Slope of a Curve

1
Slope of a Curve

• The slope of a curve at a point is defined as the
  instantaneous rate of change of the curve at that
  point.
• Slope lim f(a h) f(a)
• h?0 h

2
Find the slope of

• F(x) x2 at the point where x 1
• F(x) x2 at the point where x -2

3
Slope of a curve, redefined

• The slope of a curve at any point is given by lim
  f(x h) f(x)
• h?0 h

4
Find the slope of the curve at any point

• F(x) x2
• F(x) 3x 5
• F(x) x3
• F(x) 1/x

5
VERY IMPORTANT DEFINITION

• The derivative of f(x) is the slope of the
  tangent line to f(x) at any point.
• The derivative tells us the instantaneous rate of
  change of the function and its slope.

6
Symbols for derivatives

• The derivative of a function f(x) can be
  symbolized as
• F ? (x) y ? dy Dx
• dx

7
Find f ? (x) for

• F(x) 7 5x
• F(x) x2 3x
• F(x) ?x