1. Throwing a glass ball up into the air
h Height
Velocity = 0, Slope =0
m=0
Moving downward, slope <0
m>0
Moving upward, Slope > 0
m<0
Touch down
t Time
1 sec
2 sec
3 sec
4 sec
5 sec
6 sec

2. Use the light beam as the tangent line
A Night Ride in a Roller Coaster
m=0
Change in Concavity
Inflection Point
Use the light beam as the tangent line
m=0
Tangent line Goes Below
Tangent line Goes Above
Steepest Slope
Steepest Slope
m=0
m=0

3. Critical Points & Signs of f ’(x)
f “ (x) = 0 Inflection Point + + -
f ‘(x) < 0 Falling
f ‘(x) > 0 Rising + + + - + + - + + + - + +
f’(x)=0 + -
f “ (x) = 0 Inflection Point + + -
f’(x)=0

4. Visualizing the Derivative f(x)
m=+1
m=0
m=+2
Locate the critical points:
m = 0 ;
Inflection point
m=-1
m=+4
m=+3
m=-1.5
m=+1.5
m=-1
m=0
f’(x),m 4 3 2
Positive Slope 1 -1
m = 0
m = 0
Negative Slope -2 -3

5. Visualizing the Derivative (another method)
f(x)
Visualizing the Derivative (another method) +
Locate the critical points: - +
m = 0 ;
Inflection point + - + - +
f’(x),m
Positive Slope
m = 0
m = 0
Negative Slope

6. + - + - - + - + Inflection points Inflection points f(x) f(x) f ’(x)
f(x)
f ’(x)
f “(x)
f(x)
f “(x)
f ’(x) + - - + + -
Inflection points
Inflection points