y)= 4x2+3yx.

5. If fx(a,b) and fy(a,b) both exist, then f is differentiable at (a,b).

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Sample Questions

II. Find and sketch the domain of f (x,y)=4x2y2+ ln(xy).

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Sample Questions

III. Show that g(x,y)=

(x1)4 y4

(x1)2 +y2if (x,y) 6= (1,0)

1 if (x,y)= (1,0)

is discontinuous at

(1,0) and classify the discontinuity as removable or essential.

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Sample Questions

IV. Show that lim(x,y)(0,0)

x2y6

x4+2y12does not exist.

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Sample Questions

V. Let f (,s)=cos3

s.

1. Use the limit definition/alternative definition of the partial derivative at

a point to find fs(pi,4).

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Sample Questions

V. Let f (,s)=cos3

s.

2. If gs(,s)= f (,s), find gs(pi,4).

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Sample Questions

VI. Let z be a differentiable function of x and y implicitly defined in the

equation y3 tan(xz)= z2exy. Findz

xand

z

yat (0,2,1).

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Sample Questions

VII. Let the surface S be the graph of h(x,y)= sin(e4xy2).

1. Find the equation of the tangent plane to S at the point where x= 0 and

y= 1.

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Sample Questions

VII. Let the surface S be the graph of h(x,y)= sin(e4xy2).

2. Use a local linear approximation to approximate sin(e0.04 (0.99)2

).

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Sample Questions

VII. Let the surface S be the graph of h(x,y)= sin(e4xy2).

3. If x= ln(u2v7) and y =u32v

4, use Chain Rule to find

h

uat u= v= 2.

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Sample Questions

VIII. Use differentials to estimate the amount of metal in an open

cylindrical can that is 10 cm high and 4 cm in diameter if the metal in

the bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick.

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