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# MATA30

### Final Review

Question | Answer |
---|---|

lim [f(x) + g(x)] x->a | lim f(x) + lim g(x) x->a x->a |

lim [f(x) - g(x)] x->a | lim f(x) - lim g(x) x->a x->a |

lim [cf(x)] x->a | c lim f(x) x->a |

lim [f(x)*g(x)] x->a | lim f(x) * lim g(x) x->a x->a |

lim f(x) x->a ---- g(x) | lim x->a f(x) ---- lim g(x) x->a |

lim n x->a [f(x)] | _ _ n | lim f(x)| where n is +integer | x->a | |

lim c x->a | c |

lim x x->a | a |

lim n x->a x | n a where n is a positive integer |

lim n-rt(x) x->a | n-rt(a) where n is + integer if n is even, we assume that a>0 |

lim n-rt[(f(x)] x->a | n-rt(lim f(x)) x->a if n is even, we assume that lim x->a f(x) > 0 |

lim f(x) = L x->a | if and only if lim f(x) = lim f(x) x->a- x->a+ |

L'HOSPITAL's RULE | f and g are dif'ble on open intrvl I containing a(excpt maybe at a) AND g'(x)<>0. Suppose that lim x-a f(x)=0 & lim x-a g(x)=0 or lim x-a f(x)= +- inf & lim x-a g(x)= +- inf Then lim x->a f(x)/g(x) = lim x-a f'(x)/g'(x) if limit on the R exist(or = |

Even function (Cosine) | f(-x) = f(x) |

Odd function (Sine) | f(-x) = -f(x) |

Intermediate Value Theorem | 1. f(x) is continuous on [a,b] 2. N is any number between f(a) and f(b) , where f(a) <> f(b) Result: There exists a number c in (a,b) such that f(c) = N |

Removable discontinuity | a discontinuity that could be removed by redefining the function at a single value. |

Infinite discontinuity | a discontinuity a at which the right or left hand limit is infinity or negative infinity |

Continuous from the right | a function f(x) is continuous from the right at a number a if lim x->a+ f(x) = f(a) |

Continuous from the left | a function f(x) is continuous from the left at a number a if lim x->a- f(x) = f(a) |

jump discontinuity | a discontinuity at which the value of the function jumps. lim x->a+ f(x) = c + lim x->a- f(x) for some non-zero constant c |

continuous at an endpoint of an interval | a function is continuous at an endpoint of an interval if it is defined on only one side of an endpoint and the function is continuous from that side |

continuous on an interval | a function is continuous on an interval if it is continuous at every point in the interval |

Closed Interval Method | To find the abs max and min values of a continuous function on a closed interval [a,b] 1. Find the values of f at the critical numbers of f in (a,b) 2. Find the values of f at the interval endpoints 3. largest value is abs max, smallest value is abs mi |

optimization problem | a problem, often in differential calculus, in which a function is maximized or minimized |

local minimum | a function f has a local minimum at c if f(c) <= f(x) for all x in some open interval containing c |

local maximum value | a number f(c), where c is in the domain of a function f(x), such that f(c) >= f(x) for all x in some open interval containing c. |

critical number | a number c in the domain of f such that either f'(c) = 0 or f'(c) does not exist |

local maximum | a function f has a local maximum at c if f(c) >= f(x) for all x in some open interval containing c |

minimum value | a number f(c), where c is in the domain of a function f(x), such that f(c) <= f(x) for all x in the domain of f |

maximum value | a number f(c), where c is in the domain of a function f(x), such that f(c) >= f(x) for all x in the domain of f |

absolute / global maximum | a value c in the domain, D, of the function f such that f(c) >= f(x) for all x in D; the x-coordinate of the highest point on the graph of f(x), though there may be many points with this height. |

extreme values | the maximum and minimum values of a function f |

Extreme Value Theorem | 1. f is continuous on a closed interval [a,b] Result: f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b] |

Increasing / Decreasing test | a) if f'(x) > 0 on an interval, then f is increasing on that interval b) if f'(x) < 0 on an interval, then f is decreasing on that interval |

Concave downward | 1. If the graph of f lies below all of its tangents on an interval Then f is concave downward on I |

Second Derivative test | 1. f'' is continuous near c. a) if f'(c) = 0 and f''(c) > 0, then f has a local minimum at c. b) if f'(c) = 0 and f''(c) < 0 then f has a local minimum at c |

Inflection point | A point P on a curve y=f(x) at which f is continuous and at which f changes concavity |

Concave upward | 1. If the graph of f lies above all of its tangents on an interval Then f is concave upward on I |

First derivative test | 1. c is a critical number of a continuous function f a) if f' changes from + to - at c, then f has a local max at c b) if f' changes from - to - at c, then f has a local min at c c) if f' does not change sign at c, then f has no max or min at c |

Concavity test | a) if f''(x) > 0 for all x in an interval I, then the graph of f is concave upward on I. b) if f''(x) < 0 for all x in an interval I, then the graph of f is concave downward on I |

Rolle's Theorem | Conditions: 1. f is a continuous function on the closed interval [a,b] 2. f is differentiable on (a,b) 3. f(a) = f(b) Result: There is a number c in (a,b) such that f'(c) = 0 |

Mean Value Theorem | Conditions: 1. f is a continuous function on the closed interval [a,b] 2. f is differentiable on (a,b) Result: There is a number c in (a,b) such that f'(c) = [f(b) - f(a)] / b-a |

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xjoaniex