Mathematics MST224
How many horizontal asymptotes can a function? How many vertical asymptotes can a function have?
A function can have zero, one or two horizontal asymptotes, but potentially infinitely many vertical asymptotes.
What does it mean for a function to be continuous on an interval?
A function f is continous on interval (a,b) but doesn't have to be continous at x=a or x=b, e.g. f(x) = 1/x is continuous at (0,infinity) even though f(0) is undefined. (2 ,3) is not continuous because 0 lies in the undefined range. Many common functions are continuous: 1. All polynomials are continuous 2. All exponentials and logs are continuous 3. Trig functions are continous except where they have a vertical asymptote
Define a radian Define some common angles in degrees with the radian counterparts.
A radian is the length of the arc of a unit circle that is equal to the angle subtended at the centre of the circle. 2pi radians = 360 degrees pi radians = 180 degrees pi/2 radians = 90 degrees pi/4 radians = 45 degrees# pi/8 radians = 22.5 degrees pi/3 radians = 60 degreees pi/6 radians = 30 degrees pi/12 radians = 15 degrees
What is the general formula for finding an angle in radians?
Angle in radians = (pi/180) * angle in degrees
What happens to a rational expression as x tends towards infinity? What is the formula?
As x gets bigger, the result of the rational expression tends closer and closer to zero. lim x→∞ C / x ^n = 0 Where C is a constant and n > 0
Where does the set of complex numbers sit in the number line?
Complex numbers include Real numbers, so Real numbers are a subset of the Complex numbers
How does displacement differ from distance?
Displacement is: Final Position - Initial Position., and only involves these two measures; how much distance is actually travelled inbetween is irrelevant.
Describe the general characteristics for graphs of rational functions
For 1/x, there is a continous asymptotal line running close to the x and y axes in the A and C quadrants, i.e. mirrored about the x=y plane For 1/x^2 there is a continuous asumptotal line running close to the x and y axes in the D and A quadrants, i.e. mirrored about the y axis For 1/x^3 - as for 1/x however the line touches the x axis, mirrored about the x=y plane For 1/x^4 - as for 1/x^2 however the line touches the x-axis, mirrored about the y axis
Regarding the Angle Sum Identities, when A=B=x, what new equations are derived for cos and sin?
For sin(A+B) = sin(A)cos(B) + cos(A)sin(B); when A=B=x: sin(2x) = 2sin(x)cos(x) For cos(A+B) = cos(A)cos(B)-sin(A)sin(B); when A=B=x: cos(2x) = cos^2(x) - sin^2(x) For tan(A+B) = (tanA + tanB)/(1-tanA tanB); when A=B=x: tan(2x) = (2tan(x))/1=tan^2(x)
What is the link between differentiation and continuity?
If a function f is differentiable at x, then it is continuous at x
What is the key property of a differentiable function with regards to continuity?
If a function f is differentiable at x, then it is continuous at x.
What is the Point-Slope equation?
If a line goes through (x0, y0) and has slope m, then its equation is y − y0 = m(x − x0).
If you know two points on a line, how can you find the equation of the line?
If a line goes through (x1, y1) and (x2, y2), its slope is equal to y2 − y1 / x2 − x1 . Then use the point-slope equation to find the line equation.
What is the Intermediate Value Theorem?
If f is continuous on [a,b] and f(a) <0 and f(b) > >0, there there is at least one number c in the interval [a,b] such f(c) = 0. The same is true if instead f(a) > 0 and f(b) < 0.. The basically interprets to the fact that the continuous line of the function MUST cross the x-axis at least once.
What is the product rule with respect to differentiation of x?
If h(x) = f(x)g(x) then h prime (x) = f prime(x)g(x) + f(x)g prime(x)
How can you determine what a general polynomial's graph will look like?
