Test 3 (actually study this time, I beg)

Question 1

L’Hopitals rule

Answer 1

limₓ→꜀ of f/g = limₓ→꜀ of f’/g’

Question 2

how to use LHR

Answer 2

keep going until lim can be broken (no indeterminants)

Question 3

how to solve propogated error problems, 3 steps

Answer 3

1) Derive whatever formula you’re given
2) arrange it into differential template
3) plug in error ∆x into dx.

Question 4

How to solve rolle’s theorem problems, 2 steps

Answer 4

1. check continuity, diffibility, & if f(intvl left) = f(intvl right)
2. if good, solve f’(c) = 0 for c

Question 5

what is it asking for when it clarifies “absolute” extrema?

Answer 5

NOT THE DIP GOD DAMN IT (even tho it could be), it means the highest/lowest point on the whole interval man

Question 6

how to solve ABSOLUTE extrema problems

Answer 6

just plug in interval borders into og eqn along with any c’s you find, and pick out the lowest & highest points

Question 7

how to solve mean theorem problems on intvl [a, b], 2 steps

Answer 7

1. check continuity & diffibility

2. Solve f’(x) = (fb - fa)/(b-a)

Question 8

how to solve increasing/decreasing fn problems, 3 steps

Answer 8

1. find c’s
2. pick #’s in between c’s & intvl borders
3. test the #’s you picked on DERIVATIVE & read the sign. If -, it decreases; if +, it increases.

Question 9

what type of intervals do you use on inc/dec fn problems

Answer 9

OPEN, so (x, y). No closed intvls like [x, y] bc f’(x) doesn’t exist at a critical point

Question 10

how to solve concavity problems (3 steps)

Answer 10

1. solve f’‘(x) = 0
2. pick #’s between answer & intvl borders
3. find sign of f’’(#’s you picked). if +→ - after ans, it’s concave down; - → +, concave up.

Question 11

A field’s A must be 405,000 m². If no fence on 1 side, what dimensions use least fence? (8 freaking steps)

Answer 11

1. Draw 2 lengths y, 1 width x, (other width doesn’t exist)
2. Find primary (what you find), perimeter: P = x + 2y
3. Find 2ndary (how to find what you already know w/ what you don’t) for area: xy = 405,000
4. Solve 2ndary for a var, in this case y: y = 405,000/x
5. Sub var into primary & derive: P’ = 1 - 81,000x⁻²
6. Find crit #’s from P’: x = 900 (no “-900” fence length)
7. Put crit # thru 2nd derive test: P’‘(900) = 0.002, which is > 0, so 900 is a min length of fencing
8. Plug that crit # into 2ndary to find y: y = 450

Question 12

define point of inflection

Answer 12

the tangent point c if f(x)’s concavity changes there

Question 13

how to solve differential eqns asking for Δy for smth like y = arcsinx²

Answer 13

1. Find y’: → 2x/√(1 - x⁴)

2. Put thru differential template: dy = dx * 2x/√(1 - x⁴)

Question 14

what is the differential template

Answer 14

dy = f’(x)dx

Question 15

what indeterminants will trigger L’Hopital’s rule

Answer 15

0/0, ± ∞/∞

Question 16

2nd derivative test, x 3

Answer 16

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

Question 17

1st derivative test

Answer 17

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

Question 18

what unsolvable limits CAN’T you use L’Hopital’s Rule for?

Answer 18

infinity - or + infinity

Question 19

rel error eqn

Answer 19

dA/A x 100

Question 20

what was the trick with that one “critical values @ f’(x) and f’‘(x)” thing?

Answer 20

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

Question 21

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

Answer 21

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