ECE Special Topics - 072410

Prepared by:Joselito DL. Torculas, EcE

L’Hospital’s Rule

Suppose f and g are differentiable and . Suppose that:

or that,

Then,

0' xg

0lim0lim

xgandxfaxax

xgandxfaxax

limlim

xgxf

xgxf

axax ''

limlim

1. Evaluate:

a. 3 c. no limit

b. 11/4 d. 1/3

xgxf

xgxf

axax ''

limlim

4103

lim 2

2

2

xxx

x

2. Calculate:

.

a. 0 c. b. 1 d. 1

xx

x

2lnlim

xgxf

xgxf

axax ''

limlim

3. Find:

.

a. 1/3 c. 2/3 b. 3/4 d. 0

xx

x 3sin2sinlim

0

Algebraic Functions

0cdxd

dxdunuu

dxd nn 1

dxduv

dxdvuuv

dxd

2vdxdvu

dxduv

vu

dxd

Trigonometric Functions

dxduuu

dxd cossin

dxduuu

dxd sincos

dxduuu

dxd 2sectan

dxduuu

dxd 2csccot

dxduuuu

dxd tansecsec

dxduuuu

dxd cotcsccsc

Inverse Trigonometric Functions

dxdu

uu

dxd

21

1arcsin

dxdu

uu

dxd

21

1arccos

dxdu

uu

dxd

211arctan

dxdu

uarc

dxd

211cot

dxdu

uuarc

dxd

1

1sec2

dxdu

uuuarc

dxd

1

1csc2

Logarithmic and Exponential Functions

dxdu

ue

udxd a

alog

log dxdu

uu

dxd 1ln

dxduaaa

dxd uu ln

dxduee

dxd uu

dxdvuu

dxduvuu

dxd vvv ln1

Hyperbolic Functions

dxduuu

dxd coshsinh

dxduuu

dxd sinhcosh

dxduuhu

dxd 2sectanh

dxduuhu

dxd 2csccoth

dxduuhuhu

dxd tanhsecsec

dxduuhuhu

dxd cothcsccsc

Inverse Hyperbolic Functions

dxdu

uhu

dxd

1

1arcsin2

dxdu

uhu

dxd

1

1arccos2

dxdu

uhu

dxd

211arctan

dxdu

uuarc

dxd

211coth

dxdu

uuhuarc

dxd

21

1sec

dxdu

uuhuarc

dxd

21

1csc

4. Differentiate:

a. c.

b. d.

251 xy

2515'x

xy

2515' xxy 2515' xxy

2515'x

xy

dxdunuu

dxd nn 1

5. Differentiate:

a.b.c.d.

xxy 2cos2

xxxy 2sin2cos2' 2

xxxxy 2cos22sin2' 2

xxxxy 2sin22cos2' 2

xxxxy 2cos22sin2' 2

dxduv

dxdvuuv

dxd

6. Find the first derivative of:

a. c.

b. d.

24arcsin xy

2414arcsin8xx

2161

4arcsin8

x

x

2161

4arcsin4

x

x

21614arcsin2xx

7. Find y’ if:

a. c.

b. d.

xey 2sin

xe x 2cos2sin

xe x 2sin2cos xe x 2cos2 2sin

xe x cos2 sin

8. Find y’ if:

a. c.

b. d.

xy 42

2ln3 24 x

3ln3 22 x4ln2 23 x

2ln2 24 x

9. Find the second derivative of the function:

a. c.

b. d.

xxy ln2

xx ln21

xln2x2

xln23

D.C. APPLICATION : SLOPE OF THE TANGENT LINE

Slope of the Tangent LineThe slope of the tangent line at

the point of tangency is equal to .Equation of the Tangent Line at point (h,k) on the curve:

Slope of the Normal LineThe slope of the normal line to a

given curve is equal to .

Equation of the Normal Line at point (h,k) on the curve:

Tm'y

hxyky '

Nm

'1y

hxy

ky '1

10.Find the equation of the tangent line to

at the origin.

a. c. b. d.

023 22 yxyx

02 yx 02 yx02 yx02 yx

Equation of the Tangent Line at point (h,k) on the curve: hxyky '

At maximum or minimum points, the tangent line is horizontal or the slope is zero.

Point of InflectionThe point of inflection is a point at which

the curve changes from concave upward to concave downward, or vice versa. At the points of inflection, the second derivative of y is zero.

0'' xfydxdy

0"y

D.C. APPLICATION : MAXIMA AND MINIMA

Steps in Solving Maxima/Minima Problems:

1. Draw the diagram if needed in the problem.2. Identify the variable to be maximized or

minimized.3. Express this variable in terms of the other

relevant variable(s).4. If the function shall consist of more than one

variable, express in terms of one variable using the conditions in the problem.

