Question

The equation of the straight line representing the tangent to the curve \(y = x^2\) at the point (1, 1) is

• A. \(y = 2x - 2\)
• B. \(x = 2y - 1\)
• C. \(y - 1 = 2(x - 1)\)
• D. \(x - 1 = 2(y - 1)\)

Hint:

For the equation of a tangent, use the point-slope form, which requires the slope at the given point and the coordinates of that point.

Answer 1

The Correct Option is C

Step 1: Find the derivative of the curve.
The equation of the curve is \(y = x^2\). To find the slope of the tangent at the point (1, 1), we differentiate the equation with respect to \(x\): \[ \frac{dy}{dx} = 2x. \] Step 2: Find the slope at the point (1, 1).
Substitute \(x = 1\) into the derivative: \[ \frac{dy}{dx} = 2(1) = 2. \] Thus, the slope of the tangent at the point (1, 1) is 2. Step 3: Use the point-slope form to find the equation of the tangent.
The point-slope form of the equation of a straight line is given by: \[ y - y_1 = m(x - x_1), \] where \((x_1, y_1)\) is the point on the line, and \(m\) is the slope. Substituting \(m = 2\), \(x_1 = 1\), and \(y_1 = 1\), we get: \[ y - 1 = 2(x - 1). \] Thus, the correct equation is \(y - 1 = 2(x - 1)\), which corresponds to option (C).