How do you find the equation of the tangent and normal line to the curve #y=x^2+2x+3# at x=1?
a. Equation of tangent,
b. Equation of normal,
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To find the equation of the tangent and normal lines to the curve (y = x^2 + 2x + 3) at (x = 1), follow these steps:
- Find the slope of the tangent line by taking the derivative of the curve with respect to (x), and then evaluate it at (x = 1).
- Use the slope obtained in step 1 and the point (P(1, f(1))) (where (f(x) = x^2 + 2x + 3)) to write the equation of the tangent line in point-slope form.
- Find the slope of the normal line by taking the negative reciprocal of the slope of the tangent line.
- Use the slope obtained in step 3 and the point (P(1, f(1))) to write the equation of the normal line in point-slope form.
Let's solve it step by step:
- Find the derivative of (y = x^2 + 2x + 3): [y' = 2x + 2]
Evaluate (y') at (x = 1): [y'(1) = 2(1) + 2 = 4]
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Equation of the tangent line: [y - y_1 = m(x - x_1)] [y - (1^2 + 2(1) + 3) = 4(x - 1)] [y - 6 = 4(x - 1)] [y - 6 = 4x - 4] [y = 4x + 2]
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Slope of the normal line: [m_{\text{normal}} = -\frac{1}{4}]
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Equation of the normal line: [y - y_1 = m_{\text{normal}}(x - x_1)] [y - (1^2 + 2(1) + 3) = -\frac{1}{4}(x - 1)] [y - 6 = -\frac{1}{4}x + \frac{1}{4}] [y = -\frac{1}{4}x + \frac{25}{4}]
So, the equation of the tangent line is (y = 4x + 2) and the equation of the normal line is (y = -\frac{1}{4}x + \frac{25}{4}) at (x = 1).
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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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