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Increasing and decreasing functions
for interval (a,b) |
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Definition
F'(x) = + if f(x) increases/ ries
f'(x) = - if f(x) decreases / falls |
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Definition
critical values are the values of x in the domain of f where f'(x) = 0 or where f'(x) does not exist
- always in the domain of f
- if f is a polynomial then the critical values of f are solutions of f'(x) = 0 |
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Definition
| local extrema are the Local Max and Local Min |
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| Existence of local extrema |
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Definition
| if f is countinous on the interval (a,b) c is a number in (a,b) then either f'(c) = 0 or f'(c) does not exist |
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| First derviative test for local extrema |
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Definition
let c be a critical value of f[f(c) is defined and either f'(c) = 0 or f'(c) is not defined]
- construct a sign chart for f'(x) close to and on either side of c. |
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Definition
| - f(c) is a local minimum if f'(x) changes from - to + at c, then f(c) is a local minimum. |
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Definition
| if f'(x) changes from positive to negative at c, then f(c) is a local maximum |
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| when is f(c) is not a local extremum |
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Definition
| if f'(x) does not change sign at c, then f(c) is neither a local maximum nor a local minimum |
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| Intercepts and local extrema for polynomial functions |
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Definition
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Definition
| the graph of a function f is concave upward on the interval (a,b) if f'(x) is increasing on (a,b) and is concave downward on the interval (a,b) if f'(x) is decreasing on (a,b). |
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Definition
derivative of the first derivative.
- d^2y/dx^2 or y'' |
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Definition
| a point on the graph of a function where the concavvity changes. if f is continous on (a,b) and has an inflection point at x = c, then either f''(c) = 0 or f"(c) does not exist. |
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Definition
step 1 find the domain and intercepts. - the x intercept are the solutions to f(x) = 0 and y intercept is f(0)
Step 2 analyze f'(x)
- find the critical values of f(x). construct a sign chart for f'(x) determine the intervals where f is increasing and decreasing, and find the local maxima and local minima
step 3 analyze f"(x) construct a sign chart for f"(x), deterine the intervals where the graph of f is concave upward and concave downward, and find the inflection points.
step 4 sketch the graph of f
- locate the intercepts, local maxima and minima, and inflection points. sketch in what you know from steps 1 - 3. plot additional point a needed and complete sketch. |
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| summary for interval (a,b) |
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Definition
- f"(x) is + when f'(x) is increasing and graph of y = f(x) concave upwards
- f"(x) is - when f'(x) is decreasing and graph of y = f(x) concave downward |
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| L'Hopital's rule for 0/0 indeterminate forms: Version 1 |
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Definition
for c a real number, if lim, x --> c f(x) = 0 and lim x --> c g(X) = 0, then
lim x --> c f(x) / g(x) = lim x --> c f'(x) / g'(x)
- provided taht the second limit exists or is + infiniti or - infiniti |
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| L'Hopital's rule for 0/0 indeterminate forms: version 2 |
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Definition
(for one sided limits and limits at infinity)
the first version of the rule remains valid if the symbol x --> c is replaced everywhere it occurs with one of the following symbols.
- x --> c+,
- x --> c-,
- x --> infiniti
- x --> - infiniti |
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| L'Hopital's rule for infiniti / infiniti indeterminate forms: version 3 |
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Definition
| versions 1 and 2 of L'Hopital's rule for the indeterminate form 0/0 are also valid if the limit of f and the limit of g are both infinitie; that is both + infiniti and - infiniti are permissible for either limit |
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Definition
Step 1 find domain of f, find the intercepts, find asymptotes
Step 2 analyze f'(x)
- find critical values of f(x) and construct a sign chart for f(x). determine the intervals where f is increasing and decreasing, and find local mixima and minima
Step 2 analyze f"(x)
- contruct a sign chart for f'(x), determine the intervals where the graph of f is concave upward and concave downward, and find inflection points.
Step 3 sketch graph of f.
- draw asymptotes and locate intercepts, local maxima and minima, and inflection points. |
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Term
| Absolute maxima and minima |
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Definition
if f(c) >= f(x) for all x in the domain of f, then f(c) is called the absolute maximum value of f.
if f(c)<= f(x) for all x in the domain of f, then f(c) is called the absolute minimum value of f. |
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| existence of absolute maxima and minima |
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Definition
| a function f countinous on a closed interval [a,b] has both an absolute maximum and an absolute minimum on that interval. absolute extrema (if they exist) must always occur at critical values or at endpoints |
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| procedure for finding absolute extrema on a closed interval |
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Definition
1. check to make certain that f is countinous over [a,b]
2. find the critical values in the interval (a,b)
3. evaluate f at the endpoints a and b and at the critical values found in step 2.
4. the absolute maximum f(x) on [a,b] is the largest values found in step 3.
5. the absolute minimum f(x) on [a,b] is the smallest of the values found in step 3. |
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| Second derivative test : let c be a critical value for f(x). |
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Definition
| page 12 - 86 in solutions manual |
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| second derivative test for absolute extremum |
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Definition
let f be continuous on an interval i with only one critical value c on i
- if f'(c) = 0 and f"(c) > 0, then f(c) is the absolute minimum of f on i.
- if f'(c) = 0 and f"(c) < 0, then f(c) is the absolute maximum of f on i. |
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