Question

(b) the equation of the line tangent to ( y = 3x^2 + 3 ) at ( p(4,51) ) is ( y = square ) (simplify your answer. do not factor.)

Explanation:

Answer:

To find the equation of the tangent line to \( y = 3x^2 + 3 \) at \( P(4, 51) \):

1. First, find the derivative of \( y \) with respect to \( x \) to get the slope of the tangent line. The derivative of \( y = 3x^2 + 3 \) is \( y' = 6x \).
2. Evaluate the derivative at \( x = 4 \) to find the slope \( m \) of the tangent line at that point: \( m = 6(4) = 24 \).
3. Now use the point - slope form of a line, which is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(4,51) \) and \( m = 24 \).

• Substitute the values into the point - slope form: \( y - 51=24(x - 4) \).
• Expand the right - hand side: \( y-51 = 24x-96 \).
• Add 51 to both sides to solve for \( y \): \( y=24x-96 + 51 \).
• Simplify the right - hand side: \( y = 24x-45 \).

So the equation of the tangent line is \( y = 24x-45 \).