5. Differentiate the equation and equate to zero.

D.C. APPLICATION : MAXIMA AND MINIMA

11.Given the function . Find the coordinates of the maximum point.

a. (0,0) c. (2,0) b. (1,0) d. (1,1)

22 2 xxy

12.Given the function:

Determine the coordinates of the point of inflection.

a. (-1,-3) c. (0,5) b. (2,9) d. (1,7)

543 23 xxxy

13.Two posts, one 7 ft high and the other 8 ft high, stand 8 ft apart. They are to be stayed by wires attached to a single stake at ground level, the wires running to the tops of the posts. What is the shortest length of wire that will able to implement this setup?

a. 17 ft c. 19 ft b. 18 ft d. 20 ft

14.A man wishes to use 40 ft fencing to enclose a rectangular garden. Determine the maximum possible area of his garden.

a. 64 sq. ft c. 400 sq. ft.b. 100 sq. ft. d. 1600 sq.

ft.

15.A telephone company has to run a line from a point A on one side of a river to another point B that is on the other side, 30 km down from the point opposite A. The river is uniformly 10 km wide. The company can run the line along the shoreline to a point C and then run the line under the river to B. The cost of laying the line along the shore is P5000 per km, and the cost of laying it under water is P12000 per km. Where the point C should be located to minimize the cost?

a. 5.167 km c. 4.583 km b. 6.435 km d. 3.567 km

16.Find two numbers whose sum is 36 and the product of one by the square of the other is a maximum.

a. 13 and 23 c. 16 and 20 b. 25 and 11 d. 12 and 24

17.A rectangular box open at the top is to be formed from a rectangular piece of cardboard 3 inches by 8 inches. What side of square should be cut from each corner to form the box with maximum volume?

a. 3 in. c. 1.5 in. b. 0.67 in. d. 2 in.

18.A closed cylindrical can must have a volume of 1000 in3. Its lateral surface is to be constructed from a rectangular piece of metal and its top and bottom are to be stamped from square pieces of metal and the rest of the square discarded. What height will minimize the amount of metal needed in the construction of the can?

a. 30/pi in. c. 50/pi in. b. 40/pi in. d. 60/pi in.

Steps in Solving Time-Rate Problems:

1. Draw the diagram if needed. Label the diagram with the numerical values given in the problem.

2. Determine the given rates and the rate needed in the problem.

3. Using the diagram or conditions in the problem, find an equation relating all the given and unknown variables.

4. Find a relationship that will equate the equation into a single variable if possible.

5. Differentiate the equation.6. Substitute all the necessary given values and

solve for the unknown rate.

D.C. APPLICATION : TIME-RATES

19.A spherical toy balloon is being filled with gas at the rate of 500 mm3/sec. When the diameter is 0.5 m, find the rate (in mm2/sec) at which the surface area is increasing.

a. 2 c. 4 b. 3 d. 5

20.A man is riding his car at the rate of 30 km/hr toward the foot of the pole 10 m high. At what rate is he approaching the top when he is 40 m from the foot of the pole?

a. -5.60 m/s c. -8.08 m/sb. -6.78 m/s d. -4.86 m/s

21.A ladder 10 ft long is resting against the side of a building. If the foot of the ladder slips away from the wall at the rate of 2 ft/min, how fast is the angle between the ladder and the building changing when the foot of the ladder is 6 ft away from the building?

a. 0.5 rad/min c. 0.25 rad/minb. 0.33 rad/min d. 0.67 rad/min

STANDARDS OF INTEGRATION

cudu cauadu

duufaduuaf

zduyduxduduzyx

cdxxfa

duauf 1

Algebraic, Exponential and Logarithmic Functions

cauadu ca

aduau

u ln

111 1

nforcun

duu nn

cedue uu cuudu ln

cuuuudu lnln

Trigonometric Functions

cuudu cossin cuudu sincos

cuudu seclntan cuudu csclncot

cuuudu tanseclnsec cuuudu cotcsclncsc

cuuuudu cossin21sin2 cuuuudu cossin

21

cos2

Inverse Trigonometric Functions

c

au

ua

duarcsin

22

c

au

auadu arctan1

22

c

auarc

aauu

du sec122

Hyperbolic Functions

cuudu coshsinh cuudu sinhcosh

cuudu coshlntanh cuudu sinhlncoth

cuhudu tanharcsinsec

cuhudu2

tanhlncsc

Integration by Parts

Trigonometric Substitution

a. When the integrand involves a2 – x2, use x = asinθ .b. When the integrand involves a2 + x2, use x = atanθ.c. When the integrand involves x2 – a2,use x = a secθ.

vduuvudv

Wallis’ Formula

where:

12212311231

cossin20

ornmnmornnormm

dnm

.1

.2

otherwiseif

evenarenandmbothwhen

1. Solve:

a. c.

b. d.

cx 23

2451

2

324 x

xdx

cx 25

2451

cx 25

2451 cx 2

526

51

111 1

nforcun

duu nn

2. Integrate:

a.

b.

c.

d.

dxxx

22 2

cxxx 2ln16621 2

cxxx 2ln16621 2

cxxx 2ln16621 2

cxxx 2ln16621 2

3. Find:

a. c.

b. d.

dxxe x3

cxxe x

31

31 3 cxe x

31

31 3

cxxe x

31

31 3 cxe x

31

31 3

4. Find:

a. c.

b. d.

dxx

xxsin

sin3cot2 2

cxx coscsc2 cxx sin2sec

cxx cos3csc2 cxx 2cos2sin

5. Integrate:

a. c.

b. d.

