Suppose f(x) is defined when x is near the number a. Then we write \lim_{x\rightarrow a}f(x) = L if we can make
the values of f(x) arbitrarily close to
L by taking x to be sufficiently close to a (on either side of a) but not equal to a.
Property
\lim_{x\rightarrow c} f(x) = L \iff
\lim_{x\rightarrow c^-} f(x) = L \text{ and } \lim_{x\rightarrow c^+}
f(x) = L
Limit laws
Suppose that a,b,c are constants and
the limits \lim_{x\rightarrow c} f(x) \text{
and } \lim_{x\rightarrow c} g(x) exist. Then \lim_{x\rightarrow c} \left[a f(x) + b g(x)\right]
= a \lim_{x\rightarrow c} f(x) + b \lim_{x\rightarrow c} g(x)\lim_{x\rightarrow c}\left[f(x)g(x)\right] =
\lim_{x\rightarrow c} f(x) \cdot \lim_{x\rightarrow c} g(x) \lim_{x\rightarrow c} \frac{f(x)}{g(x)} =
\frac{\lim_{x\rightarrow c} f(x)}{\lim_{x\rightarrow c} g(x)} \text{,
if } \lim_{x\rightarrow c} g(x) \neq 0\lim_{x\rightarrow c}\left[f(x)^n\right] = \left[
\lim_ {x\rightarrow c} f(x) \right]^n
Squeeze Theorem
If f(x)\leq g(x)\leq h(x) when x is near a
(except possibly at a) and \lim_{x\rightarrow a} f(x) = \lim_{x\rightarrow a}
h(x) = L then \lim_{x\rightarrow a}
g(x) = L
Continuity
A function f is continuous at a number
a if \lim_{x\rightarrow a} f(x) = f(a) This
requires three things: f(a) is defined,
\lim_{x\rightarrow a} f(x) exists, and
\lim_{x\rightarrow a} f(x) = f(a)
Continuity
A function f is continuous on an
interval if it is continuous at every number in the interval.
A discontinuity a is called:
removable if \lim_{x\rightarrow
a}f(x) exists.
an infinite discontinuity if x = a
is a vertical asymptote of the curve y =
f(x).
a jump discontinuity if both \lim_{x\rightarrow a^-}f(x) and \lim_{x\rightarrow a^-}f(x) exist but are not
the same
Intermediate Value Theorem
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)\neq f(b). Then there exists a number
c in (a,b) such that f(c) = N.
Different techniques for finding limits
direct substitution
limit laws
rationalisation
definition of derivative
squeeze/sandwich theorem
For limits at infinity
for rational functions, only the terms with the highest degree
matter
for \infty - \infty indeterminate
forms, try multiplying by the conjugate
Some limit heuristics
\lim_{x\rightarrow 0} \frac{1}{x} =
\infty
\lim_{x\rightarrow \infty} \frac{1}{x} =
0
The line y = mx+b is a slant
asymptote or oblique asymptote of the curve y
= f(x) if \lim_{x\rightarrow
\infty}(f(x) -mx -b)=0.
Differentiation
Derivative
The derivative of a function f is f'(x) = \lim_{h\rightarrow 0}
\frac{f(x+h)-f(x)}{h} or equivantly can be defined as f'(x) = \lim_{x\rightarrow a}
\frac{f(x)-f(a)}{x - a}
Differentiatibility
A function f is differentiable at a if f'(a) exists. It is differentiable on an
open interval (a,b) [or (a,\infty), (-\infty,a), (-\infty,\infty)] if it is differentiable at
every number in the interval.
Continuity
If f is differentiable at a, then f is
continuous at a.
Derivative of a constant function
If c \in \mathbb{R}, then \frac{d}{dx} c = 0
Definition
The number e is defined as the number
such that \lim_{h\rightarrow 0} \frac{e^h -
1}{h} = 1 It can also be defined as e
= \lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x =
\lim_{x\rightarrow \infty}\left(1+x\right)^\frac{1}{x} =
2.718281828459\ldots
Definition
The natural logarithm is a logarithm to the base e, \ln x =
\log_ex
Linearity
If a, b are constants and if f, g are both differentiable functions then
\frac{d}{dx} \left[ af + bg \right] = af'
+ bg'
Power rule
If n \in \mathbb{R}, then \frac{d}{dx} x^n = nx^{n-1}
Product rule
If f and g are differentiable, then (fg)' = fg' + gf'
Quotient rule
If f and g are differentiable, then \left(\frac{f}{g}\right)' = \frac{gf' -
fg'}{g^2}
Chain rule
If g is differentiable at x and f is
differentiable at g(x), then the
composite function F = f \circ g is
differentiable at x and F' is given by F' = f'(g)\cdot g'. And if y = f(u) and u=g(x) are both differentiable, then \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
Exponentials
If a is a constant in \mathbb{R}, then \frac{d}{dx} a^x = a^x \ln x. If a = e, then we get the special case \frac{d}{dx} e^x = e^x
Logarithms
If a is a constant in \mathbb{R}, then \frac{d}{dx} \log_a x = \frac{1}{x\ln a}.
