Unit 3

Question 1

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

Answer 1

time

Question 2

volume of a sphere

Answer 2

(4/3) · π · r3

Question 3

surface area of sphere

Answer 3

(4) · π · r2

Question 4

volume of cone

Answer 4

(1/3) · π · r2 · h

Question 5

when is x=c a critical number

Answer 5

if f(c) exists and f’(c)=0 or dne

Question 6

Extreme Value Theorem

Answer 6

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

Question 7

can absolute extremas be endpoints?

Answer 7

yes

Question 8

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

Answer 8

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

Question 9

Mean Value Theorem

Answer 9

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

Question 10

conditions for MVT

Answer 10

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

Question 11

Rolle’s Theorem

Answer 11

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

Question 12

conditions for RT

Answer 12

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

Question 13

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

Answer 13

f is increasing

Question 14

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

Answer 14

f is decreasing

Question 15

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

Answer 15

f’ changes from positive to negative

Question 16

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

Answer 16

f’ changes from negative to positive

Question 17

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

Answer 17

f’ is increasing on the interval

Question 18

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

Answer 18

f’ is decreasing on the interval

Question 19

f is concave up if f”

Answer 19

f”>0

Question 20

f is concave down if f”

Answer 20

f”<0

Question 21

inflection point

Answer 21

point on the curve where curve changes concavity

Question 22

f has a local min if

Answer 22

f’=0 and f’>0

Question 23

f has a local max if

Answer 23

f’=0 and f’<0

Question 24

condition for finding local extremes using c

Answer 24

c is a critical number

Question 25

condition for finding concavity

Answer 25

f is differentiable on an open interval

Question 26

condition for inflection point

Answer 26

f is continuous

Question 27

first derivative test

Answer 27

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

Question 28

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

Answer 28

increasing; decreasing

Question 29

even graphs

Answer 29

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

Question 30

odd graphs

Answer 30

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

Question 31

when should you use L’hopital’s ruke?

Answer 31

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

Question 32

what is l’hopital’s rule?

Answer 32

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

Question 33

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

Answer 33

rewrite into indeterminant quotient and apply L’Hopital

Question 34

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

Answer 34

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