0 - Review of Fundamentals Flashcards
A function f is linear if it obeys the equation:
f(ax1 + bx2) = _________
A function f is linear if it obeys the equation:
f(ax1 + bx2) = af(x1) + bf(x<span>2</span>)
Probing a linear system with each ____ ______ is enough to completely characterize the system.
Probing a linear system with each input direction is enough to completely characterize the system.
In function notation we have the “zeros of f” whereas analogously in matrix notation we have the “_____ of T”
In function notation we have the “zeros of f” whereas analogously in matrix notation we have the “kernel of T”
Invertible transformations are one-to-one correpondences or __jections between vectors in the input space and vectors in the output space.
Invertible transformations are one-to-one correpondences or bijections between vectors in the input space and vectors in the output space.
Given f(x) = ax2+bx+c = (x-x1)(x-x2), we know that the two numbers x1 and x2 are called the roots of the function and are the points where the function f(x) touches the _-____.
Given f(x) = ax2+bx+c = (x-x1)(x-x2), we know that the two numbers x1 and x2 are called the roots of the function and are the points where the function f(x) touches the x-axis.
There is a quick test to check if a quadratic function has roots (or touches or cross the x-axis).
If b2-4ac > 0, Then _____
If b2-4ac = 0, Then _____
If b2-4ac < 0, Then _____
There is a quick test to check if a quadratic function has roots (or touches or cross the x-axis).
If b2-4ac > 0, Then f(x) has 2 roots
If b2-4ac = 0, Then f(x) has only 1 root
If b2-4ac < 0, Then f(x) has no roots or doesn’t touch x-axis
The ______ of a function is the set of allowed inputs.
The _____ or _____ of a function is the set of all possible output values.
The ______ of a function describes the type of the outputs the function has.
The domain of a function is the set of allowed inputs.
The image or range of a function is the set of all possible output values.
The codomain of a function describes the type of the outputs the function has (like integers, real numbers, etc).
A function is ________ or ______ if it maps different inputs to different outputs
(think of injecting a fluid or shot into a cylindrical pipe or arm and watching the fluid spread out with no two inputs going to the same output).
A function is one-to-one or injective if it maps different inputs to different outputs
(think of injecting a fluid or shot into a cylindrical pipe or arm and watching the fluid spread out with no two inputs going to the same output).
A function is ____ or _______ if it covers the entire output set.
In other words, the image of the function is ____ to the function’s codomain.
(think of surveying the entirety of the land)
A function is onto or surjective if it covers the entire output set.
In other words, the image of the function is equal to the function’s codomain.
(think of surveying the entirety of the land)
A function is ______ if it is both _____ and _____.
In this case, it could also be said that f is in __-__-__ _______ between the input set and output set.
In other words, for every possible output there exists exactly one input.
A function is bijective if it is both injective and surjective.
In this case, it could also be said that f is in one-to-one correspondence between the input set and output set.
In other words, for every possible output there exists exactly one input.
Given a bijective function f, there exists an ____ function f-1 that performs the ____ mapping of f.
In other words,
f-1(f(x)) = f-1 . f (x) = ____
Given a bijective function f, there exists an inverse function f-1 that performs the inverse or reverse mapping of f.
In other words,
f-1(f(x)) = f-1 . f (x) = x
The equation of a line describes input-output relationships where the change in the output is ________ to the change in the input.
The equation of a line describes input-output relationships where the change in the output is proportional to the change in the input.
The equation of the line can be reformulated in a more symmetric form as the general equation and as a relation:
Ax + __ = __
where the traditional m = ___, and b = ____
The equation of the line can be reformulated in a more symmetric form as the general equation and as a relation:
Ax + Bx = C
where the traditional m = -A/B, and b = C/B
What is the inverse function of the function f(x)=x2?
What graph does this start with and what does this inverse function transform it into? (How do they look when graphed?)
What is the inverse function of the function f(x)=x2?
f(x) = sqrt(x)
What graph does this start with and what does this inverse function transform it into? (How do they look when graphed?)
The y’s and x’s switch places. Thus, the former parabola that faces up is not turned sideways and opens to the right, with one side on the positive quadrant and one side below the x-axis in the negative quadrant. Notice how we now have to take out the negatives below y=0 to pass the vertical line test to still be considered a function.
An even polynomial contains only ____ powers of x. It has the property that f(x) = ____.
An odd polynomial contains only ____ powers of x. It has the property that f(x) = ____.
If a polynomial has both even and odd terms then it is _____.
An even polynomial contains only even powers of x. It has the property that f(x) = f(-x).