If the geaterst power (n) is even AND an > 0 then the D quadrant will have a negative slope and the A quadrant will have a positive slope. If the greatest power (n)is odd ANDan > 0 then the A quadrant will have a positive slope and the C quadrant will have positive slope. If the greatest power (n) is even AND an < 0 then the C quadrant will have a positive slope and the B quadrant will have a negative slope. If the greatest power (n) is odd AND an < 0 then D quadrant is negative slope and the B quadrant will have a negative slope.
What is the quotient rule if y = u/v?
If y = u/vf Then dy/dx = (du/dx(v) - dv/dx(u))/(v)^2
What is the other form of the product rule if y = uv?
If y = uv Then dy/dx = du/dx (v) + dv/dx(u)
What would the product rule be with three variables if y = uvw?
If y = uvw Then dy/dx = du/dx(vw) + dv/dx(uw) + dw/dx(uv)
Describe the process of implicit differentiation
Implicit differentiation allows us to differentiate expressions involving both x and y. Step 1: Express both left and right hand side of the equation in terms of d/dx Step 2: Differentiate both sides - any value of y that is differentiated put a dy/dx after it Step 3. Get dy/dxs on to same side OR isolate the single dy/dx, whichever is appropriate Step 4. dy/dx should be on left on its own - whatever is on the right side of the equation is the differentiated expression representing the gradient at any point of the original expression at Step 1.
How would you write in set notation "All real number except 2"?
R\{2}
What is the complex conjugate
See image
Describe the sandwich principle
Suppose that for all x near a, we have g(x) ≤ f(x) ≤ h(x). That is, f(x) is sandwiched (or squeezed) between g(x) and h(x). Also, let's suppose that limx→a g(x) = L and x lim→a h(x) = L. Then we can conclude that limx→a f(x) = L; that is, all three functions have the same limit as x → a. If g(x) ≤ f(x) ≤ h(x) for all x near a, and limx→a g(x) = limx→a h(x) = L, then x lim→a f(x) = L.
What is the Codomain?
The Codomain is the set of possible numbers output from a given function. The Range is a subset of the codomain.
In differentiation, explain how how changing x by a small amount leads to f prime (x) times as much change in y.
The amount h represents how much x has increased, and the difference between x and h can be called Delta x. So f prime (x) = lim as Delta x approaches 0 f(Delta (x) - f(x))/Delta x. Delta x = x new - x old, and Delta y = y new - y old; x new = Delta x + x old, and similarly y new = Delta y + y old; Delta y = y new - y old = f(x new) - f(x) = f(Delta x + x) - f(x); So, f prime (x) = lim Delta x approaches 0 Delta y/Delta x; this translates to that as you increase x by a certain amount then the change in y (or f prime (x) will be magnified - how much it will be magnified will be determined by the polynomial under evaluation. It's not EXACTLY equal, but is the LIMIT of the ratio of Delta y/Delta x and Delta x approaches 0. As Delta x gets closer and closer to zero, so the better and better will the approximation of the ratio of Delta y/Delta x become. y = f prime (x) = Delta y/Delta x. Example if y = x^2 the f prime (x) = d (x^2)/dx = d/dx (x^2) = 2x; note Delta y/Delta x is NOT a fraction but the limit of the ratio between Delta y and Delta x as Delta x approaches 0.
What is the domain of a function?
The domain of a function are all possible values that can be input into a function to generate unique outputs that makeup the range of the function.
Define the Horizontal Line Test and its purpose
The horizontal line test checks the f(x) output of any given legitimate x input for a graph of a function - if the horizontal line only intersects the graph at a single point then the function will have an inverse. If the horizontal line intersects the graph of the function at more than a single point, then that function does not have an inverse.
How would you define the Maxima and Minima of continous functions?
The min-max theorem states that if f is continous on [a,b] then f has at least one maxima and one minima on [a,b]
In the graphical interpetation of velocity, if a graph had x for time and f(x) displacement, what is the difference between the slope of a line of a linear function and the slope of a point on a curve of a polynomial function?
