Equations of Tangents and Normals to the Curve y=f(x)

Equations of Tangents to the Curve y=f(x)

Tangent to a curve at a point is a straight line that just touches the curve at that point.
The slope of the tangent to the curve y=f(x) at a point x=a is given by f’(a), the derivative of f(x) evaluated at x=a.
The equation of this tangent line can be found using the point-slope form of the equation of a line, which is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In the context of a curve y=f(x), x1=a, y1=f(a), and m=f’(a).

A normal to a curve at a point is a straight line perpendicular to the tangent at that point.
The slope of the normal to the curve y=f(x) at a point a is the negative reciprocal of the slope of the tangent at that point, i.e., -1/f’(a).
The equation of the normal can also be found using the point-slope form of the equation of a line. For a normal, x1=a, y1=f(a) and m=-1/f’(a).

It is crucial to find the correct derivative of the function, as it gives the slope of the tangent.
The negative reciprocal of the derivative at a given point will provide the slope of the normal.
Practice applying the techniques to various functions and values of x to become comfortable with deriving equations of tangents and normals.
When checking your work, ensure that the tangent touches the curve at the given point and the normal is perpendicular to the tangent.
Don’t forget to give your final answer in the form of an equation i.e., y=mx+c. Use the appropriate details from your workings to replace m, x and c.

By understanding and mastering these concepts, you’ll be able to confidently determine equations of tangents and normals for various functions.