Question

consider the parabola ( y = 5x - x^2 ). (a) find the slope of the tangent line to the parabola at the point ( (1, 4) ). (b) find an equation of the tangent line in part (a). ( y = )

Explanation:

Part (a)

Step1: Recall the derivative formula for slope

The slope of the tangent line to a function \( y = f(x) \) at a point \( x = a \) is given by the derivative \( f'(a) \). First, find the derivative of \( y = 5x - x^2 \).
Using the power rule, if \( y = ax^n \), then \( y' = nax^{n - 1} \). For \( y = 5x - x^2 \), the derivative \( y' = 5 - 2x \).

Step2: Evaluate the derivative at \( x = 1 \)

Substitute \( x = 1 \) into the derivative \( y' = 5 - 2x \).
\( y'(1) = 5 - 2(1) = 5 - 2 = 3 \).

Step1: Use the point - slope form of a line

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(1,4) \) and \( m = 3 \) (the slope from part (a)).
Substitute these values into the point - slope formula: \( y - 4 = 3(x - 1) \).

Step2: Simplify the equation

Expand the right - hand side: \( y - 4 = 3x - 3 \).
Then, add 4 to both sides to solve for \( y \): \( y = 3x - 3 + 4 = 3x + 1 \).

Answer:

3

Part (b)