Test 3 (actually study this time, I beg) Flashcards
L’Hopitals rule
limₓ→꜀ of f/g = limₓ→꜀ of f’/g’
how to use LHR
keep going until lim can be broken (no indeterminants)
how to solve propogated error problems, 3 steps
1) Derive whatever formula you’re given
2) arrange it into differential template
3) plug in error ∆x into dx.
How to solve rolle’s theorem problems, 2 steps
- check continuity, diffibility, & if f(intvl left) = f(intvl right)
- if good, solve f’(c) = 0 for c
what is it asking for when it clarifies “absolute” extrema?
NOT THE DIP GOD DAMN IT (even tho it could be), it means the highest/lowest point on the whole interval man
how to solve ABSOLUTE extrema problems
just plug in interval borders into og eqn along with any c’s you find, and pick out the lowest & highest points
how to solve mean theorem problems on intvl [a, b], 2 steps
- check continuity & diffibility
2. Solve f’(x) = (fb - fa)/(b-a)
how to solve increasing/decreasing fn problems, 3 steps
- find c’s
- pick #’s in between c’s & intvl borders
- test the #’s you picked on DERIVATIVE & read the sign. If -, it decreases; if +, it increases.
what type of intervals do you use on inc/dec fn problems
OPEN, so (x, y). No closed intvls like [x, y] bc f’(x) doesn’t exist at a critical point
how to solve concavity problems (3 steps)
- solve f’‘(x) = 0
- pick #’s between answer & intvl borders
- find sign of f’’(#’s you picked). if +→ - after ans, it’s concave down; - → +, concave up.
A field’s A must be 405,000 m². If no fence on 1 side, what dimensions use least fence? (8 freaking steps)
- Draw 2 lengths y, 1 width x, (other width doesn’t exist)
- Find primary (what you find), perimeter: P = x + 2y
- Find 2ndary (how to find what you already know w/ what you don’t) for area: xy = 405,000
- Solve 2ndary for a var, in this case y: y = 405,000/x
- Sub var into primary & derive: P’ = 1 - 81,000x⁻²
- Find crit #’s from P’: x = 900 (no “-900” fence length)
- Put crit # thru 2nd derive test: P’‘(900) = 0.002, which is > 0, so 900 is a min length of fencing
- Plug that crit # into 2ndary to find y: y = 450
define point of inflection
the tangent point c if f(x)’s concavity changes there
how to solve differential eqns asking for Δy for smth like y = arcsinx²
- Find y’: → 2x/√(1 - x⁴)
2. Put thru differential template: dy = dx * 2x/√(1 - x⁴)
what is the differential template
dy = f’(x)dx
what indeterminants will trigger L’Hopital’s rule
0/0, ± ∞/∞
2nd derivative test, x 3
f’‘(c) > 0 → f(c) is a rel min
f’‘(c) < 0 → f(c) is a rel max
f’‘(c) = 0 → test fails and is inconclusive
1st derivative test
If f’(x) switches from -→+ @ c, f(c) is rel min
If f’(x) switches from +→- @ c, f(c) is rel max
If f’(c) doesn’t switch signs, f(c) is inconclusive
what unsolvable limits CAN’T you use L’Hopital’s Rule for?
infinity - or + infinity
rel error eqn
dA/A x 100
what was the trick with that one “critical values @ f’(x) and f’‘(x)” thing?
since the graph was already f’(x), it wasn’t asking for a tangent line, it was asking for when x = 0, which would be 2 points - when x = 1 and when x = 7.
It was asking for the tangent line of that graph with f’‘(x), and that was the x=4 you put for the first
what did you do wrong on the differential question that asked about change in y when change in x was from 9 to 9.5
you didn’t sub in 9 in the derivative T-T don’t leave any variables out and use the first number they give you in the derivative, just get a # for y and NO VARIABLES
