How do you find the equation of the tangent line to the graph #f(x)=ln((e^x+e^-x)/2)# through point (0,0)?
The x-axis,
So, the equation to the tangent at (0, 0) is
graph{(e^y-(e^x+e^(-x))/2)y=0 [-10, 10, -5, 5]}
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To find the equation of the tangent line to the graph of f(x) = ln((e^x + e^-x)/2) through the point (0,0), we can follow these steps:
- Find the derivative of f(x) using the chain rule and simplify it.
- Substitute x = 0 into the derivative to find the slope of the tangent line.
- Use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point (0,0) and m is the slope found in step 2.
- Simplify the equation obtained in step 3 to get the final equation of the tangent line.
Let's go through these steps:
-
The derivative of f(x) can be found as follows: f'(x) = (1/((e^x + e^-x)/2)) * (1/2) * (e^x - e^-x) = (2/(e^x + e^-x)) * (e^x - e^-x) = (2e^x - 2e^-x)/(e^x + e^-x)
-
Substituting x = 0 into the derivative: f'(0) = (2e^0 - 2e^0)/(e^0 + e^0) = 0/2 = 0
-
Using the point-slope form with (x1, y1) = (0,0) and m = 0: y - 0 = 0(x - 0) y = 0
-
The equation of the tangent line is y = 0.
Therefore, the equation of the tangent line to the graph of f(x) = ln((e^x + e^-x)/2) through the point (0,0) is y = 0.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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