Tangent Lines and Instantaneous Velocity (Transcript Notes)

Tangent line to y = x^2 at P = (3,9)

• Question: what is the tangent line to y = x^2 at P = (3,9)?
• Conundrum: how to find a line through one point? (Through P with slope equal to the tangent slope.)
• Key idea: the tangent slope is the limit of secant slopes from P to nearby points Q on the curve y = x^2 as Q approaches P.

Secant slopes through P and a nearby point Q

• Let Q be a point on the parabola with coordinates Q = (x, x^2). The line through P = (3,9) and Q = (x, x^2) has slope
  $$m_{PQ} = rac{x^2 - 9}{x - 3} \