An odd polynomial contains only odd powers of x. It has the property that f(x) = -f(x).
If a polynomial has both even and odd terms then it is neither even nor odd.
Properties of sin(x) function:
sin(x) is an even/odd function
Domain:
Range / Image:
Roots:
Inverse function:
Properties of sin(x) function:
sin(x) is an odd function
Domain: Reals
Range / Image: y in [-1, 1]
Roots: all multiples of π (anytime it touches x-axis)
Inverse function: sin-1(x) which is NOT the same as (sin(x))-1=csc(x)
Properties of the exponential function:
Domain:
Image:
Derivative:
Properties of the exponential function:
Domain: Reals
Image: (0, +inf)
Derivative: ex
We can summarize the polynomial notation of,
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x1 + a0
…into something more succinct, like….
We can summarize the polynomial notation of,
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x1 + a0
…into something more succinct, like….
the sum of products notation:
f(x) = nΣk=0 (ak•xk)
What’s an easy way to solve for x in the following equation?
g(x) = x4 - 7x2 + 10 = 0
What’s an easy way to solve for x in the following equation?
g(x) = x4 - 7x2 + 10 = 0
Substitution trick:
Substitute y=x2
The trigonometric identity
cos2x + sin2x = 1
is really the ________ Theorem. Derive this.
The trigonometric identity
cos2x + sin2x = 1
is really the Pythagorean Theorem. Derived:
|adj|2 + |opp|2 = |hyp|2
|adj|2/|hyp|2 + |opp|2/|hyp|2 = 1
cos2x + sin2x = 1
sin(30) = sin(__/__) =____
Knowing this, use the fact that cos2x + sin2x = 1 to find cos(30)
sin(30) = sin(π/6) = 1/2
Knowing this, use the fact that cos2x + sin2x = 1 to find cos(30)
cos(30) = sqrt( 1 - sin2(30) )
cos(30) = sqrt( 1 - (1/2)2)
cos(30) = sqrt(3) / 2
Generally for non-unit circles, cos(theta) = adj/hyp = rx/r,
so rx = _____
Generally for non-unit circles, cos(theta) = adj/hyp = rx/r,
so rx = rcos(theta)
sin( a + b ) = ______
(Hint: mnemonic for this identity is sico + sico)
sin( a + b ) = sin(a)cos(b) + sin(b)cos(a)
(Hint: mnemonic for this identity is sico + sico)
cos( a + b ) = ______
(Hint: mnemonic for this identity is coco + sisi)
cos( a + b ) = cos(a)cos(b) + sin(a)sin(b)
(Hint: mnemonic for this identity is coco + sisi)
The sin rule states
a / sin(alpha) = ______ = ______
The sin rule states
a / sin(alpha) = b / sin(beta) = c / sin(gamma)
The cos rule states that:
a2 = ___ + ___ - _________
b2 = ___ + ___ - _________
c2 = ___ + ___ - _________
The cos rule states that:
a2 = b2 + c2 - 2bccos(alpha)
b2 = a2 + c2 - 2accos(beta)
c2 = a2 + b2 - 2abcos(gamma)
Circumference of a circle:
Circumference of a circle:
C = 2πr
Area of a circle:
Area of a circle:
A = πr2
Equation of a sphere:
Equation of a sphere:
x2 + y2 + z2 = r2
Surface area of a sphere:
Surface area of a sphere:
A = 4πr2
Volume of a sphere:
Volume of a sphere:
V = 4/3 πr3
Surface area of a cylinder:
Surface area of a cylinder:
A = 2(πr2) + (2πr)h
Volume of a cylinder:
Volume of a cylinder:
V = (πr2)h
Surface area of a cone with radius r and height h:
Surface area of a cone with radius r and height h:
A = πr2 + πr sqrt( r2 + h2 )
Volume of a cone:
Volume of a cone:
V = 1/3 π hr2
The word vector comes from the latin vehere which means __ _____ (similar to vehicle)
The word vector comes from the latin vehere which means to carry (similar to vehicle)
The cross product of two vectors is only defined for ___-dimensional vectors.
The cross product of two vectors is only defined for 3-dimensional vectors.
A coordinate system is the same as the _____, which is the set of vectors that can represent any vector in that system.
(for example, in colors the ____ is aRed + bGreen + Blue)
A coordinate system is the same as the basis, which is the set of vectors that can represent any vector in that system.
(for example, in colors the basis is aRed + bGreen + Blue)
One such example of vectors is found when we do operations with the ____ number system.
One such example of vectors is found when we do operations with the complex number system.
