What are the equations of the tangent lines to both functions?

In summary, the conversation is about finding two lines that are tangent to two given functions. The approach mentioned involves finding the slope where the derivatives of the two functions are equal, but this does not work as it only finds an x-value where the slopes are the same. The correct approach is to assume the lines of tangency as y = mx + c and solve for the point of intersection with each curve, keeping in mind that there is only one point of intersection with each curve. The given answers are y = 2x - 1 and y = 4x - 4.
  • #1
teken894
25
0
Here's a question from my Calc HW andI believe my approach is flawed...

Two functions
f(x) = x^2
g(x) = -x^2 + 6x -5

Find the two lines tangent to both functions


I thought I could find the slope where f' and g' are equal so I did:

f'(x) = 2x
g'(x) = -2x + 6
2x = -2x +6
x = 1.5
slope = 2(1.5) = 3

BUT that slope doesn't work as this simply finds an x on the two graphs where the two graphs have the same slope.


Help me please!

THE given answers are
y=2x-1
y=4x-4
 
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  • #2
Assume the line of tangency is y = mx +c.

Solve to find the point of intersection of y=mx + c with each of the two curves.

Keep in mind that there is only one point of intersection (with each curve) and not two x-values.
 

What does it mean for a line to be tangent to both functions?

When a line is tangent to both functions, it means that the line touches the graph of both functions at only one point. This point is called the point of tangency.

How can I determine if a line is tangent to both functions?

In order to determine if a line is tangent to both functions, you will need to find the point of tangency. This can be done by setting the equations of the functions equal to each other and solving for the values that make the equations equal at only one point. Then, plug these values into the equation of the line. If the resulting equation is true, then the line is tangent to both functions.

What is the significance of finding a line that is tangent to both functions?

When a line is tangent to both functions, it represents the point of intersection of the two functions. This point can provide valuable information, such as the solutions to a system of equations or the maximum or minimum value of a function.

Are there any special cases when dealing with tangent lines to both functions?

Yes, there are two special cases to consider when dealing with tangent lines to both functions. The first is when the functions are parallel, in which case there is no point of tangency. The second is when the functions share the same tangent line, in which case there are infinitely many points of tangency.

How can I use tangent lines to both functions in real-life applications?

Tangent lines to both functions can be used to approximate the slope of a curve at a specific point, which can be useful in fields such as engineering, physics, and economics. They can also be used to determine the optimal solution in optimization problems.

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