Transcript
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
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
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
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
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
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
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
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
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