Some exponential and log questions (1 Viewer)

[bmn]

We got a homework sheet we need to complete... I've done all that I could (we done this topic awhile ago), and I'd like help with some I cant do, and also to check answers on couple of others I'm not too sure about. Red ones are done, last time I checked this thread.

EDIT: Only 2 left...
6a) Show that d/dx (x^(2) Inx) = x(1+ 2 Inx) - Done
b) Hence evaluate 2x(1 + 2 Inx)dx (between 3 and 1)
11. Use the trapezoidal rule with 2 subintervals to find the volume of the solid formed when the curve y = e^x is rotated about the y-axis from y = 1 to y = 2. (No idea how to use trapezoidal to find volume...)

 

[kurt.physics]

10) Find the sum of the first 20 terms of the series (log2)3 + (log2)9 + (log2)27 +... Completely forget that stuff... What kind of series is it?

What kinda series do you think it is, does



or does

 

[kurt.physics]

1. Find the gradient of the tangent to the curve y= e^(1+lnx) at the point where x = e.

1. Find the derivative of y = e^{1 + lnx}

(hint: y' = f'(x)e^f(x))

2. Remember that the first derivative, ie y' is the gradient of the tangent.

3. Substitute x = e into y'

Done!

 

[kurt.physics]

2. Find the derivative of ln (1/(2x - 3))

hint: If y = log_e f(x), then y' = f'(x) X 1/f(x) = f'(x)/f(x)

hint 2: ln is log_e !

 

[kurt.physics]

3. Differentiate 2^x

This one is pretty hard (if you dont understand how to achieve the result), so i will just do it for you and you should understand the process.

We will proceed by differentiating by first principals:















Now, the only thing we need to do now is to evaluate:



Experimenting with a calulator, you will find

 

[kurt.physics]

4. Find the equation of the normal to the curve y = 2e^(2x) - e^x (I'd prob remember after Q1)

hint: the normal is perpendicular to the tangent, ie m₁m₂ = -1

hint 2: y - y₁ = m(x - x₁)

 

[lolokay]

"2. Find the derivative of ln (1/(2x - 3))"
this one can be simplified using log law

"3. Differentiate 2^x"
use e^ln[a] = a

"I've got everything clear up to integrating e^u"
err you should know how to integrate an exponential

"7) If P = 100e^(0.5t), find the value of P to the nearest unit if t = 1.7 (Is it a trick question, or just as simple as it looks?)"
yeah, i'd say it is as simple as it looks

 

[bmn]

"
"I've got everything clear up to integrating e^u"
err you should know how to integrate an exponential

I should, but I wasn't sure when intergrating to the power of u, because u = x^3... Do I just leave it at e^u or what?

 

[bmn]

hint: If y = log_e f(x), then y' = f'(x) X 1/f(x) = f'(x)/f(x)

hint 2: ln is log_e !

Would the end answer be e?

hint: the normal is perpendicular to the tangent, ie m₁m₂ = -1

hint 2: y - y₁ = m(x - x₁)

Would m1=3, m2= -1/3, and the end answer be x + 3y -3 = 0?

 

[bmn]

Due tomorrow... I'll check here again in the morning...