2

329 x

dx

cx

x

299c

x

x

29

cx

x

29c

x

x

29

9

6. Find:

a. 16/1155 c. 8/99 b. 16/1001 d.

8/1155

2074 cossin

d

7. Solve:

a. 88/3 c. 3 b. 89 d. 79/3

4

2

3

1

2 dydxyx

IC APPLICATION : PLANE AREAS

Using Horizontal Strip

Using Vertical Strip

Using Polar Coordinates

2

1

y

y LR dyxxA

2

1

x

x LU dxyyA

2

1

2

21

drA

8. Determine the area of the region bounded by the parabola y = 9 – x2 and the line x + y = 7.

a. 4.5 sq. units c. 1.5 sq. unitsb. 2.5 sq. units d. 3.5 sq.

units

9. Find the area bounded by the curves y = x4 – x2 and y = x2 - 1.

a. 16/13 sq. units c. 15/4 sq. unitsb. 16/15 sq. units d. 17/3 sq.

units

10.Find the area of the region bounded by the curve r2 = 16cos θ.

a. 16 sq. units c. 30 sq. unitsb. 32 sq. units d. 25 sq. units

I.C. APPLICATION: LENGTH OF A PLANE CURVE

in Rectangular Form

in Parametric Form

in Polar Form

dydydxdx

dxdyS

b

a

b

a

22

11

dtdtdy

dtdxS

b

a

22

d

ddr

rS

2

1

22

11.Find the length of arc of the parabola y2 = 4x from the vertex to a point where x = 4.

a. 4.92 c. 6.92 b. 5.92 d. 7.92

dydydxS

y

y

2

1

2

1

12.Find the perimeter of the curve

a. 26 c. 30 b. 28 d. 32

sin14 r

d

ddrrS

2

1

22

I.C. APPLICATION : CENTROID OF PLANE AREAS Using Horizontal/Vertical Strip

where (xc,yc) is the centroid of the strip.

For Rectangles

TOTALTOTAL AyAyA

yA

xAxAx 22112211 ;

2

1

2

1

y

yc

x

xc dyxdxxxA

2

1

2

1

y

yc

x

xc dyydxyyA

13.The given area is bounded by the curve y = x2 and the line 2x + y = 8. Determine the x-component of the centroid of the area.

a. -2 c. -0.8 b. -1 d. -0.5

14.A small square 5 cm by 5 cm is cut out of one corner of a rectangular cardboard 20 cm wide by 30 cm long. How far from the uncut longer side is the centroid of the remaining area?

a. 9.56 cm c. 9.48 cm b. 9.35 cm d. 9.67 cm

I.C. APPLICATION : MOMENT OF INERTIA OF PLANE AREAS

About the x-axis (use horizontal strip)

About the y-axis (use vertical strip)

2

1;2y

yx xdyAdAyI

2

1;2x

xy ydxAdAxI

15.Find the moment of inertia of the area bounded by the curve x2 = 8y, the line x = 4 and the x-axis on the first quadrant with respect to the y-axis.

a. 1.14 c. 15.1 b. 2.15 d. 25.6

2

1;2x

xy ydxAdAxI

I.C. APPLICATION : VOLUME OF SOLID OF REVOLUTION

Circular Disk Method

Circular Ring Method (Washer Method)

Cylindrical Shell Method

2

1

2

1

22 y

y LRx

x LU dyxxVordxyyV

2

1

2

1

2222 y

y LRx

x LU dyxxVordxyyV

2

1;2

x

x rotationLU xxrdxyyrV

2

1;2

y

y rotationLR yyrdyxxrV

16.Find the volume of the solid of revolution obtained by revolving the region bounded by y = x – x2 and the x-axis about the x-axis?

a. pi/15 cu. units c. pi/30 cu. unitsb. pi/45 cu. units d. pi/60 cu. units

17.What is the volume generated when the area in the first quadrant bounded by the curve x2 = 8y, the line x = 4 and the x-axis is revolved about the y-axis.

a. 40.13 cu. units c. 50.26 cu. unitsb. 45.78 cu. units d. 30.56 cu.

units

18.Find the volume obtained if the region bounded by y = x2, y = 8 – x2 and the y-axis is rotated about the x-axis.

a. 156pi/3 cu. units c. 254pi/3 cu. units b. 256pi/3 cu. units d. 356pi/3 cu. units

PAPPUS THEOREM

First Proposition of Pappus :

where A is area, S is the length of arc and r is the distance from the centroid of the arc to the axis of revolution.

Second Proposition of Pappus :

where V is the volume and r is the shortest distance from the centroid of the area to be revolved to the axis of revolution.

rSA 2

rAV 2

19.Find the surface area of a right circular cylinder with radius of 20 cm and height of 30 cm.

a. 1276.5 cm2 c. 3568.3 cm2

b. 2265.4 cm2 d. 4100.5 cm2

20.Determine the volume generated by rotating the curve x2 + y2 = 25 about the line x – 10 = 0.

a. 4560.1 cu. units c. 4934.8 cu. units

b. 5142.2 cu. units d. 6142.5 cu. units