Mathematics HL/SL/Studies Help

[Drake Glau]

^I'm not sure if this is right since I havent done math in a long time.

C'(x) = 1 - (10000/x²)

(Note 1/x = x^-1)

We wish to minimize the cost, so set C'(x) = 0

0 = 1 - (10000/x²)

x=100

Yes, but it might be required to do the 2nd derivative test to prove that x=100 is the minimum and not a maximum. Prove that C'' at x=100 is positive to ensure it is a minimum.

Edited August 8, 2012 by Drake Glau

[xXRainbowDashXx]

Hello fellow IBers. I would really appreciate some help with this I am stuck.

Calculate the time needed (in whole years) for a capital amount of 350 USD to reach 500 USD, if it is invested at a simple interest rate of 6 % p.a (per annum/year).

How do I solve this?

[The Economist]

Hello fellow IBers. I would really appreciate some help with this I am stuck.

Calculate the time needed (in whole years) for a capital amount of 350 USD to reach 500 USD, if it is invested at a simple interest rate of 6 % p.a (per annum/year).

How do I solve this?

Use the formula:

Final value= Initial value x (1+ r/100)^n where r is the interest paid.

Use $500 as final value, $350 as initial, r=6 and solve in terms of n which is the years.

[robot125]

Cirrito textbook:

A solid 400gm metal cube of side length 10cm expands uniformly when heated. If the lenght of its sides expand at 0.5cm/hr, find the rate at which, after 5 hrs:

A)its volume is increasing

B)its surface area is increasing

C)its density is changing

[xXRainbowDashXx]

Alex invests 3600 euros in an account that pays a nominal rate of 5.4% interest per year, compounding monthly. The interest is added to the account at the end of each month.

a)calculate the value of alex's investment after nine years.

B) find the rate of cimple interest per year that would give the same value for the investment after nine years.

(Math studies chapter 8, financial mathematics) Please help

[Fermat]

Alex invests 3600 euros in an account that pays a nominal rate of 5.4% interest per year, compounding monthly. The interest is added to the account at the end of each month.

a)calculate the value of alex's investment after nine years.

B) find the rate of cimple interest per year that would give the same value for the investment after nine years.

(Math studies chapter 8, financial mathematics) Please help

I think you would be able to use this formula:

A=P(1+(r/n))^nt

where:

A is the amount of money after the time n years (including interest)

r is the interest rate

P is the amount invested

t is the number of years the money is deposited for

n number of times the interest is compounded / year

[The Economist]

Alex invests 3600 euros in an account that pays a nominal rate of 5.4% interest per year, compounding monthly. The interest is added to the account at the end of each month.

a)calculate the value of alex's investment after nine years.

B) find the rate of cimple interest per year that would give the same value for the investment after nine years.

(Math studies chapter 8, financial mathematics) Please help

I think you would be able to use this formula:

A=P(1+(r/n))^nt

where:

A is the amount of money after the time n years (including interest)

r is the interest rate

P is the amount invested

t is the number of years the money is deposited for

n number of times the interest is compounded / year

wonderful, thanks!

The time should be in months. So for instance, at a) the time would be 9 (years)*12=108 months.

[MariusIBDP]

Matrices problem:

If A³=I, simplify A²(A+I)²

Please show me a step by step solution. The answer is suppose to be A²+A+2I, but I, for some reason, end up with a different answer no matter how I do it. (I is an identity matrice)

[Fermat]

A^2(A+1)^2

=

A^2(A^2+I^2+2AI)

=

A^4+A^2I^2+ 2A^3I

= A^3(A)+A^2I^2+2I^2

=IA+A^2+2I^2

=A+A^2+2I

I know all the ^ are confusing, but if you write in down a paper, you'll understand better.

EDIT:Remember that multiplying a matrix by its identity matrix gives you the same matrix.

Edited August 22, 2012 by Athymy

[MariusIBDP]

A^2(A+1)^2

=

A^2(A^2+I^2+2AI)

=

A^4+A^2I^2+ 2A^3I

= A^3(A)+A^2I^2+2I^2

=IA+A^2+2I^2

=A+A^2+2I

I know all the ^ are confusing, but if you write in down a paper, you'll understand better.

EDIT:Remember that multiplying a matrix by its identity matrix gives you the same matrix.

Thanks, now I see my mistakes. Did you know that there is a X² button?

