# How do you find the equation of the tangent and normal line to the curve #y=1+x^(2/3)# at (0,1)?

Any line through the point

Substitute, the slope of the tangent line,

Usually we would use the form given in the above answer:

Evaluate at the x coordinate:

The procedure for the normal line is similar.

Evaluate at the x coordinate:

Substitute 0 for the slope:

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To find the equation of the tangent and normal lines to the curve (y = 1 + x^{2/3}) at the point (0,1), you can follow these steps:

- Find the derivative of the function (y = 1 + x^{2/3}) to get the slope of the tangent line at any point.
- Evaluate the derivative at the given point to find the slope of the tangent line at that point.
- Use the point-slope form of the equation of a line to write the equation of the tangent line.
- Use the negative reciprocal of the slope of the tangent line to find the slope of the normal line.
- Use the point-slope form of the equation of a line to write the equation of the normal line.

Let's go through the steps:

- The derivative of (y = 1 + x^{2/3}) is (dy/dx = (2/3)x^{-1/3}).
- Evaluate the derivative at (0,1): [dy/dx = (2/3)(0)^{-1/3} = \text{undefined}] Therefore, we need to use the limit definition to find the slope at (0,1): [\lim_{h \to 0} \frac{(1+(0+h)^{2/3}) - 1}{h} = \lim_{h \to 0} \frac{h^{2/3}}{h} = \lim_{h \to 0} h^{-1/3} = \infty] So, the slope of the tangent line at (0,1) is infinity.
- The equation of the tangent line at (0,1) is (x = 0).
- The slope of the normal line is the negative reciprocal of the slope of the tangent line, which is 0.
- Since the slope is 0, the equation of the normal line is (y = 1).

Therefore, the equations of the tangent and normal lines to the curve (y = 1 + x^{2/3}) at the point (0,1) are (x = 0) and (y = 1), respectively.

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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.

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