Know Cold

Question 1

limits: vertical asymptote (x=c)

Answer 1

lim f(x) = +/-infinity
x–>c

Question 2

horizontal asymptote (y=L)

Answer 2

lim f(x) = L
x–> +/-infinity

Question 3

L’Hospital

Answer 3

f(c)/g(c) = 0/0 or infinity/infinity then lim f(x)/g(x) = lim f’(x)/g’(x)
x–> c

Question 4

When is a function continuous?

Answer 4

f(c) is defined
lim(-) f(x) = lim(+) f(x)
lim f(c) = f(c)
x–> c

Question 5

d/dx[C]

Answer 5

0

Question 6

d/dx[sin(x)]

Answer 6

cos(x)

Question 7

d/dx[cos(x)]

Answer 7

-sin(x)

Question 8

d/dx[tan(x)]

Answer 8

sec^2(x)

Question 9

d/dx[sec(x)]

Answer 9

sec(x)tan(x)

Question 10

d/dx[e^x]

Answer 10

e^x

Question 11

d/dx[a^x]

Answer 11

ln(a)a^x

Question 12

d/dx[ln(x)]

Answer 12

1/x

Question 13

d/dx[cot(x)]

Answer 13

-csc^2(x)

Question 14

d/dx[csc(x)]

Answer 14

-csc(x)cot(x)

Question 15

d/dx[x^(n-1)]

Answer 15

nx^(n-1)

Question 16

IVT

Answer 16

if f(x) is continuous on the the interval [a,b] and Y is between f(a) and f(b) there exist a c value within [a,b] where f(c) = Y exists

Question 17

MVT

Answer 17

if f(x) is continuous on [a,b] and differentiable on (a,b) then there exists some C value within (a,b) such that f’(c) = f(b)-f(a)/b-a

Question 18

EVT

Answer 18

if f(x) is continuous on [a,b] then f(x) has an absolute maximum and absolute minimum on [a,b]

Question 19

total distance traveled

Answer 19

integral of the absoulte value of v(t)

Question 20

displacement

Answer 20

integral of v(t)

Question 21

volume of solids (cross sections) - square

Answer 21

integral S^2

Question 22

volume of solids (cross sections) - rectangles

Answer 22

integral B * H

Question 23

volume of solids (cross sections) - isosceles right leg

Answer 23

1/2 integral L^2

Question 24

volume of solids (cross sections) - isosceles right hypotenuse

Answer 24

1/4 integral H^2

Question 25

volume of solids (cross sections) - equilateral triangle

Answer 25

sqrt3/4 integral S^2

Question 26

volume of solids (cross sections) - semicircles

Answer 26

pi/8 integral D^2

Question 27

volume of solids (discs and washers) - discs

Answer 27

pi integral r^2

Question 28

volume of solids (discs and washers) - washers

Answer 28

pi integral (R^2 - r^2)

Question 29

Arc Length

Answer 29

s = integral sqrt(1+[f’(x)]^2)

Question 30

critical number

Answer 30

f’(x) = 0 or DNE

Question 31

increasing

Answer 31

f’(x) > 0

Question 32

decreasing

Answer 32

f’(x) < 0

Question 33

concave up

Answer 33

f’‘(x) > 0

Question 34

concave down

Answer 34

f’‘(x) < 0

Question 35

relative min

Answer 35

f’(x) changes - to +

Question 36

relative max

Answer 36

f’(x) changes + to -

Question 37

inflection point

Answer 37

f’‘(x) changes sign or f’(x) has a relative min or max

Question 38

fraction decomposition

Answer 38

integral 1/(x-a)(x-b) = integral A/(x-a) + B/(x-b)

Question 39

partial decomposition

Answer 39

intergral udv = uv - integral vdu

Question 40

euler’s method when change in x and (x(0), y(0)) given

Answer 40

dy/dx = F(x,y)

Question 41

euler’s method when y(n+1) = y(n) + (change in x)F(x(n), y(n))

Answer 41

x(n+1) = x(n) + change in x

Question 42

Exponential Model

Answer 42

dy/dx = ky = Ce^kt

Question 43

Logistic Model

Answer 43

dy/dx = kLy (1-k/L)
dy/dx = ky (L-y)
y = L/(1+Ce^(-kLt))

Question 44

Logistic –> L = ?

Answer 44

carrying capacity

Question 45

Logistic –> 2/L = ?

Answer 45

inflection point