- #1
MarkFL
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MHB
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Here is the question:
I have posted a link there to this topic so the OP can see my work.
I have posted a link there to this topic so the OP can see my work.
Find Tangent Point of 2 Functions: Mikey's Calculus Help
In summary, to find the point where the graphs of f(x)=x^3-2x and g(x)=0.5x^2-1.5 are tangent to each other, we need to find the common tangent line by setting the derivatives of the two functions equal to each other. We then solve for the common point of tangency, which in this case is x=1.
- #2
MarkFL
Gold Member
MHB
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Hello Mikey,
If the two given functions are tangent to each other, then the difference between the two functions will have a repeated root:
\(\displaystyle f(x)-g(x)=(x-a)^2(x-b)\)
\(\displaystyle x^3-\frac{1}{2}x^2-2x+\frac{3}{2}=x^3-(2a+b)x^2+\left(a^2+2ab \right)x-a^2b\)
Equating corresponding coefficients, we find:
\(\displaystyle 2a+b=\frac{1}{2}\)
\(\displaystyle a^2+2ab=-2\)
\(\displaystyle a^2b=-\frac{3}{2}\)
The third equation implies:
\(\displaystyle b=-\frac{3}{2a^2}\)
Using this, the first equation gives:
\(\displaystyle 4a^3-a^2-3=0\implies a=1,\,b=-\frac{3}{2}\)
And so the two functions are tangent to one another at $x=1$. Here is a plot:
View attachment 1527
If the two given functions are tangent to each other, then the difference between the two functions will have a repeated root:
\(\displaystyle f(x)-g(x)=(x-a)^2(x-b)\)
\(\displaystyle x^3-\frac{1}{2}x^2-2x+\frac{3}{2}=x^3-(2a+b)x^2+\left(a^2+2ab \right)x-a^2b\)
Equating corresponding coefficients, we find:
\(\displaystyle 2a+b=\frac{1}{2}\)
\(\displaystyle a^2+2ab=-2\)
\(\displaystyle a^2b=-\frac{3}{2}\)
The third equation implies:
\(\displaystyle b=-\frac{3}{2a^2}\)
Using this, the first equation gives:
\(\displaystyle 4a^3-a^2-3=0\implies a=1,\,b=-\frac{3}{2}\)
And so the two functions are tangent to one another at $x=1$. Here is a plot:
View attachment 1527
Attachments
Related to Find Tangent Point of 2 Functions: Mikey's Calculus Help
1. How do I find the tangent point of two functions?
To find the tangent point of two functions, you first need to graph both functions on the same coordinate plane. Then, find the point where the two functions intersect. This point is the tangent point.
2. What is the significance of the tangent point in calculus?
The tangent point represents the point where two curves have the same slope, indicating that they are tangent to each other at that point. This is useful in calculus as it allows us to find the slope of a curve at a specific point, which is important in determining the rate of change of a function.
3. Can two functions have more than one tangent point?
Yes, two functions can have more than one tangent point. This occurs when the two functions intersect at multiple points with the same slope.
4. Are there any shortcuts or formulas for finding the tangent point of two functions?
There are no specific shortcuts or formulas for finding the tangent point of two functions. However, you can use the derivative of the functions to determine the slope at a specific point and then find where the slopes of the two functions are equal, indicating the tangent point.
5. How does finding the tangent point of two functions relate to real-world applications?
In real-world applications, finding the tangent point of two functions can help in determining the optimal point for a given situation. For example, in economics, it can help in finding the equilibrium point where supply equals demand. In physics, it can help in determining the maximum or minimum value of a function.
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