Test 3

Question 1

population growth equation

Answer 1

y=ce^kt ; k is greater than 0

Question 2

radioactive decay equation

Answer 2

y=ce^kt ; k is less than 0

Question 3

Recall:
1. ln(AB) =
2. ln(A/B) =
3. ln(A^n)=

Answer 3

1. ln (A) + In (B)
2. ln (A) - In (B)
3. n ln A

Question 4

Indeterminate Forms?

Answer 4

“0/0”, “∞/∞”, “0-∞”, “∞-∞”

Question 5

L’Hopital’s Rule

Answer 5

If lim x->a = 0/0 or ∞/∞ forms, then lim x->a f (x)/g(x) ?= f ‘(x)/g’(x).

Question 6

Review of some limits
1. lim x->∞ . e^x=
2. lim x->-∞ e^x=
3. lim x->∞ ln(x)=
4. lim x->0+, ln(x)=

Answer 6

1. e^x= ∞
2. e^x= 0
3. ln(x)= ∞
4. ln(x)= -∞

Question 7

critical numbers

Answer 7

C is a critical number of f if
f ‘(c) = 0 , f ‘(c) DNE and c is in the
domain of f.

Question 8

steps to finding critical numbers

Answer 8

1. find derivative of equation given
2. solve for zero
   *if it is 0, it DNE/ undefined.

Question 9

first derivative test

Answer 9

if c is a critical number of f and the sogn of f chnages from positive to negative at c, then its a local max. If the sign of g changes from negative to positive at c, the f (c) is a local min.

*if the sign doesn’t change at c then f (c) is neither a local max or a local min.

Question 10

theorem 19

Answer 10

if f(c) is a local max or min, then c is a critical number of f.

Question 11

steps on finding the local min/max

Answer 11

one: find critical numbers
two: look at sign of f’ in between critical nmbers (so create table with test value, sign,)

Question 12

extreme value theorom

Answer 12

if f is continous on a closed interval [a,b] them there are number c and d m[a,b] such that f(c) is abs max and f(d) is abs min)

Question 13

Steps to find absolute maximums and absolute minimums

Answer 13

1. Find critical numbers in (a, b)
2. Plug in the critical number in (a, b), a, and b into f .
3. The largest value of f is the absolute maximum and the smallest, the absolute minimum

Question 14

Concavity?

Answer 14

F is concave down if f’ decreases and f’’ < 0.

f is concave up is if f’ increases and f’’ > 0

Question 15

inflection point

Answer 15

(d, f(d)) is an inflection point of f if the concavity of f i.e the sign of f’’ changes at d and d is in the domain of the function.

Question 16

Rolles Theorom

Answer 16

Let f be a continuous function at [a,b] and differentiable in (a,b). If f(a)=f(b) then theres c in (a,b) such that f’c=0.

Question 17

Mean Value Theorom

Answer 17

let f be a cont. function at [a,b] and differentiable in (a,b). Then there is c in (a,b) such that f’c= f(b)-f(a)/ b-a.

Question 18

second derivative test:

Answer 18

if c is a critical number of f and f’‘c > 0, then f(c) is a local min of f.

Question 19

Optimization steps:

Answer 19

1. read steps
2. assign variables/ draw diagram
3. condition equation, function to max/min
   4.solve condition for 1 variable and plug into function
4. critical numbers of function
5. show that critical numbers yields max/min
6. answer question