1) Using differentials, find the approximate values of the following:
a) √25.02. 5.002
b) ³√(0.009). 0.208333
c) ³√(0.007). 0.1916667
d) √401. 20.025
e) ⁴√(15). 1.96875
f) ⁴√255. 3.9961
g) 1/(2.002)². 0.2495
h) 1/√(25.1). 0.198
i) √25.2. 5.02
j) ³√127. 5.026
k) √0.037. 0.1925
l) (80)¹⁾⁴. 2.9907
m) (29)¹⁾³. 3.074
n) (66)¹⁾³. 4.0416
o) √26. 5.1
p) √37. 6.083
q) √0.48. 0.693
r) (82)¹⁾⁴. 3.009
s) 1/√100.5. 0.09975
2) Using differentials, find the approximate values of the following:
a) log(4.01) given log 4= 1.3863.
1.3888
b) log(4.04) given log 4= 0.6021 and log e= 0.4343. 1.396368
c) log(10.02) given log 10=2.3026 2.3046
d) log(10.1) given log 2= 0.4343. 1.004343
e) log 1005, given log e = 0.4343. 3.0021
f) log 40.05, log 4= 0.6021and loge = 0.4343. 1.6026
g) log (25.02), given log 25= 3.2189
3.2197
3) Using differentials, find the approximate values of the following:
a) tan 46°, given 1°= 0.01745. 1.03490
b) tan 44°, given 1°= 0.01745. 0.965
c) cos61, given sin60= 0.86603 and 1= 0.01745. 0.4849
d) sin61=, given that 1= 0.01745. 0.874725
e) sin(21/14). 1
f) sin(11π/20). 1
g) cos(11π/36). 0.575575
h) cos(11π/24), given π= 3.14159. 0.1309
4) Find the approximate value of:
a) f(2.01) where f(x)= 4x²+5x+2. 28.21
b) f(5.001) where f(x)= x³-7x²+15. -34.99
5) The radius of a balloon is 7cm. if an error 0.01cm. is made in measuring the radius. find the error in measuring the volume of the balloon. 616cm³
6) Using the method of Differential find approximate the difference between
A) The area of two circle of radii 7 cm and 7.02cm. 0.88cm²
B) the volume of a sphere of radius 10cm and 9.99cm. 12.57cm³
7) the radius of a sphere is found by measurement to be10cm. if there are be a maximum probable error of 0.05cm. in the measurement of the radius, find the maximum possible error in the computation of the surface area of the sphere. 4πcm²
8) Due to heating of the side of a metallic cube expand from 4 to 4.05cm. using differential find approximate the increase in volume of the cube. 2.4cm³
9) if there is an error of 1% in measuring the radius of a sphere, what is the approximate percentage error in the measurement of the volume of the sphere. 3%
10) The volume of Sphere, radius r cm is 4/3 πr³ cm³. Find the approximate decrease in volume of a sphere when radius decreases from 3 to 2.98 cm. 072π cm³
11) find the approximate change in 1/x when x= 1, δx= 0.2. -0.2
12) A sphere of radius 10 cm shrinks to radius 9.8 cm. Find approximately the decrease in
a) volume. 80π cm³
b) surface area 16π cm²
13) A circular plate expands when heated from a radius of 5 to 5.06 cm. Find the approximate increase in area. 0.6π cm²
14) if the side of a cube is 10.01 cm, find approximately the volume of the cube. 1003 cm³
15) find the approximate change in the volume of a cube of side x cm caused by increasing the sides by 1%. 0.03x³ cm³
16) Find the approximate mass of a length of copper tubing if the inside diameter is 2.5cm and the thickness is 0.25 cm given that density is 8800 kg/cm³.
17) differentiate 1/√x with respect to x, and use the result to find an approximate value of 1/√100.5. 0.09975
18) The shape of a bowl is such that V= h³+ 3h²+ 11h where V cm³ is the volume of water in the bowl and h cm the depth of water. If, when h= 7, an additional small volume δV cm³ of water is poured into the bowl, prove that the level of the water rises approximately δV/200 cm
19) A closed circular cylinder has height 16 cm, and radius r cm. The total surface area is A cm². prove that dA/dr = 4π(r+8).
hence calculate an approximate increase in area if the radius increases from 4 to 4.02 cm, the height remaining constant (you may leave your answer in terms of π).
20) A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10 cm. 4π
21) find the percentage error in calculating the volume of cubical box if an error of 1% is made in measuring the length of edges of the cube. 3%
22) The time T of a complete oscillation of a simple pendulum of length is given by the equation T= 2π√(l/g) , where g is constant. what is the percentage error in T when l is increased by 1% ? 1/2%
23) Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 2%. 0.06x³ m³
24) if the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximating error in calculating its volume. 9.72π cm³
25) the radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume. 80π cm³
26) A circular metal plate expands under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of a plate before heating is 10cm. 2kπ
27) find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube. 2%
28) If there is an of 0.1 % in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere. 0.3%
29) the pressure p and the volume v of a gas are connected by the relation pv¹·⁴= constant. find the percentage error in p corresponding to a decrease of 1/2% in v. 0.7%
30) the height of a cone increases by k%, its semi Vertical angle remaining the same. What is the approximate percentage increase
a) in total surface area. 2k%
b) in the volume, assuming that k is small. 3k%
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