How do you find the equations of the tangent to the curve #y = (1 - x)(1 + x^2)-1# that pass through the point (1, 2)?
Find the equation for the line tangent to the curve at
.
The tangent line has equation:
The slope-intercept form is
Use whatever tools you have to solve this cubic. I got
So the equation of the tangent line is (approximately)
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To find the equations of the tangent to the curve y = (1 - x)(1 + x^2)^-1 that pass through the point (1, 2), we need to follow these steps:
- Differentiate the given curve equation with respect to x to find the derivative dy/dx.
- Substitute the x-coordinate of the given point (1, 2) into the derivative to find the slope of the tangent line at that point.
- Use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope found in step 2.
- Simplify the equation obtained in step 3 to find the equation of the tangent line.
Let's go through these steps:
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Differentiating y = (1 - x)(1 + x^2)^-1: dy/dx = -[(1 + x^2)^-1] + (1 - x)(-1)(2x)(1 + x^2)^-2 Simplifying: dy/dx = -[(1 + x^2) + 2x^2(1 - x)] / (1 + x^2)^2
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Substituting x = 1 into the derivative: dy/dx = -[(1 + 1^2) + 2(1)^2(1 - 1)] / (1 + 1^2)^2 Simplifying: dy/dx = -2/4 = -1/2
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Using the point-slope form with (x1, y1) = (1, 2) and m = -1/2: y - 2 = (-1/2)(x - 1)
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Simplifying the equation: y - 2 = (-1/2)x + 1/2 y = (-1/2)x + 5/2
Therefore, the equation of the tangent line to the curve y = (1 - x)(1 + x^2)^-1 that passes through the point (1, 2) is y = (-1/2)x + 5/2.
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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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