How do you find the slope of the tangent line to the parabola defined by the equation y = 4x – x² at the point (1, 3)?

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Finding the Slope of the Tangent Line

To find the slope of the tangent line to the parabola described by the equation y = 4x – x² at the point (1, 3), we’ll use calculus, specifically the concept of derivatives.

Step 1: Differentiate the function

First, we need to find the derivative of the function, which gives us the slope of the tangent line at any point x on the curve. The derivative of y = 4x – x² with respect to x is calculated as follows:

Using the power rule, we differentiate:

  • If y = 4x, the derivative is 4.
  • If y = -x², the derivative is -2x.

So, the derivative y’ becomes:

y' = 4 - 2x

Step 2: Evaluate the derivative at x = 1

Now that we have our derivative, we will evaluate it at the point where x = 1:

y'(1) = 4 - 2(1) = 4 - 2 = 2

Thus, the slope of the tangent line at the point (1, 3) is 2.

Step 3: Conclusion

In conclusion, the slope of the tangent line to the parabola y = 4x – x² at the point (1, 3) is 2.

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