The concept of a tangent to a curve is one of the foundations of differential calculus.

Intuitively, a tangent line should only touch a curve at one point, locally. How can we ensure this, for an arbitrary curve?

By considering secant lines through a fixed point on a curve

m_[a,b] = slope = [ (f(a)-f(b))/(a-b) ] the slope of the tangent to the curve y=f(x), at the point x=a, is given as m_a = lim _b->a [ (f(a)-f(b))/(a-b) ]

In order for this to make sense, such a limit must be same as b approaches a from both sides!

In terms of motion, the secant is analogous to average velocity and the tangent is analagous to the instantaneous velocity.

For motion along a parameterized curve,

r (t) = < x(t), y(t) > the average velocity (which is a vector!) is given by r (t+dt) - r (t) / dt

Certain questions arise:

• When can a curve fail to have a tangent?
• How can we ensure that the limit is the same from both sides?
• How can we compute the tangent exactly?

The answers to these, and other questions, will be found in later lectures.

Last modified Wed Sep 11 23:42:58 CDT 1996