Exam Questions - Tangents

Key concepts in Tangents:

A tangent to a curve at a point is the straight line that just touches the curve at that point.
It has the same gradient as the curve at the point of contact.
The equation of a tangent to a curve can be found using the point-gradient form of a straight line: y - y1 = m(x - x1), where m is the gradient and (x1, y1) is the point.
The gradient of a curve at a given point can be found by differentiating the equation of the curve and substituting the x-coordinate of the given point.

A typical question may ask for the equation of a tangent at a particular point on a curve.
Alternatively, you may need to use the concept of tangents to solve a problem, such as finding the x-coordinate where a tangent intersects a curve.

To find the equation of a tangent at a point, first find the derivative of the curve’s equation to work out the gradient at the given point.
Plug the point into the point-gradient formula alongside the gradient to find the equation of the tangent.
Consider using implicit differentiation if the equation of the curve isn’t easily differentiable.
In problems where you need to find where a tangent intersects a curve, you may need to solve simultaneous equations.
Always remember to double-check your work for errors as simple sign or arithmetic mistakes can easily occur.

A tangent at a point has the same gradient as the curve at that point.
You need to take the derivative of the curve’s equation to find the gradient at a specific point.
The point-gradient form of a straight-line equation is essential for finding the equation of a tangent.
Many problems involving tangents require the use of algebraic techniques such as solving simultaneous equations.