The tangent and velocity problems
MAT 135: Calculus
14 Sept 2009
How fast is the student going?
Always traveling at the same rate
= Slope
= 50 yards/min
Rate of change can be different 
at different points

Rate of change at a single point = 
INSTANTANEOUS RATE OF CHANGE at that point
Rate of change in a function
is measured by SLOPE (steepness)
Instantaneous velocity at exactly 1 minute? 
Measuring instantaneous ROC of 
a CURVE at a point = measuring the 
SLOPE of a curve at that point
HOW?
Zoom on x=1
Zoom on x=1
Zoom on x=1
Function almost = tangent line
when very near x = 1
*** INSTANTANEOUS RATE OF CHANGE AT X = 1
= SLOPE OF THE CURVE AT X = 1
= SLOPE OF THE TANGENT LINE TO THE CURVE AT X = 1

TO MEASURE RATE OF CHANGE AT A POINT: 
MEASURE SLOPE OF TANGENT LINE TO GRAPH AT THAT POINT