The slope of a linear function on a displacement - time graph represents the average velocity, whereas the slope of a point on a curve of a function on the same axes would represent instantaneous velocity. This represents the difference between the slope of a secant line measured between two points, and the slope of a tangent line at a point.
In the context of limits and graphing of the associated function, if the denominator is zero but the numerator isn't zero, what is the outcome?
There is always a vertical asymptote at a as x approaches a. There are four potential behaviours - I'll call them 'the four infinities'
What happens to the denominator when multiplying by the complex conjugate?
This process always reduces the denominator to a real number because the product of a complex number a + bi and its complex conjugate a − bi is always real: Note that a^2 + b^2 is always (a + bi) × (a − bi) = a^2 + b^2.
How would we classify a stationary point?
Use the second derivative test.
What is the difference between speed and velocity?
Velocity has a directional vector. Velocity can be negative, whereas speed must be non-negative. If you replace distance with displacement in average speed = distance/time, then you get velocity, i.e., average velocity = displacement/time
If classifying a stationary point using second derivative test is messy, how else could we classify the stationary point?
We could use the first derivative test.. It basically translates to: 1. If you punch in a value slightly less than the stationary point value into f'(x) and the output is negative AND if you punch in a value slightly more than the stationary point value into f'(x) and the output is positive, THEN you have a local maximum. Just reverse the logic and you get a local minimum.
Differentiate sqrt(x) with respect to x using limits
We know that f prime (x) = (f(x+h) - f(x))/h; replace x with sqrt(x); f prime (sqrt(x)) = (f(sqrt(x) + h) - f(sqrt(x))/h; use conjugate (sqrt(x) +h) + f(x)/(sqrt(x) + f(x)) f prime (sqrt(x)) = lim x approaches 0 (f((sqrt(x) + h) - f(sqrt(x))/h x (f(sqrt(x) +h) +(f(sqrt(x))/(f(sqrt(x) +h) +(f(sqrt(x)) = lim approaches x ((x+h)-x) / (h(sqrt(x) + h) + sqrt(x)) - lim x approaches 0 h/(h(sqrt(x) + h) + sqrt(x)) = lim x approaches 0 1/(sqrt(x)+h) + sqrt(x) = lim x approaches 0 1/(2sqrt(x) + h; let h = 0 f prime (sqrt(x)) = lim x approaches 0 1/2x
Differentiate 1/x with respect to x using limits
We know that f prime (x) = lim x approaches h (f(x + h) - f(x))/h; replace x with 1/x: f prime(1/x) = lim x approaches 0 ((f((1/x) + h) - f(1/x+h))/x; using common denominator x(x+h) we get: lim x approaches 0 ((x - (x+h))/x(x+h))/h = lim x approaches 0 (-h/x(x+h))/h = lim x approaches 0 (-h/x(x+h) / h = lim x approaches 0 (-h/x(x+h) * 1/h = lim x approaches 0 -h/hx(x+h) = lim x approaches 0 -1/x(x+h); let h = 0; lim x approaches 0 -1/x(x) = -1/x^2, that is d/dx (1/x) = -1/x^2
Define the Double Angle Identities.
We know that in the unit circle using Pythagoras Theorem x^2 + y^2 = r^2 = 1^2 = 1. x = cos(x) and y = sin(x), so cos^2(x) + sin^2(x) = 1. This leads to the following identities: cos^2(x) = 1 - sin^2(x) sin^2(x) = 1 - cos^2(x) Using sin(A+B) = sin(A)cos(B) + cos(A)sin(B) and cos(A+B) = cos(A)cos(B) - sin(A)sin(B); where A = B = x: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x) But using the unit circle identities above we get the following additional Double Angle Identities for cos(2x): cos(2x) = (1-sin^2(x)) - sin^2(x) = 1 - 2sin^2(x) and cos(2x) = cos^2(x) - (1 - cos^2(x)) = cos^2(x) - 1 + cos^2(x) = 2cos^2(x) - 1
How many points do you need to plot a linear graph, and what is the equation for a linear equation?
