Thread: Finding critical points and equations of tangents.

1. Determine the critical points (Clasify as maxima, minima), Inflection points and trace the y=16x+4x^2-x^4
2. Find the equation of tangent to the circle x^2 + y^2 = 9 and parallel to 2x + y= 10.

Originally Posted by banana_banana

1. Determine the critical points (Clasify as maxima, minima), Inflection points and trace the y=16x+4x^2-x^4
2. Find the equation of tangent to the circle x^2 + y^2 = 9 and parallel to 2x + y= 10.

2.
First find a slope form for the given circle by implicitly differentiating with respect to x:



and solve for y':



Because the slopes are equal, the derivative form above, when evaluated at the point of tangency, should be equal to the slope of the given line, which is is -2. Note:




which means


Substitute this condition into the equation of the circle to solve for x, then find y by taking half of x.

1.y=16x+4x^2-x^4

Differentiate


Now solve for zeros (or critical numbers to f(x)) then study the sign to determine types of extrema.

Good luck!!

Originally Posted by apcalculus

2.

1.y=16x+4x^2-x^4

Differentiate


Now solve for zeros (or critical numbers to f(x)) then study the sign to determine types of extrema.

Good luck!!

, Why is that using using quadratic formula and completing square in getting the CPs would not yield the same value such as this:
Completing Square:
0=x²-2x+2

0= x²-2x+1=-2+1
(x-1)²=-1
x-1=±√-1
x=1±√-1

Quadratic Formula:
-1±√(-2)²-4(1)(2)
(2)(1)
-1±√-4
2

Originally Posted by banana_banana

, Why is that using using quadratic formula and completing square in getting the CPs would not yield the same value such as this:
Completing Square:
0=x²-2x+2

0= x²-2x+1=-2+1
(x-1)²=-1
x-1=±√-1
x=1±√-1 Mr F says: Do you realise that this value of x is not real?

Quadratic Formula:
-1±√(-2)²-4(1)(2) Mr F says: b = -2, not -1 (which is what you substituted in the bit of have highlighted).
(2)(1)
-1±√-4
2

Your equations would look much better and be easier to post if you learnt some basic latex: http://www.mathhelpforum.com/math-help/latex-help/

Originally Posted by banana_banana

, Why is that using using quadratic formula and completing square in getting the CPs would not yield the same value such as this:
Completing Square:
0=x²-2x+2

0= x²-2x+1=-2+1
(x-1)²=-1
x-1=±√-1
x=1±√-1

Quadratic Formula:
-1±√(-2)²-4(1)(2)
(2)(1)
-1±√-4
2

When you apply the quadratic formula, b = -2, so -b is 2. For some reason you start with a -1 in the numerator in your work.

Pull out the 4 from the square root as a 2, and you get:



Did you need the complex zeros?

so from that I can conclude that from X+2=0, i can have X= -2, and for 1±√-1, how should I have this If it has plus and minus sign, how could i fit it in the parenthesis because both will yield a different value of y.

First Critical Point (-2, -48)
SEcond critical point (1±√-1,?)

TANGENT EQUATION OF CIRCLE

Originally Posted by apcalculus

First find a slope form for the given circle by implicitly differentiating with respect to x:



and solve for y':



Because the slopes are equal, the derivative form above, when evaluated at the point of tangency, should be equal to the slope of the given line, which is is -2. Note:




which means


Substitute this condition into the equation of the circle to solve for x, then find y by taking half of x.

I dont understand what you mean about this? If i substitute X=2y to the X^2+Y^2=9 , i will get a y=(√9/5). then what will be the next?

Originally Posted by banana_banana

so from that I can conclude that from X+2=0, i can have X= -2, and for 1±√-1, how should I have this If it has plus and minus sign, how could i fit it in the parenthesis because both will yield a different value of y.

First Critical Point (-2, -48)
SEcond critical point (1±√-1,?)

Have you realised yet that are not real (I did point this out in my earlier post)? Surely the implied domain of is real numbers.

Originally Posted by banana_banana

[snip]TANGENT EQUATION OF CIRCLE
I dont understand what you mean about this? If i substitute X=2y to the X^2+Y^2=9 , i will get a y=(√9/5). then what will be the next?

What you do next is recall that x=2y and substitute the value of y ....

(By the way, please don't edit posts that have been replied to by adding slabs of new stuff. Make a new post in the thread).