2.1 the derivative and the tangent line problem the definition of a derivative
TRANSCRIPT
2.1 The Derivative and The Tangent Line Problem
The Definition of a Derivative
WARM UP: Find the slope of the tangent line to the curve at the given point
)12,3(;3)( 2 xxf
The DERIVATIVE finds the slope of the tangentline to a given function at a given point.
x
xfxxfxf
x
)()(lim)(
0
'
Know the different notations:
][)],([,,),( '' yDxfdx
dy
dx
dyxf x
Comparison of SLOPE and DERIVATIVE:
SLOPE:Slope between 2 points: Average “Rate of Change”
DERIVATIVE: Slope at 1 point Instantaneous Rate of Change
Places where there is NO derivative: • Discontinuity• Vertical Tangent• Cusp
Theorem: If f(x) is differentiable, it IS continuous.
***BUT, all continuous functions are NOT differentiable.
Alternate form of derivative to the graphat x=c:As long as the one-sided limits from the right and from the lift exist and are equal,
cx
cfxfxf
cx
)()(lim)('
Ex) Find the derivative by the limit process.
2
1)(x
xf
Plan: 1) Find the slope of the tangent line to the
graph at any point (x,f(x))
Ex) Describe the x-values at which f is differentiable.
Plan: 1) Omit parts of graph where there are
discontinuities, vertical asymptotes and cusps.
Ex) Describe the x-values at which f is differentiable.
Plan: 1) Omit parts of graph where there are
discontinuities, vertical asymptotes and cusps.
Ex) Describe the x-values at which f is differentiable.
Plan: 1) Omit parts of graph where there are
discontinuities, vertical asymptotes and cusps.
Ex) Use the alternate form of derivative to find the derivative at x=c, if it exists.
1);1()( cxxxf
Plan: 1) Use the alternate form of derivative…
check that limit exists.