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Formulae for Calculus 151 p. 1
Differentiation
Definition:
f (a + h) − f (a)
f (a) = lim
h→0
h
General formulas:
0
Graphing
Symmetry:
Even: f (−x) = f (x) (symmetric)
Odd: f (−x) = −f (x) (skew symmetric)
Product: (uv)0 = u0 v + uv 0
Asymptotes:
Quotient: (u/v)0 = [u0 v − uv 0 ]/v2
Horizontal:
Chain rule: [f (u)]0 = f 0 (u) · u0
Vertical: lim± f (x) = ±∞ (four cases)
lim f (x) = a (two cases)
x→±∞
x→a
Constant multiple: (cu)0 = cu0
Increasing/Decreasing:
Inverse: dx/dy = 1/(dy/dx)
f 0 Positive on interval: Increasing
Special functions:
f 0 Negative on interval: Decreasing
Constants: c0 = 0
Local maxima/minima:
d
n
n−1
Powers:
x = nx
Critical numbers: f 0 (x) = 0, or undefined;
dx
Exponential, logarithmic:
or Endpoints
d ln(x) = 1/x
d ex = ex ;
Concavity, inflection:
dx
dx
d ax = ln a · ax
f 00 (x) > 0: upward; f 00 (x) < 0: downward
dx
Trigonometric:
f 00 (x) changes sign: Inflection (∼)
d sin x = cos x;
d cos x = − sin x
Differentials and Newton’s method
dx
dx
d sec x = sec x tan x; d csc x = − csc x cot x
dy = y 0 dx; y ≈ y0 + dy
dx
dx
d tan x = sec 2x; d cot x = − csc 2x
f (a + ∆x) ≈ f (a) + f 0 (a)∆x
dx
dx
−1
Newton’s method: xnew = x − f (x)/f 0 (x)
d sin−1 x = √ 1
; d cos−1 x = √
dx
dx
1 − x2
1 − x2 (iterate)
1
d tan−1 x =
d cot−1 x = −1
;
Intermediate Value Theorem:
dx
dx
1 + x2
1 + x2
If f (x) is continuous on [a, b], f (a) < N < f (b),
d sec−1 x = √ 1
then the equation f (x) = N is solvable, with
dx
x x2 − 1
a < x < b.
d csc−1 x = √ −1
dx
x x2 − 1
Limits
Mean Value Theorem:
f (x)
f 0 (x)
L’Hospital: lim
= lim 0
f (x) cont. on [a, b], diff. on (a, b): one can solve
x→a g(x)
x→a g (x)
f (b) − f (a)
– (when applicable)
f 0 (x) =
,
b−a
lim (1 + 1/n)n = lim (1 + h)1/h = e
n→∞
with a < x < b.
h→0
Squeeze Theorem:
If g1 (x) ≤ f (x) ≤ g2 (x) near a,
and lim g1 (x) = lim g2 (x) = L,
x→a
x→a
then lim f (x) = L.
x→a
limx→0
sin x
=1
x
Formulae for Calculus 151 p. 2
Interpretations of derivatives
1st: Velocity or rate of change
2nd: Acceleration
Logarithmic: Relative rate of change
Slope of tangent line
Integration
Trigonometry
Values:
θ 0 π/6 π/4 π/3 π/2
√ √
1
2
3
sin 0
1
2
2
2
√ √
3
2 1
cos 1
0
2
2
2
Integration gives the signed area between the
Right Triangles:
curve and the x-axis (above−below).
sine: opposite/hypotenuse
Fundamental Theorem
of Calculus:
Z b
cosine: adjacent/hypotenuse
(f continuous:)
f (x)dx is F (b) − F (a),
a
tangent: opposite/adjacent
with F (x) an antiderivative.
Formulas:
Z
xn+1
xn dx =
+ C (n 6= −1)
n
+
1
Z
sec xdx = ln | sec(x) + tan(x)| + C
Read the differentiation formulas from
right to left!
Algebra
p
secant: 1/cosine
Multiples:
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos 2(x) − sin 2(x)
1 − cos x
1 + cos x
sin 2(x/2) =
; cos 2(x/2) =
2
2
More identities:
sin(x ± y) = sin(x) cos(y) ± cos(x) sin(y)
Slope: ∆y/∆x; Distance: (∆x)2 + (∆y)2
cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y)
Quadratic formula:
ax2 + bx + c = 0: then
tan(x) + tan(y)
√
tan(x + y) =
−b ± b2 − 4ac
1 − tan(x) tan(y)
x=
2
2a
2
sin x + cos x = 1
(a − b)(a + b) = a2 − b2
tan2 x + 1 = sec2 x
ln(ab ) = b ln(a)
Co-functions:
ln(ab) = ln(a) + ln(b); ln(1/b) = − ln(b)
cos(x) = sin(π/2 − x); cot(x) = tan(π/2 − x)
ln(1) = 0; ln(e) = 1
csc(x) = sec(π/2 − x)
ln x
loga (x) =
ln a
Numbers (rough approximations)
Area:
√
√
π ≈ 3.14 e ≈ 2.7
2 ≈ 1.4 3 ≈ 1.7
Triangle: 1/2 base×altitude
Circle: πr2;
Sphere (surface): 4πr2
Volume:
Box: Product of dimensions.
Sphere (inside):
4
3 πr3
Cylinder: Base area × Height
Cone:
1
3
Base area × Height
ln 2 ≈ .7
ln 10 ≈ 2.3