Summary of some Calculus I Formulas - Yahdi
Equations for lines
Equation of the Tangent line y = f 0 (a)(x − a) + f (a)
Equation of a line with slope m through (a, b): y = m(x − a) + b
Quadratic Formula
2
ax + bx + c = 0 ⇐⇒ x =
−b −
√
√
b2 − 4ac
−b + b2 − 4ac
or
2a
2a
Some Algebraic Formulas:
√
1
n
A = A n A1n = A−n An Am = An+m
An
Am
= An−m
ln AB = B ln A ln AB = ln A + ln B
Formulas related to Economics:
Cost function: C(x)
Average Cost: C(x) =
C(x)
x
Revenue = (Price per unit) × (Quantity)
Profit: P(x) = R(x) − C(x)
Geometric Formulas: C=circumference, A=area & V= volume
Circle: C = 2πR, A = πR2
Sphere: A = 4πr2 , V = 34 πR3
Cylinder: Area of the sides A = 2πRh, V = πR2 h
Triangle: Area= 12 base × height
Gravitational attraction
g = −9.8m/s2 = −32f t/s2
Derivative using the limit:
f (x) − f (a)
f (a + h) − f (a)
f (a) = lim
= lim
x→a
h→0
x−a
h
0
L’Hospital’s Rule:
f (x)
f 0 (x)
f (x)
lim
= I.F. =⇒ lim
= lim 0
x→a g(x)
x→a g(x)
x→a g (x)
Using the derivatives:
Second Derivative Test: if f 0 (p) = 0 and
• Local maximum for f at p ⇐⇒ f 00 (p) < 0 (negative).
• Local minimum for f at p ⇐⇒ f 00 (p) > 0 (positive)
What does f 0 say about f ?
• The first derivative f 0 is positive (+) ⇐⇒ f is increasing (%)
• The first derivative f 0 is negative (–) ⇐⇒ f is decreasing (&)
Relationships between f , f 0 and f 00
• f concave up ∪ ⇐⇒ f 0 increasing % ⇐⇒ f 00 positive.
• f concave down ∩ ⇐⇒ f 0 decreasing & ⇐⇒ f 00 negative.
1
2
Formulas Related to Differentiation:
(x)
Definition: f 0 (x) = limh→0 f (x+h)−f
h
Tangent line: y = f 0 (a)(x − a) + f (a)
sum rule: (f + g)0 = f 0 + g 0
Multiply by constant: (cf )0 = c(f 0 )
Product rule: (f g)0 = f 0 g + f g 0
Quotient rule: ( fg )0 =
f 0 g−f g 0
g2
Chain rule: (f [g(x)])0 = f 0 [g(x)] · g 0 (x)
Basic Derivatives and Chain Rules short-cuts:
Power rules: (xn )0 = n x(n−1) (un )0 = n u(n−1) · u0
Exponential rules: (ex )0 = ex (eu )0 = eu · u0
(ax )0 = ln a · ax (au )0 = ln a · au · u0
Logarithmic rules: (ln x)0 =
1
x
(ln u)0 =
u0
u
Trig. rules: [sin(x)]0 = cos(x) [sin(u)]0 = cos(u) · u0
[cos(x)]0 = − sin(x) [cos(u)]0 = − sin(u) · u0
[tan(x)]0 =
1
cos2 (x)
Inverse Trig: [arcsin(x)]0 =
= sec2 (x) [tan(u)]0 =
√ 1
1−x2
[arccos(x)]0 =
0
√−1
1−x2
[arctan(x)]0 =
1
1+x2
[arcsin(u)]0 =
[arccos(u)]0 =
[arctan(u)]0 =
u0
cos2 (u)
= u0 sec2 (u)
0
√u
1−u2
0
√−u
1−u2
u0
1+u2
Formulas Related to
R antiderivatives:
Definition: F (x) = f (x) dx = antiderivative of f with respect to x .
R
R
R
sum rule: (f + g) dx = f dx + g dx
R
R
Multiply by constant:
kf dx = k f dx
R n
n+1
Power rules for n 6= −1:
x dx = xn+1 + c
R 1
Logarithmic rules:
dx = ln |x| + c
x
R x
R x
Exponential rules:
e dx = ex + c
a dx = ln1a ax + c
R
R
Trig. rules:
cos xdx = sin x + c
sin xdx = − cos x + c
R
R
sec2 xdx = cos12 x dx = tan x + c
R 1
R 1
√
Inverse Trig:
dx
=
arcsin
x
+
c
dx = arctan x + c
2
1+x2
1−x
Fundamental Theorem of Calculus:
Rb
f (t) dt = [F (t)]ba = F (b) − F (a) where F is antiderivative of f .
a
Rx
If F (x) = a f (t) dt then F 0 (x) = dFdx(x) = f (x).