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!
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
-------
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.
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)
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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)
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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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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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