Summer Review Flashcards
What are the two branches of calculus and what are the defining characteristics of the two?
Differential calculus divides things into small (different) pieces and tells us how they change from ine moment to the next.
Integral calculus joins (integrates) the small pieces together and tells us how much of something is made, overall, by a series of changes.
*taken off the wikipedia page for calculus.
Explain what limits are and how they tie into the fundamental ideas of calculus.
A limit is the value that a function or sequence “approaches” as the input approaches some value. Limits are essential to calculus (and mathematical analysis in general), and are used to define continuity, derivatives, and integrals.
*from the wikipedia page on limits (mathematics)
From what concept are all the trigonometric identites derived?
All trigonometric identities are derived from both the pythagorean theorem (and, loosely off the unit circle).
Simplify [sin^2(t)/cos^2(t)] + [cos^2(t)/cos^2(t)] = [1/cos^2(t)] , which was derived from the proof of the pythagorean identity.
*t stands for theta
tan^2(t) + 1 = sec^2(t)
Pythagorean Identity
Sin(t) + cos(t) = 1
Proof: we know that a^2 + b^2 = c^2, and SOH CAH TOA describes the relationship of the angles in a right triangle. (a) and (b) represent the legs, and (c) represents the hypotenuse. Graphically, either a or b could be the opposite leg. Divide the entire pythagorean theorem by c^2 (which could also equal h^2, by meaning) and you get the angles described in SOH CAH TOA. Simplify.
*t stands for theta
Double Angle Identity
The double angle identities refer to these two formulas respectively:
Sin(2a) = 2•sin(a)•cos(a)
Cos(2a) = cos^2(a) - sin^2(a)
These are derived from the formulae that adds the sine/cosine of two angles:
Sin(a+b) = [sin(a)•cos(b)] + [sin(b)•cos(a)]
Cos(c+d) = [sin(c)•cos(d)] - [sin(d)•cos(c)]
*see the proof videos on khan academy
Graphically, what do the following equations mean:
Sin(a) = cos[a-(pi/2)] Cos(a) = sin[a+(pi/2)]
The cosine of (a) shifted to either the left or the right (depending on the sign of the inner equation) is the same as its sine.
The same explanation goes for the cos(a) = sin[a+(pi/2)] formula.
Explain the idea if the reciprocal identities.
We know that the “opposites” of the three functions sine, cosine, and tangent are cosecant, secant, and cotangent respectively.
If we take, for example, the inverse of one of the functions (secant, for example), we get its “opposite”:
[1/secant(t)] = sin(t)
This rule applies to all the trig functions.
Which functions are odd (meaning, the function would change signs if theta was negative), and which ones are even?
Cosine and secant are even functions:
cos(-t) = cos(t)
sec(-t) = sec(t)
Sine, cosecant, tangent, and cotangent functions are all odd.
sin(-t) = -sin(t), and so on…
What are the quotient identities?
The quotient identities are:
tan(a) = [sin(a)/cos(a)] cot(a) = [cos(a)/sin(a)]
What is L’Hospitals rule?
L’Hospital’s rule states that under certain conditions (where the limit is found to be x/0 or 0/0) the limit of the quotient [f(x)/g(x)] is determined by the limit of the quotient of the derivatives of the parts; so this becomes [f’(x)/g’(x)].
*Both functions must be differentiable on an open interval.
Explain the idea of a derivative.
Derivatives describe a rate of change (e.g. Instantaneous rates of change), found by finding the limit of the following formula:
[f(x+∆x) - f(x)]/ ∆x
This is true provided that the limit exists.
Graphically, think of it as a slope in a graph; the derivative of the equation {mx+b} is m (x and b are constants).
Find d/dx [u/v]
d/dx [u/v] = (vu’ - uv’)/v^2
*{denominator times the derivative of the numerator} minus the {numerator times the derivative of the denominator} all over the denominator squared.
Find d/dx [c]
The derivative of any constant is always 0.
Find d/dx [a±b]
Addition/subtraction within derivative equations is the same as finding the derivative of the parts; in other words:
d/dx [a±b] = a’±b’
