Calculus Flashcards
What are ZEROS OF POLYNOMIALS
For any function f , a zero or root of f is a number r
such that f (r) = 0
Theorem: if r is an integer that is a root of f , then r must be a divisor of the constant term of f.
Theorem: A number r is a root of the polynomial if and only if f (x) is divisible by the polynomial x − r.
(Fundamental Theorem of Algebra): If repeated roots are counted multiply, then every polynomial of degree n has exactly n roots.
What are odd and even functions?
A function f is even if and only if the graph of f is symmetric with respect to the y-axis i.e. f (−x) = f (x)
A function f is called odd if, for any x in the domain of f , −x is also in the domain of f and f (−x) = −f (x).
If f (x) is defined as a polynomial that involves only odd powers of x (and lacks a constant term), then f (x) is an odd function. E.g. f (x) = 3x5 − 2x3 + x
Behaviour of limits is governed by behaviour of f
The value of L—or the very existence of L—is determined by the behavior of f near a, not by its value at a (if such a value even exists).
Conditions for a continuous function
Definition: A function f is said to be continuous at a if the following three conditions hold:
(i) lim x→a f (x) exists.
(ii) f (a) is defined.
(iii) lim x→a f (x) = f (a).
Rules for Determining Limits - f(x)/g(x), g(x) of greater degree
GENERAL RULE A. If f (x) and g(x) are polynomials and the degree of g is greater than the degree of f , then
lim x→+∞ f(x)/g(x) = 0 and
lim x→−∞ f(x)/g(x) = 0.
Rules for Determining Limits - f(x)/g(x), g(x) and f(x) of equal degree
GENERAL RULE B.
If f (x) and g(x) are polynomials and the degree of g is the same as the degree of f , then
lim x→+∞ f(x)/g(x) and
lim x→−∞ f(x)/g(x)
are both equal to the quotient of the leading coefficients of f and g.
Rules for Determining Limits - f(x)/g(x), g(x) of lesser degree - how to tell sign of limit
GENERAL RULE C.
If f (x) and g(x) are polynomials and the degree of g is smaller than the degree of f , then
lim x→+∞. f(x)/g(x) = ±∞.
The result is +∞ when the leading coefficients of f and g have the same sign,
and the result is −∞ when the leading coefficients of f and g have opposite signs.
Rules for Determining Limits - f(x)/g(x), g(x) of lesser degree - size of limit
GENERAL RULE D. If f(x) and g(x) are polynomials and the degree of g is smaller than the degree of f , then
lim x→−∞ f(x)/g(x) = ±∞
Rules for Continuity over a Closed Interval
Definition: A function f is continuous over [a, b] if:
(i) f is continuous at each point of the open interval (a, b).
(ii) f is continuous on the right at a.
(iii) f is continuous on the left at b.
Rules for Differentiation - Derivatives of Sums and Differences
Dx(f(x) + g(x)) = Dx f(x) + Dx g(x)
The derivative of a sum is the sum of the derivatives.
Dx(f(x) − g(x)) = Dx f(x) − Dx g(x)
The derivative of a difference is the difference of the derivatives.
Product Rule of Differentiation
Dx(f(x)·g(x)) = f(x)·Dx g(x) + g(x)·Dx f(x)
Condition of Derivative f’(x) - Right and Left Hand Derivatives
Define f’+(x), the right-hand derivative of f at x, to be lim h→0+ (f(x + h) − f(x))/h
and f’−(x), the left-hand derivative of f at x,
to be lim h→0− (f(x + h) − f(x))/h
the derivative f’(x) exists if and only if both f’+(x) and f’−(x) exist and are equal.
Odd / Even - Function and Derivative
If f is differentiable:
If f is even, f’ is odd;
If f is odd, f’ is even.
Quotient Rule of Differentiation
(Quotient Rule). If f and g are differentiable at x and if g(x) <>0, then
Dx(f(x)/g(x))
= (g(x).Dx f(x) − f(x).Dx g(x))/[g(x)]^2
Extreme Value Theorem
Any continuous function f over a closed interval [a, b] has an absolute maximum and an absolute minimum on [a, b].
