The equation of the tangent to the curve y = x^2 at the point (2, 4) is:

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Question: The equation of the tangent to the curve y = x^2 at the point (2, 4) is:

Options:

Correct Answer: y = 2x - 4

Solution:

The derivative f\'(x) = 2x. At x = 2, f\'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 2x - 4.

The equation of the tangent to the curve y = x^2 at the point (2, 4) is:

The equation of the tangent to the curve y = x^2 at the point (2, 4) is:

Step 1: Identify the function. The function is y = x^2.
Step 2: Find the derivative of the function. The derivative f'(x) = 2x.
Step 3: Calculate the slope of the tangent line at the point (2, 4). Substitute x = 2 into the derivative: f'(2) = 2 * 2 = 4.
Step 4: Use the point-slope form of the equation of a line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is the point (2, 4) and m is the slope (4).
Step 5: Substitute the values into the point-slope form: y - 4 = 4(x - 2).
Step 6: Simplify the equation. Distribute the 4: y - 4 = 4x - 8. Then add 4 to both sides: y = 4x - 4.
Step 7: The final equation of the tangent line is y = 4x - 4.

Derivative – Understanding how to find the slope of the tangent line to a curve at a given point using differentiation.
Point-Slope Form – Using the point-slope form of a linear equation to write the equation of the tangent line.
Quadratic Functions – Recognizing the properties of quadratic functions and their graphs.