Common tangent line Calculus
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Common tangent line Calculus

[From: ] [author: ] [Date: 12-09-28] [Hit: ]
6], find the slope of the common tangent line between the two. In other words, I need to find the tangent line of f(x) that intersects g(x) in only one location. PLEASE HELP ME!-On the curve y= 1-x^2,......
For the two functions: f(x)=1-x^2 where x is between [-1,1] and g(x)=4-(x-4)^2 where x is between [2, 6], find the slope of the common tangent line between the two. In other words, I need to find the tangent line of f(x) that intersects g(x) in only one location. PLEASE HELP ME!

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On the curve y= 1-x^2, let the point of tangency be (a, 1-a^2)
On the other curve, let the point of tangency be (b, 4-(b-4)^2)

On the first curve, y' = -2x ; at x= a, the slope is -2a

On the second curve, y' = -2(x-4); at x= b, the slope is -2(b-4)

So -2a = -2(b-4)
a= b-4

Also, using slope formula, the slope of the common tangent is [4-(b-4)^2 -(1-a^2)] / (b-a)
Substitute a= b-4:

= [ 4-(b-4)^2 -1+ (b-4)^2]/(b -(b-4))

= 3/4
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If you need the line also, -2x= 3/4, when x= -3/8 on f(x)

And -2(x-4)= 3/4
X-4= -3/8
X= 29/8 on g(x)

Then plug the x coordinates into the functions to find the y coordinates.
(-3/8, 55/64)
(29/8, 247/64)
Then m= 3/4, which we already found

Then the tangent line is Y= (3/4)x + 73/64

Hoping this helps!

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The common tangent to two functions always has the same slope, so:
The derivative of f(x) is -2x.
The derivative of g(x) is -2x+8
Setting them equal with different variables:
-2a=-2b+8
divide by -2
a=b-4.
Looking at the two functions in the given domains, and trying a few, it looks like the line touches f(x) somewhere in (-0.4,-0.3).
You could make a program to run through all possible scenarios (there's not that many for a calculator to do), or there might be some more algebra you can do. I tried a calculator program, but it's unoptimized and still searching at -0.3999994214 - I don't want to wait; you can try. I'm leaving it overnight, so I'll have it by tomorrow morning if I remember to check it.
So try lines tangent to f(x) at (-0.4,-0.3) and they should touch g(x) at (3.6,3.7) - it's somewhere in those domains.

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ƒ(x) = 1 - x², ......... x in [-1,1] ............. (1)

g(x) = 4 - (x-4)², ... x in [2,6] ............... (2)
__________________________

At their Common Point : ƒ(x) = g(x)

∴ 1 - x² = 4 - (x-4)²

∴ 1 - x² = 4 - x² + 8x - 16

∴ 13 = 8x

∴ x = 13/8 ........................................… (3)
_____________________________

Now : x = 13/8 ∉ [ -1, 1 ] and x = -1 ∉ [ 2, 6 ]

Hence, the two curves y = ƒ(x) and y = g(x)

do Not Have any Common Point and, hence,

do Not Have any Common Tangent. .................................. Ans.
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