CALCULUS I TOOLBOX |
1) INEQUALITIES:
basic rule: reverse the direction of the inequality when multiplying or dividing by a negative.
a) absolute value: | f(x) | < b becomes -b < f(x) < b (sandwich)
b) quadratics: relate to zero, locate zeroes, then solve with a number line.
c) fractions: denominator a constant, multiply through by the lowest common multiple of the denominators and solve as a linear inequality.
denominator some f(x) , combine the fractions and relate to zero. Then solve as a quadratic inequality.
2) LIMITS:
a) general limits: lim _{x}_{d }_{a} f(x): factor, cancel and substitute x = a.
b) trig limits: lim_{h}_{d}_{0} = 1 | lim_{h}_{d}_{0} 0 |
c) limit theorem: lim _{x}_{d }_{a}_{ }
d) limits involving infinity:
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3) DERIVATIVES:
a) power rule: if f(x) = ax^{n} then f '(x) = nax^{n-1}
b) product rule: (uv)' = u'v + v'u, where u and v are fuctions of x.
c) quotient rule: (u/v)' = (u'v - v'u)/ v^{2}
d) chain rule: (f[g(x)])' = f ' [g(x)]g'(x)
e) implicit: assume that y = f(x) exists and use the chain rule on that function to find y'.
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f) trig functions:
(sin u)' = du cos u | (cos u)' = -du sin u |
(tan u)' = du sec^{2} u | (cot u)' = -du csc^{2} u |
(sec u)' = du sec u tan u | (csc u)' = -du csc u cot u |
Note_{1}: If the function's name includes the "co" syllable (ex: cotan), the derivative is negative!
Note_{2}: Notice the symmetry of the pairs.
For instance, (tan u)' = du sec^{2} u, so (cot u)' = -du csc^{2} u
It's the same for the other 2 pairs. Just add the minus sign and the "co" syllable.
g) inverse trig functions:
(arcsin u)' = | (arccos u)' = |
(arctan u)' = | (arcsec u)' = |
h) exponential and log functions:
(e^{u})' = du e^{u} | (a^{u})' = du a^{u} ln a |
(ln u)' = | (log_{a}u)' = |
i) increments and differentials:
y = f(x) u dy = f '(x) dx u y_{2} = f(x) + f '(x) dx
4) GRAPHING TECHNIQUES:
a) first derivative test:
b) second derivative test:
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c) M_{2}I_{2}ACIDS
max, min, intercepts, inflection points, asymptotes, concave up, concave down, increasing, decreasing, and symmetry.
* see hand-out titled "NOTES ON GRAPHS".
5) RELATED RATES PROBLEMS:
6) MAX-MIN PROBLEMS:
7) MEAN VALUE THEOREM:
If f(x) is continuous on [a,b] and differentiable on ]a,b[,
then there exists a "c" on ]a,b[ such that f'(c) =
which means that the slope of the tangent at (c, f(c)) is equal to
the slope of the chord joining (a, f(a)) to (b, f(b)).
8) ANTIDIFFERENTIATION:
a) power rule: if f '(x) = ax^{n} then f(x) = + C
b) differential equations: Antidifferentiate until you reach f(x), plugging in boundary values.
f ' ' '(x) d f '' (x) d f '(x) d f(x)
Use boundary values to establish the values of C at each level of antidifferentiation.
Use different letters to indicate the constant in each level of derivative.
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