Expert: Paul Klarreich - 3/28/2008

Question
QUESTION: Show that the family of curves described by x^2 + 3y^2 = b are orthogonal y=ax^3?

ANSWER: Questioner:   mary
Category:  Calculus
Private:  No
 
Subject:  calculus 1
Question:  Show that the family of curves described by x^2 + 3y^2 = b are orthogonal y=ax^3?
..............................................
Hi, Mary,

The word 'orthogonal' means (I assume you looked it up.) 'meeting at right angles.'  

For curves, that means 'tangents meeting at right angles'.  

And that means 'slopes of tangents are negative reciprocals.'  

That, in turn, means 'product of derivatives is -1.'
..........................
Let's try that:

x^2 + 3y^2 = b

2x + 6y dy/dx = 0
dy/dx:1 = -x/(3y)

y = ax^3

dy/dx:2 = 3ax^2.

Now if (x,y) is a point of intersection, then y = ax^3, and so;

dy/dx:1 = -x/(3(ax^3)) = -1/(3ax^2)

And: dy/dx:1 times dy/dx:2 = -1/(3ax^2) times 3ax^2 = -1.

That's it.


---------- FOLLOW-UP ----------

QUESTION: thanks a lot!!

i have few more questions! tmw is my big exam..so im getting prepared for it!

1) find the derivative of f(x)=cos(tanx^3) + ln(cot^3)

2)find the lim sin 7x/3x (under the lim sign, it states f(x)--> 0)

3)The function f(x)= x^3 + 2x + 1 has an inverse g(x) fine g'(-2)          (the answer is g(-2)=1/5) please show steps!

4)Find the second derivative of f(x) = sinxlnsinx

5) find the derivative of x^2 + 3squareroot of x - 5x/squareroot of x

PLEASE REPLY ASAP!! ill B VERY GREATFUL!! THANKS

Answer
Questioner:   mary
Category:  Calculus
Private:  No
 
Subject:  Implicit differentiation.

---------- FOLLOW-UP ----------

QUESTION: thanks a lot!!

i have few more questions! tmw is my big exam..so im getting prepared for it!

1) find the derivative of f(x)=cos(tanx^3) + ln(cot^3)

2)find the lim sin 7x/3x (under the lim sign, it states f(x)--> 0)

3)The function f(x)= x^3 + 2x + 1 has an inverse g(x) fine g'(-2)          (the answer is g(-2)=1/5) please show steps!

4)Find the second derivative of f(x) = sinxlnsinx

5) find the derivative of x^2 + 3squareroot of x - 5x/squareroot of x

PLEASE REPLY ASAP!! ill B VERY GREATFUL!! THANKS
 
.......................
A few more questions.  This begins to look as if you are asking me to do your homework and be your tutor at the same time.  I'll get you started, but I'm not going to do all your work for you.


1) find the derivative of f(x)=cos(tanx^3) + ln(cot^3)

Use the chain rule (more than once).  Also, don't write cot^3 by itself.  Pay attention to detail.



2)find the lim sin 7x/3x (under the lim sign, it states f(x)--> 0)

Use lim sin t/t = 1 and a substitution  t = 7x.  Do the algebra right.


3)The function f(x)= x^3 + 2x + 1 has an inverse g(x) find g'(-2)          (the answer is g(-2)=1/5) please show steps!

g'(x) = 1/f'(x)


4)Find the second derivative of f(x) = sinxlnsinx

Use product and chain rules.

5) find the derivative of x^2 + 3squareroot of x - 5x/squareroot of x

Write it as x^2 + 3x^1/2 - 5x/x^1/2. The rest should be easy.