Another problem:

Find all 2x2 matrices A for which A²=A. [Hint: Let A = 2 x 2 Matrix a, b, c, d.]

[Fermat]

A^2(A+1)^2

=

A^2(A^2+I^2+2AI)

=

A^4+A^2I^2+ 2A^3I

= A^3(A)+A^2I^2+2I^2

=IA+A^2+2I^2

=A+A^2+2I

I know all the ^ are confusing, but if you write in down a paper, you'll understand better.

EDIT:Remember that multiplying a matrix by its identity matrix gives you the same matrix.

Thanks, now I see my mistakes. Did you know that there is a X² button?

Another problem:

Find all 2x2 matrices A for which A²=A. [Hint: Let A = 2 x 2 Matrix a, b, c, d.]

Oh lol, I didn't notice

I think you could solve the problem (it's probably not the best way but w/e) by multiplying A*A and just let it equal A (A² =A) And then you can set each entry in matrix A² equal the corresponding entry in matrix A. So you should get 4 different equations.

I haven't solved it yet so I can't tell if it's the correct way but you can always try it

[MariusIBDP]

A^2(A+1)^2

=

A^2(A^2+I^2+2AI)

=

A^4+A^2I^2+ 2A^3I

= A^3(A)+A^2I^2+2I^2

=IA+A^2+2I^2

=A+A^2+2I

I know all the ^ are confusing, but if you write in down a paper, you'll understand better.

EDIT:Remember that multiplying a matrix by its identity matrix gives you the same matrix.

Thanks, now I see my mistakes. Did you know that there is a X² button?

Another problem:

Find all 2x2 matrices A for which A²=A. [Hint: Let A = 2 x 2 Matrix a, b, c, d.]

Oh lol, I didn't notice

I think you could solve the problem (it's probably not the best way but w/e) by multiplying A*A and just let it equal A (A² =A) And then you can set each entry in matrix A² equal the corresponding entry in matrix A. So you should get 4 different equations.

I haven't solved it yet so I can't tell if it's the correct way but you can always try it

That part I understand. I get 4 different equations:

a²+bc=a ab+bd=b ac+dc=c cb+d²=d

From this I find that a+d=1 and that's it, I'm stuck

[Procrastination]

Hello people. I need some help. What's the derivative for 2x - (1/x)? Thanks

[Fermat]

the derivative of f(x)+g(x) is equal to f'(x)+g'(x)

So let f(x)=2x and g(x)=(-1/x)

The derivative of 2x is 2 (f'(x)=2)

The derivative of g(x)=-1/x is the same thing as writing g(x)=-1(x^-1) and the derivative of that [using the derivative rule that the derivative of x^n is equal to nx^(n-1)] is (1/x^2)

So combining these we get that the derivative is (1/x^2)+2

EDIT:Oops, noticed I wrote x instead of 2, sorry about that, corrected it now.

Edited August 31, 2012 by Fermat

[Drake Glau]

Hello people. I need some help. What's the derivative for 2x - (1/x)? Thanks

f(x)=2x-(1/x)

f'(x)=g'(x)-h'(x)

g(x)=2x therefore g'(x)=2

h(x)=1/x therefore h'(x)=-1/(x^2)

g'(x)-h'(x)=2+(1/x²)

[Drake Glau]

Find an equation of the plane.

The plane through the point (7, −8, −4)

and parallel to the plane 7x − y − z = 7

I missed the day we went over planes and can't figure it out on my own =/

Edit because I don't know anything about planes apparently...including how to even make the equation for one. How do I find the line that indicates the intersection of 2 planes.

I keep getting questions about finding the plane containing some point and containing some line, how would I go about that too?

All I know so far is I need a point on the plane and a vector that is perpendicular to the plane in order to form a plane equation.

Edited September 12, 2012 by Drake Glau

[macrofire]

I've never really looked at planes either until now, but let's try something.

First of all, re-write the equation to get z = 7x-y-7

Now note that when z = 0, you have your line on the x-y axis

now try at z = 1. On the 3-d graph, you will have a different line (but its image is parallel to the line at z = 0), only this line is at the plane z = 1.

Now you can tell where the orientation of the plane is.

