Question

Find the equations of the tangent lines at the points (1, 7) and (4, 7²).

Answer 1

To find the equations of the tangent lines at given points on a curve, take the derivative of the function to find the slope at those points, then use the point-slope form to write the tangent line equations. However, without the specific function, the actual equations cannot be determined with the information provided.

Step-by-step explanation:

To find the equations of the tangent lines at the points (1, 7) and (4, 7²), we need to know the function they are tangent to, which seems to be a power function based on the points given. However, this information is not provided, so we'll use the general process, assuming we have the function and its derivative.

Here's the step-by-step process:

1. Calculate the derivative of the function, which gives us the slope of the tangent line at any point.
2. Evaluate the derivative at the given x-coordinates (1 and 4 in this case) to find the slopes of the tangents at the points (1, 7) and (4, 7²).
3. Use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the given point, to write the equation for each tangent line.

Without the function, we can't find the actual equations, but this is the method you would use.