If a = e, then we get the special case
\frac{d}{dx} \ln x = \frac{1}{x} Also,
\frac{d}{dx} \ln |x| = \frac{1}{x}
Inverse
The derivative of the inverse of a function is (f^{-1})'(x) =
\frac{1}{f'(f^{-1}(x))}
\begin{aligned}
\frac{d}{dx} \sin x &= \cos x \\
\frac{d}{dx} \cos x &= -\sin x \\
\frac{d}{dx} \tan x &= \sec^2 x \\
\frac{d}{dx} \csc x &= -\csc x \cot x \\
\frac{d}{dx} \sec x &= \sec x \tan x \\
\frac{d}{dx} \cot x &= -\csc^2 x \\
\end{aligned}
\begin{aligned}
\frac{d}{dx} \sin^{-1} x &= \frac{1}{\sqrt{1-x^2}} \\
\frac{d}{dx} \cos^{-1} x &= \frac{-1}{\sqrt{1-x^2}} \\
\frac{d}{dx} \tan^{-1} x &= \frac{1}{1+x^2} \\
\frac{d}{dx} \csc^{-1} x &= \frac{-1}{x\sqrt{x^2-1}} \\
\frac{d}{dx} \sec^{-1} x &= \frac{1}{x\sqrt{x^2-1}} \\
\frac{d}{dx} \cot^{-1} x &= \frac{-1}{x^2+1} \\
\end{aligned}
Logarithmic differentiation
Suppose you need to differentiate y =
f(x) where f(x) is a relatively
complicated function (usually involving exponentials). Logarithmic
differentiation can help:
y = f(x)\ln y = \ln f(x)\frac{d}{dx} \ln y = \frac{d}{dx} \ln f(x)\text{Solve for } y'
Since, \frac{d}{dx}\ln f(x) =
\frac{f'(x)}{f(x)}, we can use the following shortcut to
calculate the derivative using logarithmic differentiation: f'(x) = f(x) \frac{d}{dx} \ln f(x).
Optimisation
Important to distinguish between absolute and local minima.
Extreme value theorem
If f is continuous on a closed interval
[a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in
[a,b].
Fermat’s theorem
If f has a local maximum or minimum at
c, and if f'(c) exists, then f'(c)=0.
Stationary point
A stationary point of a function f is a
value c in the domain of f such that f'(c)=0. A stationary point may be a
minimum, maximum, or inflection point.
Critical point
A critical point of a function f is a
value c in the domain of f such that either f'(c)=0 or f'(c) does not exist (i.e. f is not differentiable).
The closed interval method
To find the absolute maximum and minimum values of a continuous function
f on a closed interval [a,b], you need
to find the values at the critical points of f in (a,b)
and find the values at the endpoints a,b. The
largest of those values is the absolute maximum, and the smallest is the
absolute minimum value.
Rolle’s theorem
If a function f satisfies all of these:
continuous on [a,b]
differentiable on (a,b)
f(a) = f(b).
Then there is a number c in (a,b) such that f'(c) = 0
Mean value theorem
If a function f satisfies all of these:
continuous on [a,b]
differentiable on (a,b)
Then there is a number c in (a,b) such that f'(c) = \frac{f(b)-f(a)}{b-a}
Theorem
If f'(x)=0 for all x in an interval (a,b), then f is constant on (a,b)
Corollary
If f'(x) = g'(x) for all x in an interval (a,b), then f-g is constant on (a,b). That is, f(x)=g(x) + c, where c is a constant.
Newton’s method formula
The recursive formula to solve for f(x)=0 is x_{n+1}
= x_n - \frac{f(x_n)}{f'(x_n)}
Integration
Theorem (definition, first principles)
If f is integrable on [a,b] then \int_{a}^{b} f(x)dx \geq \int_{a}^{b} g(x)dx,
if f(x)\geq g(x), for a\leq x\leq b\int_{a}^{b} f(x)dx = \lim_{n \rightarrow \infty}
\sum_{i=1}^{n}f(x_i)\Delta x, where \Delta x = \frac{b-a}{n} and x_i = a + i\Delta x (right-point).
Fundamental Theorem of Calculus
Suppose f is continuous on [a,b]:
If g(x) = \int_{a}^{x} f(t) dt,
then g'(x)=f(x).
\int_{a}^{b}f(x)dx = F(b) - F(a),
where F is any antiderivative of f, i.e., F' =
f.