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| f^-1(f(x))=x f(f^-1(x))=x |
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| lim f(x)/lim g(x) if g(x)!=0 |
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| The formal definition of a limit |
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| Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write *lim(x->a) f(x) = L* if for every number µ>0 there is a number ´>0 such that if 0<|x-a|<´ then |f(x)-L|<µ |
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| If f(x)d"g(x)d"h(x) when x is near a (except possibly at a) and lim(x->a) f(x)= lim(x->a) h(x)= L then lim(x->) g(x)= L |
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| lim(x->a) f(x)= f(a) (therefore f(a) exists and is defined) |
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| If f(x) and g(x) are continuous, then what can be said of the result of them when different opperators are applied(+-*%) |
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| lim(x->a) f(g(x))=(not chain rule- continuity rule) |
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| Intermediate Value theorem |
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| Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a)!=f(b). Then there exists a number c in (a,b) such that f(c)=N. |
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| The first formula for a derivative(if you have to find a value) |
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| f'(a)=lim(x->a)[(f(x)-f(a))/(x-a)] |
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| The second formula for a derivative(if you have the formula and no points) |
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| f'(a)=lim(h->0)[(f(x+h)-f(x))/h] |
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| The formal definition of e |
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| lim(h->0) ((e^h)-1)/h = 1 (Means that the slope at x=0 is 1) |
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| Def of absolute maximum and minimum values |
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| f has a maximum at c if f(c)e"f(x) for all x in the interval, f has a minimum at c if f(c)d"f(X) for all x in the interval |
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| The extreme value theorem |
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| If f is continuous on a closed intrval[a,b], then f atains an absolute maximum value f(c) and an absolute minimum alue f(d) at some numbers c and d in [a,b]. |
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| If f has a local max or min at c, and if f'(c) exists, then f'(c)=0 |
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| A number c in the domain of f where f'(c)=0 or DNE |
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| the closed interval method |
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| 1)Find the values of f at the critical numbers, 2)Find the values of f at the endpoints, 3)The largest of the values from steps 1 and 2 is the max, the smallest is the min |
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| 1)F is continuous on [a,b], 2)F is differentiable on (a,b), 3)f(a)=f(b)- Then there is a number c in (a,b) where f(c)=0 |
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| 1)F is continuous on [a,b], 2)F is differentiable on (a,b)- then there is a numberc in (a,b) such that f'(c)=(f(b)-f(a))/b-a (or in other words, the slope of f at c has to equal the slope between (a,f(a)) and (b,f(b))) |
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| If f'(x)=0 for all x in the interval then ____ |
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| the function is constant on (a,b) |
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| if f'(x)>0 then f is increasing, if f'(x)<0 then f is decreasing |
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| The first derivitave test |
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| if f' changes from pos to neg, then there is a local max, if f' changes from neg to pos, then there is a local min, if there is no sign change for f', then there is no max or min |
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| up-tangents of f lie below the graph of f, down-tangents of f lie above graph |
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| when the second derivative of f equals 0 (f"=0) |
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| The second derivative test |
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| a)If f'(c)=0 and f"(c)>0 then there is a local min at c, b)If f'(c)=o and f"(c)<0 then there is a local max at c |
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| lim(x->a) f(x)/g(x)= lim(x->a) f'(x)/g'(x) |
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| 0/0, "/ ", 0* ", "- ", 0^0, "^0, 1^ " |
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| x_n+1=(x_n)-(f(x_n)/f'(x_n)) |
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