Help with three questions... need them for my test tomorrow (1 Viewer)

[Brodie28]

I have an extension test tomorrow and there are three questions I couldn't get out... I'm sure I just made a stupid mistake when doing them though because they arn't that difficult. The second one I got one correct solution but couldn't get the other for some reason...

ANGLE BETWEEN TWO CURVES:

What is the obtuse angle between the curves f(x) = x² - 4x and g(x) = x² - 12 at the point where they meet.

Find the acute angle between the curve f(x) = x² -1 and the line g(x) = 3x -1 at their two points of intersection.

MATHEMATICAL INDUCTION:
n
∑ 3^-r = 3^n - 1 ÷ [2(3^n)]
r=1



Prove that n(n+1) is divisible by two when n is equal or greater than one.




These are just some example questions from the text book but I was hoping someone could explain how to do them better because the text book version isn't very clear (it blows).

 

[klaw]

What is the obtuse angle between the curves f(x) = x² - 4x and g(x) = x² - 12 at the point where they meet.

Equs meet when x=3

f'(x)=2x-4
f'(3)=2
g'(x)=2x
g'(3)=6
tan @=|4/(1-12)|
@=tan^-1 (4/11)
Obtuse angle = 180-tan^-1 (4/11)

 

[klaw]

Find the acute angle between the curve f(x) = x² -1 and the line g(x) = 3x -1 at their two points of intersection.

Equs meet when x=0 or 3

f'(x)=2x
f'(0)=0
f'(3)=8
g'(x)=3

tan @=3 or 5/23
@=tan^-1 3 or tan^-1 (5/23)

 

[Stan..]

For the Mathematical Induction exercise,
The second is
Test for 1 -> 2 True for 1
Assume true for n=k
k(k+1) = Q (Q is an integer)
Assume true for n=k+1
(k+1)k + 2(k+1) = P
Q + 2(k+1) = P
P is divisible by 2 thus statement true for all values of 2.

 

[klaw]

MATHEMATICAL INDUCTION:
n
∑ 3^-r = 3^n - 1 ÷ [2(3^n)]
r=1

When n=k+1,
LHS=3^k-1/[2(3^k)]+3^(-k-1)
=(3^k-1)[3^(k+1)]+2(3^k)]/{2[3^k][3^(k+1)}
=[3^k][3^(k+1)-3+2]/{2[3^k][3^(k+1)]
=[3^(k+1)-1]/{2[3^(k+1)]}
=RHS