Overview
Chapter Four
Increasing and Decreasing
• This is slope so you use first derivative.
• If it asks for intervals of increasing and
decreasing f’(x)=0 to find stationary points and
also find critical #s.
• Then make an interval table and test in the first
derivative.
Concavity
• This is the second derivative.
• If it asks for intervals of concave up and concave
down, use f “(x)=0 to solve for possible inflection
points.
• Then make an interval table and test in the
second derivative.
Inflection Point
• To find inflection points, set f “(x)=0 to solve for
possible inflection points.
• Then make an interval table and test in the
second derivative.
• Where f”(x)=0 AND concavity changes, then
that x value is an inflection point.
Relative Extrema
• Relative extrema are relative maximums (top of any
mountain on a local level) and relative minimums
(bottom of any valley on a local level).
• These occur at critical numbers.
• Critical numbers include cusps, locations of vertical
tangent lines, stationary points, etc.
• Stationary points occur where the slope of the
tangent line is horizontal (f’(x)=0) and changes from
increasing to decreasing or vice versa.
• Critical numbers occur where f’(x) is undefined.
Are they Minimums or Maximums?
• First derivative test (test in first derivative using an interval
table)
▫ If f’(x) on your interval table goes from increasing to decreasing,
then you have a relative maximum.
▫ If f’(x) on your interval table goes from decreasing to increasing,
then you have a relative minimum.
• Second derivative test (take the critical numbers, including
stationary points) and substitute them into the second
derivative (NO INTERVAL TABLE)
▫ If the result of your substitution is a positive number, then f(x) is
concave up at that value of x which means you have a relative
minimum (local minimum).
▫ If the result of your substitution is a negative number, then f(x) is
concave down at that value of x which means you have a relative
maximum (local maximum).
Absolute Maximums and Minimums
• Absolute maximums and minimums are the overall
highest peak in the mountain range and the overall
lowest valley.
• Therefore, we need to know position. We find
position by substituting the following numbers into
f(x).
• Take the critical numbers, including stationary
points from the first derivative and substitute them
into the original function.
• You must also test endpoints (if there are any) and
end behavior if there is not a given interval [a,b].
Optimization
• Step 1: Draw and label
• Step 2: Formula to maximize or minimize
• Step 3: Write that formula as a function of one
variable
• Step 4: Interval of possible values
• Step 5: Find the relative extrema
• Take the derivative, set=0, solve for x, determine
whether the results are maximum(s) or
minimum(s).
Rectilinear Motion
• Velocity is the derivative of position
▫ v(t) = s’ (t) = ds/dt
• Acceleration is the derivative of velocity
▫ a(t) = v’ (t) = dv/dt
• Speed is the absolute value of velocity
▫ speed = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
Speeding Up vs. Slowing Down
• speeding up when its speed is increasing
• slowing down when its speed is decreasing
• SIGNS of velocity and acceleration:
▫ same sign
speeding up
▫ different sign
slowing down
• Make a timeline with velocity and
acceleration and compare their signs as we
did in the example on the next slide
Speeding Up/Slowing Down Analysis
Example
Mean Value Theorem
1. Plug end points into original function to find
the y values of your points
2. Plug two points into slope formula y2-y1/x2-x1
to get the slope of your secant line.
3. Take the derivative of f(x) to get the slope of
your tangent line.
4. Parallel lines have equal slopes, so set the slope
of your secant line=slope of your tangent line:
y2-y1/x2-x1 = f’(x)
and solve for x.
Rolle’s Theorem
• Rolle’s Theorem is a special case of the Mean
Value Theorem, so you can follow the same
steps.
1.
2.
3.
4.
Plug end points into original function to find the y values of your
points
Plug two points into slope formula y2-y1/x2-x1 to get the slope of
your secant line which for Rolle’s Theorem will always = 0.
Take the derivative of f(x) to get the slope of your tangent line.
Parallel lines have equal slopes, so set the slope of your secant line
which for Rolle’s Theorem will always = 0 =slope of your tangent
line:
y2-y1/x2-x1 = 0 = f’(x) and solve for x.
Good luck on test!!!