Midterm Review Flashcards
Define Function
A rule that assigns each x in a set D a unique element f(x) in a set R.
Rules for Inequalities
- If a < b, then a+c < b+d
- If a < b, c < d, then a+c < b+d
- If a < b, then ca < cb if c > 0
- If a < b, then ca > cb if c < 0
4a) If 0 < a < b, then 1/a > 1/b
4b) If a < b < 0, then 1/a > 1/b
How to solve Inequalities with double solution? (x-a)(x+b)>0
Use sign chart, ie test each interval (x < b, b < x < a, x > a) to see which is
Properties of Absolute Value
√(x^2) = |x|
- |ab|=|a||b|
- |a/b|=|a|/|b| (b != 0)
- |x| = a, or x = +/-a
- |x| < a, where -a < x < a
- |x| > a, where x > a or x < -a
Define Even Function
Where f(-x) = f(x) for every x in the domain. Even functions are symmetric wrt y-axis.
Define Odd Function
Where f(-x) = -f(x) for all x in the domain. Odd functions are symmetric wrt the origin.
1:1 Functions
If a function never maps to the same value twice. Horizontal line test.
Inverse of Functions
Inverse DNE for non 1:1 functions. To find inverse, sub y for f(x) and solve function for x, then switch x with y. Draw the graph of inverse by reflecting function wrt the line y=x.
Domain f = Range f^-1, Range f = Domain f^-1
Periodic Functions
f(x) is periodic if f(t+nT)=f(t) for all integer n.
Frequency: f= 1/T Hz
Angular Frequency: w=2πf
Hyperbolic Functions
cosh(x) = (e^x + e^-x) / 2, sinh(x) = (e^x - e^-x) / 2
Logarithmic Functions
If a^c = b, then log[a]b = c If f(x) = a^x, then f^-1(x) = log[a]x If f(x) = e^x, then f^-1(x) = lnx
Polynomials
Degree = largest power term.
Rational function is proper if degree top < degree bot
Partial Fraction Expansion
Proper rational function can be written as sum of partial fractions with form A/(ax+b)^2 and (Cx+D)/(ax^2+bx+c)^2 ~ irreducible factor.
For any factor repeated n times, you need n terms of the form given distinguished by exponents 1-n.
Solve by multiplying out the denominators and setting x to be a value which eliminates one constant.
How to go from Improper to Proper?
long division until you get poly + proper
Distance between points
√((x2-x1)^2+(y2-y1)^2) or √(Δx^2+Δy^2)
Equation of a Circle
x^2 + y^2 = r^2 or (x-h)^2 + (y-k)^2 = r^2 whose center is at point (h,k) and radius r.
Arcsinx
Domain= [-1, 1], Range= [-π/2, π/2]
Arccosx
Domain= [-1, 1], Range= [0, π]
Arctanx
Domain= (-∞, ∞), Range= [-π/2, π/2]
How to add two sin/cos waves of same frequency?
If asin(wt) + bcos(wt):
Asin(wt + θ)
A = √(a^2+b^2)
tanθ = b/a
Sequences
A sequence has limit L if for every +ive real # ε there exists an integer N such that for all n > N we have
|a[n] - L| < ε
Squeeze Theorem
If f(x) ≤ g(x) ≤ h(x) for x≥N and lim[x→∞]f(x) = y, lim[x→∞]h(x) = y, then lim[x→∞]g(x) = y
Important Limit and Indeterminate Forms
lim[x→∞] sinx / x = 1
∞-∞, 0/0, ∞/∞, 0x∞
Continuity
f(x) is continuous at a IFF lim[x→a]f(x) = f(a).
f(x) is continuous at interval I if it is continuous at all points in the interval.
If f and g are continuous at point a then f±g, fg, f/g (g≠0) are continuous.
secx and cscx
1/cosx and 1/sinx respectively
e^x
Domain = (-∞, ∞) and Range = (0, ∞)
lnx
Domain = (0, ∞) and Range = (–∞, ∞)
Domain of gof(x)
{xeD[f] | f(x)eD[g]}
Heaviside Formula
When given f(x) = { C1 (-∞, a), x < a ; C2 (a, b), a < x < b ; C3 (b, ∞), x >b }
f(x) = C1[1-H(x-a)] + C2[H(x-a) - H(x-b)] + C3[H(x-b)]
