To find the equation of the tangent line to the curve given by the equation y = 5x2 + 2x + 1 at a specific point, we need to follow these steps:
- Determine the point: Identify the specific point on the curve at which you want to find the tangent line. For this example, let’s say we want to find the tangent line at the point (1, y). To get the y-coordinate, plug in x = 1 into the curve’s equation:
- Calculate y:
y = 5(1)2 + 2(1) + 1 = 5 + 2 + 1 = 8.
So, the point is (1, 8). - Find the derivative: The slope of the tangent line at any point on the curve is the value of the derivative of the function at that point. We first need to find the derivative of the function y = 5x2 + 2x + 1.
- Calculate the derivative:
The derivative is given by:
y' = d(y)/dx = 10x + 2. - Evaluate the derivative at x = 1: Substitute x = 1 into the derivative to find the slope of the tangent line:
y'(1) = 10(1) + 2 = 12.
So, the slope of the tangent line at the point (1, 8) is 12. - Write the equation of the tangent line: Using the point-slope form of a line, which is given by:
y - y1 = m(x - x1),
where m is the slope and (x1, y1) is the point of tangency: - Substituting the values of the slope and the point (1, 8) into the equation:
y - 8 = 12(x - 1). - Simplify the equation: Distributing and rearranging gives:
y - 8 = 12x - 12
y = 12x - 4.
Thus, the equation of the tangent line to the curve at the point (1, 8) is y = 12x – 4.