Final answer:
To find the tangent line at a point on a curve, use differentiation to find the slope of the curve at that point, then write the equation of the tangent line using the point-slope form.
Step-by-step explanation:
To find the tangent line at a point, we need to determine the slope of the curve at that point. This can be done by finding the slope of a straight line that is tangent to the curve at the given point. To do this, we can use the concept of differentiation. Let's take an example to illustrate:
If we have a curve defined by the equation y = x^2, and we want to find the tangent line at the point (2, 4), we can find the slope of the curve at that point using differentiation:
- Differentiate the equation: y' = 2x
- Plug in the x-coordinate of the point to get the slope: y'(2) = 2(2) = 4
- Use the point-slope form of a line (y - y1 = m(x - x1)) to write the equation of the tangent line: y - 4 = 4(x - 2)
- Simplify the equation: y - 4 = 4x - 8
- Write the equation in slope-intercept form (y = mx + b): y = 4x - 4
So, the equation of the tangent line to the curve y = x^2 at the point (2, 4) is y = 4x - 4.