Calc unit 2 derivatives

Question 1

find slope of tangent using limits

Answer 1

f(x+h) - f(x)

h

Question 2

find slope of tangent using limits at point x=a

Answer 2

f(x)-f(a)

x-a

Question 3

sum rule

Answer 3

f’(x+y) = f’(x) + f’(y)

Question 4

scalar rule

Answer 4

f’(c . y) = c . f’(x)

Question 5

horizontal tangent

Answer 5

numerator= 0

Question 6

vertical tangent

Answer 6

denomerator=0

Question 7

continuity

Answer 7

continous: make sure orig function is continous. its differentiable is derivitive is contious

Question 8

absolute value

Answer 8

set limit of both side for orig. if theyre the same, then continous at that point
then take derivative
plug in x to find the two ys. if they’re the same, then the function is differentiable at that point

Question 9

IVT

Answer 9

if function if differentiable then f’ has very y value btw y1 and y2

Question 10

with variable and continuity n absolute value

Answer 10

set both orig equation to eachother solve (A)
set derivative of the equation to eachother solve (B)
solve for intersection of A n B- elimination/substitution

Question 11

critical number

Answer 11

when derivative = o or undefined

veritcal tangent

Question 12

orig increasing

Answer 12

f’ positive

Question 13

orig decreasing

Answer 13

f’ negative

Question 14

orig max

Answer 14

f’ posit to negat

Question 15

orig mi

Answer 15

f’ neg to posit

Question 16

how to find relative max/min and intervals of ncrease/decrease

Answer 16

find critical numbers

plot on sign chart

Question 17

points of inflection

Answer 17

orig graph changes concavity
relative max/min of f’

f” changes sign

when f”=0/undefined

can’t b a point of discontinuity on orig

Question 18

how to find points of inflection/interval of concavity (always open)

Answer 18

find f”
find points of inflection
make sign chart

Question 19

orig concave up

Answer 19

f” is postive

Question 20

orig concave down

Answer 20

f” is negative

Question 21

2nd derivative

relative min at x

Answer 21

f’ at x is zero

f” at x is positive

Question 22

2nd derivative

relative max at x

Answer 22

f’ at x is zero

f” at x is negative

Question 23

chain rule

Answer 23

derivative at outside

derivative on inside

Question 24

product rule

find derivative of two function multiplying eachother

Answer 24

f(x)g(x)

f.g’ + g.f’

Question 25

quotient rule

Answer 25

lo dHi - Hi dlo

lo.lo

Question 26

sinx

Answer 26

cosx

Question 27

cosx

Answer 27

-sinx

Question 28

tanx

Answer 28

sec^x

Question 29

secx

Answer 29

tanxsecx

Question 30

cotx

Answer 30

-csc^x

Question 31

cscx

Answer 31

-cotxcscx

Question 32

how many times can you take the derivative until its zero

Answer 32

highest degree +1 = when derivative is zero

4+1=5

5th derivative=0

Question 33

mean value theorem

Answer 33

f(x) continuous on [a,b]
differentiable on (a,b) and c is btw a n b
then

slope of secant= derivative at c

slope of tangent=secant slope

Question 34

how to find c in mvt

Answer 34

verify that mtv applies on (x1,x2)

find secant slope: y2-y1 / x2,x1

set secant slope= derivative equation

Question 35

rolle’s theorem

horizontal tangent exist

Answer 35

f(x) continuous on [a,b] and differentiable on (a,b) and y value of endpoints are the same.

then there’s a point called C btw a n b where tangent at c is 0 and secant slope is 0

Question 36

L Hopital’s rule

only when limit is indeterminate

Answer 36

lim of f(x)/g(x) is f’(x)/g’(x)
take derivative of top and bottom
if answer still indeterminate then take derivative again

Question 37

types of indeterminate

Answer 37

0/0

infinity/infinty

Question 38

extreme value theorem

Answer 38

if f is continuous at closed intereval then it must have absolute max and min

Question 39

limit of

constant/ 0

Answer 39

infinity

Question 40

limit of

0/ contant

Answer 40

0

Question 41

limit of 1/infinty

Answer 41

0

Question 42

f’ (e^x)

Answer 42

e^x

Question 43

f’ (e^u)

Answer 43

u’ . e^u

Question 44

f’ (a^x)

Answer 44

x’ . a^x . lna

Question 45

f’ (lnx)

Answer 45

x’/x

Question 46

f’ (loga X)

Answer 46

x’ / x . lna

Question 47

how high does something go

Answer 47

velocity=0

plu back in position function

velocity tells direction

at rest: velocity=0

Question 48

optimalization

EVT

Answer 48

to find absolute max

find critical numbers on (a,b)
make a sign chart with endpoints too
see where relative maxes are
then plug values into original function, see which relative is highest

Question 49

applied optimalization

Answer 49

set up contrainst equation and optimization equation(which would equal what is asked for). plu constainst into optimization equation