2. • Concave up: Second derivative is a positive number.
• Concave down: Second derivative is a negative number.
• To find the concavity find the second derivative and set it
equal to zero to find critical numbers. Place critical
numbers on a number line and test points in between each
region (test in second derivative equation). When the
values are positive then that interval is concave upward.
When the values are negative then that interval is concaved
downward.
4. Definition – The point on a graph where the concavity
changes
*Cannot be asymptote or undefined point
How to Find:
1. Find the second derivative
2. Set the second derivative equal to zero, find the critical
values (f’(x) =0 or where f(x) or f’(x) is undefined)
3. Place on number line and test regions in the second
derivative
4. Locate critical numbers where sign changes POI
5. Plug P.O.I. into original equation to find y-value
6. • Lowest point on the graph/equation
• Check endpoints and local minimum heights in table
of values to compare/determine
• Careful with endpoints that are not included, they
cannot be answers
8. • Highest point on the graph/equation
• Check endpoints and local maximum heights in table
of values to compare/determine
• Careful with endpoints that are not included, they
cannot be answers
10. What is increasing and decreasing?
-When the graphs slope is positive/negative
-Steps
1. Find the derivative of the function
2. Set the derivative equal to zero, find critical numbers
3. Test regions on the number line into the derivative (slope)
4. If regions are positive its increasing, if negative its decreasing
12. • The highest point in a neighborhood of points
• Most cases f’(x)=0, but f’(x) can be undefined
there like at a cusp or corner
• Slope goes from positive to negative
14. • The lowest point in a neighborhood of points
• Most cases f’(x)=0, but f’(x) can be undefined
there like at a cusp or corner
• Slope goes from negative to positive
16. 1. Find the derivative
2. Set derivative = 0 and find critical numbers
3. Set up a # line with critical #s on it
4. Test each section by plugging the #s surrounding
critical #s into the derivative
5. When there is a change in signs you have relative
extrema (negative – positive is minimum) ;
(positive – negative is maximum)
6. Plug those critical values into the original to find
y value for the relative extrema
20. • Special case of the Mean Value Theorem
• Case in which, with respect to points where x=a
and x=b, f(A) = f(B)
– When the above is true, a point c in between a and b
exists so that f’(C)= 0
• Rolle’s Theorem only exists under the same
conditions as the MVT
22. • Skips doing # line for first derivative to determine
if local min/max, can determine using second
derivative as well:
– Find critical values from the first derivative
– Plug those values into the second derivative
– If the second derivative is +, then concave up, so that
value is a local min
– If the second derivative is -, then concave down, so
that value is a local max
– If second derivative is 0, then inconclusive
26. • IVT states that if you have a continuous
function/equation [a,b] then there must exist
some c value where f(a)≤f(c)≤f(b).
• Basically states if continuous, all values in
between f(a) and f(b) must be reached
• Can help prove you have a root if f(a) is + and
f(b) is – or vice versa.