You need two points to plot a linear graph, and the equation for a linear graph is f(x) = mx + c
What is the descriminant for quadratic equations?
b^2-4ac if b^2-4ac = 0 then there is a single solution for x if b^2-4ac > 0 then there are two solutions for x if b^2 -4ac <0 then there are no real solutions for x
What is the power rule and working out for differentiation of a constant?
d/dx (x^n) = nx^(n-1) note: if n = 0 then d/dx(x^0) = d/dx(1) = 0*x^(0-1) = 0 - this confirms that the derivative of a constant is 0 So d/dx(C) = 0 Note: if n = 1 then d/dx(x^1) = 1*x^(1-1) = 1*x^(0) = 1*1 = 1 So d/dx(x) = 1
Prove that the derivative of a linear function is equal to zero using limits
f(x) = mx + c; so f prime (x) = m; since m is the slope of the function, we can show this as follows: f prime (x) = lim h approaces 0 (f(x+h) - f(x))/h; lim h approaches 0 (m(x+h) + c) - (mx + c))/h lim h approaches 0 (((mx +mh) +c) - (mx +c))/h = lim h approaches 0 mh/h = lim h approaches 0 mx + c = m. So if f(x) = c then f prime (x) = 0 because a constant in this context is a horizontal line parallel to the x axis, therefore the derivative of a constant function is 0 identically.
Where p and q are polynomials, for p/q what are the limits for rational numbers involving polynomials?
for lim x to infinity p(x)/q(x) If p = q then the limit is finite and non-zero If p>q then the limt is positive or negative infinity If p<q then the limit is zero
Define the chain rule where y = (x)^n
for y = (x)^n then Let y = x^x and let u = x dy/dx = (dy/du) * (du/dx) Once differentiated out then replace u with the original values.
for y = f(x/c) what happens to the graph for different values of c?
for y = f(x/c), as c get larger and positive the graphs flattens horizontally - for each point on the graph, move it away from the y-axis c units. As c gets larger and negative, each point in the graph moves away from the y axis by c units and the graph is then reflected in the y-axis - the graph seems to become 'stretched'. For values of c between 1 and -1, the same principle applies but the graph seems to become more 'squashed'
What is i^2?
i^2 = (sqrt(-1))^2 = -1
Define the chain rule with respect to differentiation
if h(x) = f(g(x)) Then h prime (x) = f prime(g(x))g prime(x)
What is the quotient rule with respect to differentiation as a function of x?
if h(x) = f(x)/g(x) Then h prime (x) = (f prime (x)g(x) - f(x)g prime(x))/(g(x))^2
In the context of |x| show that a derivative does not exist.
lim x approaches 0 |x + h|/h; replace x by 0; lim x approaches 0 |h|/h: on right side of limit the value of f prime (x) is 1 and the left side of the limit of f prime (x) is -1, so lim x approaches 0 of |h|/h DNE
What is the mathematical notation for "f has a left-hand horizontal asymptote at y = M"?
lim x→−∞ f(x) = M
What is mathematical notation for "f has a right-hand horizontal asymptote at y = L"?
lim x→∞ f(x) = L.
Utilising the fact that rational expressions tend towards one as x tends towards infinity, what expression involving polynomials uses this principle, and why is this the case?
lim x→∞ p(x)/ leading term of p(x) = 1 This is because the leading term of p(x) divided by the leading term of p(x) is 1, and every other term tends towards 0 , so the sum of the limits of a polynomial expression is 1.
