Question

The equation of the tangent to the curve \[ y = x^3 - 2x + 7 \] at the point \( (1,6) \) is:

• A. \( y = x + 5 \)
• B. \( x + y = 7 \)
• C. \( 2x + y = 8 \)
• D. \( x + 2y = 13 \)

Hint:

To find the equation of a tangent, differentiate the function and use point-slope form.

Answer 1

The Correct Option is A

Step 1: Finding the derivative \[ \frac{dy}{dx} = 3x^2 - 2. \] Step 2: Evaluating at \( x = 1 \) \[ m = 3(1)^2 - 2 = 3 - 2 = 1. \] Step 3: Using point-slope form Equation of the tangent: \[ y - 6 = 1(x - 1). \] \[ y = x + 5. \]

Answer 2

Given the curve:
\[ y = x^3 - 2x + 7 \] Find the equation of the tangent at the point \( (1, 6) \).

Step 1: Find the derivative \( \frac{dy}{dx} \) which gives the slope of the tangent:
\[ \frac{dy}{dx} = 3x^2 - 2 \]

Step 2: Calculate the slope at \( x = 1 \):
\[ m = 3(1)^2 - 2 = 3 - 2 = 1 \]

Step 3: Use point-slope form of the line:
\[ y - y_1 = m (x - x_1) \] where \( (x_1, y_1) = (1, 6) \) and \( m = 1 \):
\[ y - 6 = 1 \times (x - 1) \] \[ y - 6 = x - 1 \] \[ y = x + 5 \]

Therefore, the equation of the tangent is:
\[ \boxed{y = x + 5} \]