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].
Critical Number - Definition
A critical number of a function f is a number c in the domain of f for which either f’(c) = 0 or f’(c) is not defined.
Chain Rule of Differentiation
(Chain Rule): Assume that f is differentiable at x and that g is differentiable at f(x). Then the composition g◦f is differentiable at x, and its derivative (g◦f)’ is given by
(g◦f)’(x) = g’(f(x))f’(x)
that is,
Dx(g(f(x))) = g’(f(x))Dx f(x)
Rolle’s Theorem
(Rolle’s Theorem): If f is continuous over a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b) = 0, then there is at least one number c in (a, b) such that f’(c) = 0.
What is Mean-Value Theorem
(Mean-Value Theorem): Let f be continuous over the closed interval [a, b] and differentiable on the open interval (a, b). Then there is a number c in the open interval (a, b) such that
f ‘(c) = (f(b) − f(a))/(b − a)
In graphic terms, the mean-value theorem states that at some point along an arc of a curve, the tangent line is parallel to the line connecting the initial and the terminal points of the arc.
Increasing and Decreasing Functions
A function f is said to be increasing on a set A if, for any u and v in A ,
u < v implies f(u) < f(v).
Similarly, f is decreasing on a set A if, for any u and v in A ,
u < v implies f(u) > f(v).
If f’(x) > 0 for all x in the open interval (a, b), then f is increasing on (a, b).
If f’(x) < 0 for all x in (a, b), then f is decreasing on (a, b).
What is Intermediate Value Theorem
(Intermediate-Value Theorem):
Let f be a continuous function over a closed interval [a, b], with f(a)<>f (b). Then any number between f (a) and f (b) is assumed as the value of f for some argument between a and b.
Basically, the function could not “skip” an intermediate value unless there were a break in the graph; that is, unless the function were discontinuous.
What is Newton’s Method?
Let us assume that we are trying to find a solution of the equation
f(x) = 0. and let us also assume that we know that x0 is close to a solution. If we draw the tangent line T to the graph of f at the point with abscissa x0, then T will usually intersect the x-axis at a point whose abscissa x1 is closer to the solution than x0.
A point-slope equation of the tangent line T is
y − f(x0) = f’(x0)(x − x0)
If T intersects the x-axis at the point (x1, 0), then
0 − f(x0) = f’(x0)(x1 − x0)
If f’(x0)<>0,
x1 − x0 = − (x0)/f’(x0)
x1 = x0 − f (x0)/f’(x0)
By repeating the above, using x1 as the starting point of the next x0, to generate x0, x1, x2, x3…and so on , it is possible to get progressively closer solutions for the problem
E.g. Using the approximation principle, estimate the value of √ 62.
Letting f (x) = √ x and c = 62, choose x = 64 (the perfect square closest to 62). Then,
dx = c − x = 62 − 64 = −2
f (x) = √ 64 = 8
f’(x) = 1/2 √ x = 1/2 √ 64 = 1/2(8) = 1/16
√ 62 ≈ 8 + (-2)1/16 = 8 - 1/8 =
16 (−2) = 8 − 1 = 7.875
Actually, √ 62 = 7.8740 . . ..
Key Features for Graph Sketching
GRAPH SKETCHING
The most important features of graphs are:
(i) Relative extrema (if any)
(ii) Inflection points (if any)
(iii) Concavity
(iv) Vertical and horizontal asymptotes (if any)
(v) Behavior as x approaches +∞ and −∞