What trigonometric equation in the unit circle is derived from Pythagoras Theorem?
sin^2(x) + cos^2(x) = 1 (Derived from x^2 = y^2 = r^2 = 1^2 = 1)
What is sin(x)/cos(x)? cos(x)/sin(x)?
tan(x) cot(x)
If we divide the unit circle base equation derived from Pythagoras Theorem by cos^2(x), what new equation do we get?
tan^2(x) + 1 = sec^2(x)
What is u(v+w)?
uv + uw
What are the predominant characteristics of polynomial graphs with a coefficient of 1? (think odd and even powers)
x^1 is a horizontal straight line x^2 is a parabola with a negative slope in the D quadrant and a positive slope in the A quadrant, touching the x-axis at the origin x^3 is a loose 's' shape with a positive slope in the C quadrant and a positive slope in the A quadrant, touching the x-axis at the origin x^4 is a slightly squashed parabola (with slopes as for x^2) and the base of the parabola touching the x-axis at multiple points up to about half a unit on the axis x^5 is a slightly squashed loose 's' shape (with slopes as for x^3), and the middle of the 's' touching the x-axis at multiple points up to about half a unit on the axis x^6 is a more squashed parabola (with slopes as for x^2) and the base of the parabola touching the x-axis at multiple points up to about 3/4 of a unit on the axis x^7 is a more squashed loose 's' shape (with slopes as for x^3) and the middle of the 's' touching the x-axis at multiple points up to about 3/4 of a unit on the axis ....rinse and repeat for additional odd and even powers.....
Describe the characteristics of the graph of an exponential function: (a). for values of x (b). for values of -x. What is the inverse of an exponential function?
y = b^x cuts the y axis at 1. The left hand tail of the graph is an asymptote to the x-axis, and the domain of the function is all real numbers and the range of the function is [0, infinity) y = b^-x is a mirror image of y=b^x in the y-axis, also cutting the y-axis at 1 The inverse of the function y=b^x is the mirror in the y=x plane and is equal to y = log2 (x)
for y = f(x), what is the equation for scaling the graph vertically?
y = c f(x) where c scales every value of x, c times away vertically from the x axis. If c is negative the scale every value | c | times vertically from the x axis and then reflect the whole graph in the x axis.
for y = f(x), what is the equation for translating a graph left or right?
y = f(x -c) to shift whole graph right by c units (c > 0), and y = f(x - (-c)) to shift the whole graph left by - -c units (i.e. + c units) (c < 0)
for y = f(x), what is the equation for translating a graph up or down?
y = f(x) + c to shift the graph up by c units, (or y = f(x) -c to shift the whole graph down by - c units)
What is the form of a complex number?
z = a + bi where a is the Re(a) and b is Im(b)
What is the general relationship of angles when talking about complementary angles?
<trig function>(x) = co <trig function>(pi/2 - x)
What does it mean for a function to be continuous at a single point?
For a function to be continous at a point: 1. The two-sided limit must exist 2. The function is defined at x = a 3. f(x) must equal f(a) i.e. lim x approaching a f(x) = f(a)
What is the modulus of a complex number?
G√iven any complex number z = a + bi, its modulus |z| is defined to be sqrt(a^2+b^2), so we have |z|^2 = a^2 + b^2 = z*conjugate z.
Define the range of a function
The range of a function is all actual outputs from that function
What is a point of inflection?
This is where f'(x) = 0 but is neither a local maximum or local minimum.
To divide one complex by another, how would we calculate u/v?
To divide one complex number by another, as in u/v, we multiply many textbooks. top and bottom by the complex conjugate v of the denominator.
What is i
i = sqrt(-1)
If we divide the unit circle base equation derived from Pythagoras Theorem by sin^2(x), what new equation do we get?
1 + cot^2(x) = cosec^2(x)
How do you find stationary points of function, and then classify each stationary point?
1. Change function so that variables are expressed as powers and not rationals. 2. Find the first derivative of the rearranged function. 3. Solve for dy/dx = 0; solutions will give the stationary points on the horizontal axis. 4. To find the stationary point types, take the second derivative using the first derivate function 5. Use the second derivative function and plug in the solution(s) from step 3 to get outputs. 6. If the result from the output is >0 , then the solution is a local minimum. 7. If the result from the output is <0, then the solution is a local maximum
What should you be careful/aware of when dealing with square roots in the context of limits?
