Quiz 1 Flashcards
(26 cards)
average velocity
∆ in position / ∆ in time
[ s(b) - s(a) ] / b - a
uses secant line (from point a to b) to calculate slope which is the average velocity
tangent
a tangent to a curve is the instantaneous velocity
Lim
x -> c+
[ s(x) - s(a) ] / x - a
Limit notations
L’ or lim : limit
c- : limit as approaching from the left
c+ : limit as approaching from the right
Existant, non-existant & indefinite limits
limit exists: if L’ from left = L’ from right
limit DNE : if L’ from left ≠ L’ from right
indefinite: if numerator and denominator equal 0, limit could be anything
NB: undefined is when only the denominator equals 0
Limit definition
what f(x) equals as it approaches that point
what y-value occurs when approaching a certain x-value
y-value is the limit
Finding the limit in a polynomial equation
plug the value x = a into the polynomial
f(x) = f(a) = aₙaⁿ + … + a₁x + a₀
Limit rules
lim ( f(x) ± g(x) = lim f(x) ± g(x) *
lim f(x) . g(x) = lim f(x) . lim g(x)
lim (f(x))ⁿ = (lim f(x))ⁿ
constant multiple:
lim c f(x) = c lim f(x)
quotient rule:
lim f(x)/g(x) = [ lim f(x) ] / [ lim g(x) ] **
root rule:
lim ᶜ√k(x) = ᶜ√ lim k(x)
- rules apply given the limit exists
** the quotient rule can not be used if the limit makes the denominator 0, meaning the equation must be simplified or rewritten
Squeeze theorem
g(x) ≤ f(x) ≤ h(x)
lim g(x) = L , lim h(x) = L
f(x) = L
Indefinite limits
if limit results in 0/0 then the limit is indefinite meaning ∞ or -∞
NB: if limit from one side goes to ∞ and limit from the other side goes to -∞ then limit DNE
Cancellation rule
never cancel zero with zero in maths
Determining if indefinite limit goes to ∞ or -∞
Test numerator and denominator against limit and if positive it goes to ∞
If negative it goes to -∞
Indefinite limits with log/ln
lim ln(x) = -∞
x -> 0+
NB: ln = logₑ
Vertical asymptotes
when the denominator equals zero and the numerator is not zero
When the limit can not be simplified or cancelled
Test limit values against the equation, if - and coming from the left assume limit value is slightly smaller, if coming from the right assume limit value is slightly larger and evaluate for positive and negative and determine if answer if positive or negative infinity
cosec, sec and cot graphs
Tip:
for sin and cos peaks and troughs of wave hit max and min of inverse function graphs
cosec:
negative parabola reaching to -1 from -180º to 0º
positive parabola reaching to 1 from 0º to 180º
asymptotes at 0º, 180º and 360º
sec:
negative parabola reaching to -1 from -270º to -90º
positive parabola reaching to 1 from -90º to 90º
asymptotes at 90º and 270º
cot:
flipped tan line from 0º to 180º and from 180º to 360º
asymptotes at 0, 180º and 360º
When the limit is going to ∞ or -∞
this value is a horizontal asymptote
Squeeze theorem applications
lim sinx/x = 0
x -> ∞
-1 ≤ sin x ≤ 1
-1/x ≤ sin ≤ 1/x
when x goes to ∞, -1/x , goes to 0
when x goes to ∞, 1/x , goes to 0
hence
sinx also goes to 0 when x goes to infinity
you can apply squeeze theorem here..
sinx/x² or sinx/√x or cosx/x³
you can not apply squeeze theorem here..
lim sinx/x
x -> 0
Inverse function
graphically: reflect in line y = x
algebraically: swap x and y, solve for y, swap x and y
Note: tan⁻¹ x ≠ 1/tan x
∞/∞
indefinite
Rational limits
p(x)/q(x)
factor out highest xⁿ from numerator and denominator, resulting coefficient (quotient of leading coefficients) is the limit as x approaches ∞ or -∞
Rational function takes the form ex.
p(x) = pₙxⁿ + pₙ-₁xⁿ-¹ … + p₁x + p₀
Rational limit at +∞ & -∞ rule
n = m , quotient of leading coefficient
n > m , +- ∞
n < m , 0
If limit at ∞ and -∞ is the same then there it is a horizontal asymptote
c / +/- ∞ = 0
if x -> -∞ then substitute it for t -> ∞
Types of discontinuity
- hole/removable
- jump
- vertical asymptote/infinite discontinuity
Continuity
A function f(x) is continuous at an interval [a,b] if the function is continuous at every point of that interval
Intermediate value theorem (IVT)
The function f(x) is continuous on the interval [a,b] and f(a) = A < f(b) = B. Then for A<L<B there is a c in [a,b] so that f(c) = L