So here's my hypothesis: parallel planes are related by the coefficients of the x,y,z variable. This implies that one can get infinitely many parallel planes by changing the constant value. This is because of a small test I did. I imagined the plane ax +by+cz +d = 0 as infinitely many lines put together to create the plane. Then I isolated one line at z = 0. I noted that by increasing the z or d value by one, the resulting line would move up the z-axis. The only difference is that by increasing the d value by one, the line moves immediately up, while increasing the z value moves it up and moves it proportional to the orientation of the plane.

At any rate, just by changing the constant value will you be able to find infinitely many parallel planes to the one given. So you know the solution plane has to be in the form 7x-y-z=n for some n and it contains the point (7, -8, -4). Ergo...

So try it out and tell me I got it right or wrong. I'll work on finding the lines parallel to the plane later.

[MariusIBDP]

Find the equation of the tangent and the normal to the curve f(x)=x+x^{-1 ,} x≠0 at the point (1;2). Find the coordinates of the points where the tangent and the normal cross the x- and y- axis, and hence determine the area enclosed by the x-axis, the y-axis, the tangent and the normal

Edited October 10, 2012 by MariusIBDP

[macrofire]

Cirrito textbook:

A solid 400gm metal cube of side length 10cm expands uniformly when heated. If the lenght of its sides expand at 0.5cm/hr, find the rate at which, after 5 hrs:

A)its volume is increasing

B)its surface area is increasing

C)its density is changing

A) If the volume is changing, then solve for dV/dt. Remember, V = r^3 and dr/dt is 0.5cm3/hr

B) Surface area is changing, then solve for dA/dt for A = 6r^2

C) D is density, and D = m/V. Or DV = m and m is a constant. Differentiating both sides in terms of t, one gets D'V + DV' = 0. So D' = -V' (D/V) and you already know V' (V' = dV/dT).

A last hint is that you might need the r when t = 5. So use dr/dt accordingly to find the appropriate r, where r is the length of the cube.

Find the equation of the tangent and the normal to the curve f(x)=x+x^{-1 ,} x≠0 at the point (1;2). Find the coordinates of the points where the tangent and the normal cross the x- and y- axis, and hence determine the area enclosed by the x-axis, the y-axis, the tangent and the normal

Take the derivative of f(x). Plug in the x coordinate and get the slope. That is the tangent. The normal is the one that is perpendicular to the tangent. Use the slope intercept form to find the equations. Now solve for when x = 0 and when y = 0 for both equations. Plot on graph and determine the area. Because they're lines, you can outline a rectangle and take away various triangles until you get the area bounded by the x-axis. y-axis. tangent and normal.

Really, the hardest part is to figure out what to do. If you can do that, then the math exam will be a joke...if you manage your time.

[robot125]

Cirrito textbook:

A solid 400gm metal cube of side length 10cm expands uniformly when heated. If the lenght of its sides expand at 0.5cm/hr, find the rate at which, after 5 hrs:

A)its volume is increasing

B)its surface area is increasing

C)its density is changing

A) If the volume is changing, then solve for dV/dt. Remember, V = r^3 and dr/dt is 0.5cm3/hr

B) Surface area is changing, then solve for dA/dt for A = 6r^2

C) D is density, and D = m/V. Or DV = m and m is a constant. Differentiating both sides in terms of t, one gets D'V + DV' = 0. So D' = -V' (D/V) and you already know V' (V' = dV/dT).

A last hint is that you might need the r when t = 5. So use dr/dt accordingly to find the appropriate r, where r is the length of the cube.

Find the equation of the tangent and the normal to the curve f(x)=x+x^{-1 ,} x≠0 at the point (1;2). Find the coordinates of the points where the tangent and the normal cross the x- and y- axis, and hence determine the area enclosed by the x-axis, the y-axis, the tangent and the normal

Take the derivative of f(x). Plug in the x coordinate and get the slope. That is the tangent. The normal is the one that is perpendicular to the tangent. Use the slope intercept form to find the equations. Now solve for when x = 0 and when y = 0 for both equations. Plot on graph and determine the area. Because they're lines, you can outline a rectangle and take away various triangles until you get the area bounded by the x-axis. y-axis. tangent and normal.

Really, the hardest part is to figure out what to do. If you can do that, then the math exam will be a joke...if you manage your time.

Yes that is the obvious approach but it does not lead to the correct answer...the answer you get from that is about 4 times the answer you should...for A,B, and C.