1. If numerator evaluates to zero when dividing by leading term then look to use the conjugate and determine the leading term that way. 2. Make sure to add all like terms together - this is easy to miss so take care; look at root value and make sure the result is not the same as any remaining terms (for both the numerator and denominator) 3. Simplify then apply lim x to infinity C/x^n = 0 principle 4. When dealing with negative infinity make sure you are using the negative value of any leading term that forms part of the main ratio; this is because we want to understand what happens as x approaches negative infinity - not infinity. 5. You have to be careful for all even powered roots. In general if x <0 and you are looking to find nth root of something ^m , the ONLY time you need a minus sign in from of x^m is when n is EVEN and m is ODD 6. When using absolute values, |x| = { x if x >= 0; -x if x < 0
How do we translate a complex number to Polar coordinates?
1. Take complex number; 2. Take coefficients of x and y and plug them into unit circle formula; 3. Map coefficients onto graph to get θ; 4. Write out polar coords. Remember: radius of circle MUST be positive, and -pi < θ <= pi
Define the Angle Sum Identities
1. sin(A+B) = sin(A)cos(B) + cos(A)sin(B) 2. cos(A+B) = cos(A)cos(B) - sin(A)sin(B) 3. tan(A+B) = (tan(A) + tan(B))/1 - tan(A)tan(B)
What are the Angle Difference Identities?
1. sin(A-B) = sin(A)cos(B) - cos(A)sin(B) 2. cos(A-B) = cos(A)cos(B) + sin(A)sin(B) 3. tan(A-B) = (tan(A)-tan(B))/1 + tan(A)tan(B)
List some common complentary angles in general relationship terms
1. sin(x) = cos(pi/2 - x) 2. tan (x) = cot(pi/2 -x) 3. sec(x) = cosec(pi/2 - x) and conversely: 1. cos(x) = sin(pi/2 -x) 2. cot(x) = tan(pi/2 - x) 3. cosec(x) = sec(pi/2 -x)
Describe the graphs of: 1. y = sin(x) 2. y = cos(x) 3. y = tan(x) 4. y = sec(x) 5. y= cosec(x) 6. y = cot(x)
1. y = sin(x) repeats periodically between 0 and 2pi. The sin function is an odd function with symetry around the origin. Values are constantly changing between 1 and -1 2. y = cos(x) repeats periodically between pi/2 and 5pi/2 - i.e. shifted to the left by pi/2 compared to y = sin(x). The symettry is reflected in the y-axis so y = cos(x) is an even function. Values are constantly changing between 1 and -1 3. y = tan(x) repeats distinctly between odd values of pi/2, e.g. between -pi/2 and pi/2 the tangent wave has a positive gradient is asymptotic from negative infinity at -pi/2 slowly crossing the origin and then increasing to negative infinity but again being asymptotic at pi/2. This pattern infinitely repeats. The graph is symmetrical about the origin and is therefore an odd function 4. y = sec(x) (1/cos(x)): repeats distinctly with alternating positive and negative values between odd values of -pi/2 and pi/2. Each repetition of the graph is asymptotic at -pi2 and pi/2 (i.e., at pi-spaced intervals). Similar in shape to a parabola, the apex of the shape is 1 for positive values of y and -1 for negative values of y. This is an even function. 5. y = cosec(x) (1/sin(x)): repeats distinctly with alternating positive and negative values between even values of 0 and pi. Each repetition of the graph is asymptotic at 0 and pi (i.e. at pi-spaced intervals). Similar in shape to a parabola, the apex of the shape is 1 for positive values of y and -1 for negative values of y. y = cosec(x) is shifted pi/2 to the right compared to y = sec(x). This is an odd function. 6. y = cot(x) (1/tan(x)): As for tan(x) but the repetition of the graphs occurs between even values of pi at pi intervals. The graph is a reflection in the x = y plane where the graph crosses the x-axis. This is an odd function. So sin(−x) = − sin(x), tan(−x) = − tan(x), and cos(−x) = cos(x) for all real numbers x.