Section 2 Flashcards
tangent line
slope of a line that best approximates a curve
how to approximate a curve?
- find slope of secant line
- bring those 2 points closer and closer until there is no distance between them. the result is the slope of the tangent line
vertical asymptote
where denominator = 0
hole
x-values that make the numerator and denominator 0
horizontal asymptote
look at leading coefficient of numerator and denominator
- if n + d have same degree, divide coefficients
- if n > d, no HA/oblique asymptote
- if n < d, HA is x-axis
definition of a limit
if f(x) approaches L as x approaches c, then the limit of f(x) as x approaches c is L
division by 0 =
undefined, limit DNE
define continuity
a function is continuous at a point c if:
1. f(c) is defined
2. lim as x->c of f(x) exists
3. lim as x->c of f(x) = f(c)
intermediate value theorem rules
function f(x) on interval [a,b] and f(c)=k. k is any number between a and b.
- f must be continuous on the closed interval [a,b]
- f(a) cannot = f(b)
knowing this, there is at least one number c in [a,b] such that f(c)=k.
extreme value theorem
if f is continuous on the closed interval from a to b, then f has both a max and min on [a,b]c
critical number meaning
when f’(x) = 0
*relative extrema only occur at critical numbers
how to find extrema on a closed interval
- find critical numbers of f
- evaluate f at each critical number and at each endpoint of [a,b]
- the least of these values is the minimum, the greatest is the maximum
rolle’s theorem
given f is continuous on [a,b] and differentiable on (a,b), f(a) = f(b)
conclusion: there is at least one c on (a,b) such that f’(c) = 0
mean value theorem
if f is continuous on [a,b] and differentiable on (a,b), there is a point c in the interval where f’(c) = the slope of the line containing the interval
*f’(c) = f(b) - f(a)/b-a
test for increasing/decreasing functions
- if f’(x) > 0, then f is increasing
- if f’(x) < 0, then f is decreasing
- if f’(x) = 0, then f is constant
finding INTERVALS on which f is increasing/decreasing
- find critical numbers
- use critical numbers to determine intervals by plugging in test values into f’(x) and determining the sign
test for concavity with the 2nd derivative
- if f’‘(x) > 0, concave up
- if f’‘(x) < 0, concave down
- if f’‘(x) = 0, f is linear or there is an inflection point
2nd derivative test for max/min
- find f’(x) and set to zero
- plug that value into f’‘(x)
- if f’‘(x) > 0, it is a min, if it’s < 0, it is a max
limits to infinity: bottom heavy functions
limit = 0
limits to infinity: degree of numerator = degree of denominator
divide coefficients
L’Hospital’s rule
If taking the limit of a function produces the indeterminate form (0/0), take the derivative of the function and try again
what is the process for optimization
- identify what needs to be maximized
- set up equations (primary and secondary)
- substitute so that there is only one variable in the equation
- differentiate and set equal to 0 to find x
