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Calculus Rules
Rules Learned in Calc I
| Fill in blank | |
|---|---|
| If f is ____ on the closed interval [a,b], then f attains ___ and ___ in [a,b] | continuous, absolute max, absolute min |
| f(x) has an inflection point at x=c if ___. | f changes concavity at c. |
| f(x) has a critical point at x=c, if ___ or ___. | f'(c)=0 or DNE |
| If f'(x) is greater than 0 on Interval I, then f is ___. | increasing on I |
| If f'(x) is less than 0 on interval I, then f is ___. | decreasing on I |
| If f''(x) is less than 0 on Interval I, then f'(x) is ___ and f is ___. | decreasing, concave down |
| If f''(x) is greater than 0 on Interval I, then f'(x) is ___ and f is ___. | increasing, concave up |
| x=c is an inflection point if ___. | the concavity changes |
| f(c) is an absolute max on I if ___. | f(c) is greater than f(x) for all values x in I. |
| f(c) is an absolute min on I if ___. | f(c) is less then f(x) for all values x in I. |
| f(c) is a local max if ___. | f(c) is greater than f(x) for all x near c. |
| f(c) is a local min if ___. | f(c) is less than f(x) for all x near c. |
| Assume that c is a critical point of f and that f is continuous at x=c. 1. If f'(x) changes from positive to negative, then c is ___. 2. If f'(x) changes from negative to positive, then c is ___. 3. If f'(x) does not change sign, then c is ___. | 1. Local max 2. Local min 3. Neither local max nor local min |
| Assume f'' is continuous near c. 1. If f''(c) is greater than 0, then f(c) is ___. 2. If f''(c) is less than 0, then f(c) is ___. | 1. Local max 2. Local min |
| If f is ___ on a closed interval (a,b), then there exists a c in (a,b), such that ___. | continuous, f'(c)=f(b)-f(a)/b-a |
| If f and g are ___ and ___ near a, and the limit of f(x)/g(x) as x approaches a gives an indeterminate form of ___ or ___, then ___. | continuous, g does not equal 0, infinity/infinity. 0/0, limit as x approaches a is f'(x)/g'(x) |
| A function f is continuous at a number a if ___. | The limit of f(x) as x approaches a = f(a) |
| Suppose that f is ___ on the closed interval [a,b] and let N be any number between f(a) and f(b) where ___. Then there exists a number c in (a,b), such that ___. | continuous, f(a) does not equal f(b), f(c)=N |
| The line x=a is a vertical asymptote when ___. | The limit of f(x) as x approaches a is infinity or negative infinity. |
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