MTH101 (Fall 2010) Calculus And Analytical Geometry first assignment solution

[Xpert]

Solve the two sided inequality and show the solution on real line7 < 1-2x ≤ 10
Step 1: subtract 1 from both sides and we get
6<-2x<9
Step2: divide by -2 on both sides and we get
-3>x>-5
(note: symbols are reversed with a negative operation)
So x can have a value between -3 and or equal to -5
ß--|---|-----|-----|----|---0--------------------------------à
-5 -4 -3 -2 -1
[IMG]file:///C:%5CDOCUME%7E1%5Csalman%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image001.gif[/IMG]

Question 2; Marks: 10



Given two functions as:
f(x) = [IMG]file:///C:%5CDOCUME%7E1%5Csalman%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image003.gif[/IMG] and g(x) = [IMG]file:///C:%5CDOCUME%7E1%5Csalman%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image005.gif[/IMG]
Find fog(x) also find the domain of f, g and fog
Solution:
Fog(x)=f(g(x))
fog(x)=(g(x))2-(g(x))-1
fog(x)=(3/x)2-(3/x)-1
fog(x)=(9/x2)-3/x-1
fog(x)= -(x2-3x+9)/x2
(note: domain and ranges are to be found out yourselves, listen to lecture No. 6 for this)


Question 3; Marks:10

Simplify, then apply the rules of limit to evaluate
[IMG]file:///C:%5CDOCUME%7E1%5Csalman%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image007.gif[/IMG]
Solution:
Lim(x->3) x(x2-5x+6) / x2-32
Lim(x->3) x(x2-3x-2x+6) / (x-3)(x+3)
Lim(x->3) x(x(x-3)-2(x-3)) / (x-3)(x+3)
Lim(x->3) x((x-3)(x-2)) / (x-3)(x+3)
Lim(x->3) x(x-2) /(x+3)
Now applying limits lim x->3
=3(3-2)/3+3
=3(1)/6
=3/6
=½