Unit 4 - Contextual Application of Differentiation Flashcards
If f’(x) > 0, then the function is
increasing
If f’(x) < 0, then the function is
decreasing
Position function
represents position of an object
often notated as s(t) where t is time
Velocity function
represents velocity of an object
v(t) = s’(t)
Acceleration function
represents acceleration of an object
a(t) = v’(t)
Velocity
Rate of change of position
If v(t) < 0, then the particle is
Moving left (x-axis) Moving down (y-axis)
If v(t) > 0, then the particle is
Moving right (x-axis) Moving up (y-axis)
If v(t) = 0, then the particle is
at rest/ not moving
Avg velocity
s(b) - s(a) / b-a on interval [a, b]
Speed
|velocity|
If velocity and acceleration has the same sign, then the particle is
Speeding up
If velocity and acceleration has different sign, then the particle is
Slowing down
Displacement
Net change in position
Initial position - final position
To know if something is increasing or decreasing,
Check its derivative
If the sign of derivative is greater than zero, then
the function is increasing
If the sign of derivative is less than zero, then
the function is decreasing
Related rates
Changing dimensions relate to each other
Differentiate relationship to one variable
To solve a related rates problem
1- Draw a picture
2 - Make list of all known and unknown rates and quantities
3- relate variables in an equation
4 - Differentiate with respect to time
5- Substitute know quantities & rates and solve
Substituting non-constant quantity before differentiating is
NOT ALLOWED
If a function is concave up, the points on a tangent line would be
Underestimate
If a function is concave down, the points on a tangent line would be
Overestimate
L’Hopital Rule
Suppose f(a)=0 and g(a)=0 and lim f(x) / g(x) = 0/0 or infinity/ infinity x->a
Then, lim f(x) / g(x) = f’(a) / g’(a)
x->a
The derivative of a function can be interpreted
as the
instantaneous rate of change with respect to its independent variable.
The unit for f ‘(x) is the
unit for f divided by the unit for x.
