Unit 3 Flashcards

1
Q

With what do we differentiate with respect to in related rates questions?

A

time

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2
Q

volume of a sphere

A

(4/3) · π · r3

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3
Q

surface area of sphere

A

(4) · π · r2

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4
Q

volume of cone

A

(1/3) · π · r2 · h

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5
Q

when is x=c a critical number

A

if f(c) exists and f’(c)=0 or dne

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6
Q

Extreme Value Theorem

A

if f continuous on a closed interval [a,b], then f has both an absolute minimum and an absolute maximum on the interval

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7
Q

can absolute extremas be endpoints?

A

yes

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8
Q

3 steps for justifying absolute extrema of continuous function f on [a,b]

A
  1. derivative of f
  2. critical number
  3. function values evaluated at the endpoints and critical endpoints
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9
Q

Mean Value Theorem

A

If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c between a and b that

f’(c) = [f(b) -f(a)]/ b-a

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10
Q

conditions for MVT

A
  1. continuous [a,b]
  2. differentiable (a,b)
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11
Q

Rolle’s Theorem

A

Let f be continuous n the closed interval [a.b] and differentiable on the open interval; (a,b), and f(a)=f(b), then there is at least one c between a and b such that f’(c) = 0

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12
Q

conditions for RT

A
  1. continuous [a.b]
  2. differentiable (a,b)
  3. f(a)=f(b)
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13
Q

what is happening to f if f’(x)>0 on [a,b]

A

f is increasing

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14
Q

what is happening to f if f’(x)<0 on [a,b]

A

f is decreasing

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15
Q

suppose that c is a critical number of a continuous function f, then f has a local maximum at c if

A

f’ changes from positive to negative

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16
Q

suppose that c is a critical number of a continuous function f, then if has a local minimum at c if

A

f’ changes from negative to positive

17
Q

let f be differentiable on an open interval, the graph of f is concave upward on f if

A

f’ is increasing on the interval

18
Q

let f be differentiable on an open interval, the graph of f is concave donward on f if

A

f’ is decreasing on the interval

19
Q

f is concave up if f”

A

f”>0

20
Q

f is concave down if f”

A

f”<0

21
Q

inflection point

A

point on the curve where curve changes concavity

22
Q

f has a local min if

A

f’=0 and f’>0

23
Q

f has a local max if

A

f’=0 and f’<0

24
Q

condition for finding local extremes using c

A

c is a critical number

25
Q

condition for finding concavity

A

f is differentiable on an open interval

26
Q

condition for inflection point

A

f is continuous

27
Q

first derivative test

A

suppose that c is a critical number of a continuous function f
then f has a local maximum at c if f’ changes from positive to negative
then if has a local minimum at c if f’ changes from negative to positive

28
Q

what happens to then slope of f’ when the graph is concave up? concave down?

A

increasing; decreasing

29
Q

even graphs

A

symmetry at y-axis
f(-x)=f(x)

30
Q

odd graphs

A

symmetry at origin
f(-x)=-f(x)

31
Q

when should you use L’hopital’s ruke?

A

when the limit of f(x) and g(x) as x approaches c both equal zero OR infinity/negative infinity

32
Q

what is l’hopital’s rule?

A

lim of f(x)/g(x) as x approaches c = lim of f’(x)/g’(x) as x approaches c

33
Q

what should we do for indeterminate products (0 x infinity) or differences (infinity - infinity)?

A

rewrite into indeterminant quotient and apply L’Hopital

34
Q

what should we do for indeterminant powers (1^infinty, 0^0, 0^infinity)?

A

rewrite as e raised to natural log and then indeterminant quotient and apply L’Hopital