Unit 2

Question 1

what does it mean when the function is differentiable at a point?

Answer 1

the derivative of the function exits at that point

Question 2

what is a differentiable function?

Answer 2

a function that is differentiable at every point

Question 3

what are derivatives used for?

Answer 3

used to measure the rates at which things change

Question 4

what is the first derivative of a function?

Answer 4

the slope of the tangent to the curve

also the instantaneous rate of change

Question 5

derivatives are in other word what of average changes?

Answer 5

limiting values of average changes

(slope of the secant line approaches the slope of the tangent line)

Question 6

what four cases do not have a derivative?

Answer 6

corner, cusp, vertical tangent, discontinuity

Question 7

what is a corner

Answer 7

one sided derivatives differ

Question 8

what is a cusp?

Answer 8

slopes of secant lines approach infinity from one side and negative infinity from the other

Question 9

what is a vertical tangent?

Answer 9

where the slopes of the secant line approach either infinity or negative infinity from both sides

Question 10

what is a discontinuity?

Answer 10

one or both of the one-sided derivatives are nonexistent

Question 11

what does differentiability imply?

Answer 11

local linearity and continuity

Question 12

what does continuity not imply?

Answer 12

differentiability

Question 13

d/dx [cosx]

Answer 13

-sinx

Question 14

d/dx [sinx]

Answer 14

cosx

Question 15

d/dx [tanx]

Answer 15

sec^2(x)

Question 16

d/dx [cotx]

Answer 16

-csc^2(x)

Question 17

d/dx [secx]

Answer 17

secxtanx

Question 18

d/dx [cscx]

Answer 18

-cscxcotx

Question 19

product rule

Answer 19

d/dx (uv)= uv’+vu’

Question 20

quotient rule

Answer 20

d/dx (u/v)= vu’-uv’/v2

Question 21

chain rule

Answer 21

d/dx (f(g(x))) = f’(g(x)) (g’(x))

Question 22

dy/dx =

Answer 22

(dy/du) (du/dx)

Question 23

average velocity on the interval [t1,t2]

Answer 23

Δs/Δt = [s(t1)-s(t2)]/ (t1-t2)

Question 24

average acceleration on the interval [t1,t2]

Answer 24

Δv/Δt = [v(t1)-v(t2)]/ (t1-t2)

Question 25

velocity

Answer 25

v(t) = s’(t) = ds/dt

Question 26

what does it mean when the sign changes for velocity

Answer 26

particle reverses direction

Question 27

when is v(t) +/-?

Answer 27

+ when s(t) is increasing
- when s(t) is decreasing

Question 28

accerleration

Answer 28

a(t) = v’(t) = dv/dt= s”(t)= d2s/dt

Question 29

speed

Answer 29

magnitude of velocity: |v(t)|

Question 30

when is a particle speeding up?

Answer 30

v and a have the same sign

Question 31

when is a particle slowing down?

Answer 31

v and a have different signs

Question 32

jerk

Answer 32

a’(t)

Question 33

at what velocity is a particle at max height?

Answer 33

0

Question 34

when does particle hit ground?

Answer 34

h=0

Question 35

at what velocity does a particle hit ground?

Answer 35

find zero and plug into velocity equation

Question 36

when is a particle moving in positive direction?

Answer 36

v(t)>0

Question 37

when is a particle at rest?

Answer 37

v(t)=0

Question 38

given f(x) is differntiable, g(x) = f-1(x) , and f’(x) does not equal 0 the,

Answer 38

g’(f(x)) = 1/f’(x)

Question 39

pi/2-arcsinx=

Answer 39

arccosx

Question 40

pi/2-arctanx=

Answer 40

arccotx

Question 41

pi/2-arcsecx=

Answer 41

arccscx

Question 42

d/dx arcsinu

Answer 42

(1/√1-u^2)(du/dx)

Question 43

d/dx arctanu

Answer 43

(1/1+u^2)(du/dx)

Question 44

d/dx arcsecu

Answer 44

(1/|u|√u^2-1)(du/dx)

Question 45

d/dx arccosu

Answer 45

(-1/√1-u^2)(du/dx)

Question 46

d/dx arccotu

Answer 46

(-1/1+u^2)(du/dx)

Question 47

d/dx arccscu

Answer 47

(-1/|u|√u^2-1)(du/dx)

Question 48

d/dx (e^x)

Answer 48

e^x

Question 49

d/dx (lnx)

Answer 49

1/x

Question 50

d/dx (a^x)

Answer 50

(a^x)(lna)

Question 51

d/dx (logax)

Answer 51

1/xlna

Question 52

d/dx (e^u)

Answer 52

e^u(du/dx)

Question 53

d/dx (lnu)

Answer 53

1/u(du/dx)

Question 54

d/dx (a^u)

Answer 54

(a^u)(lna)(du/dx)

Question 55

d/dx (logau)

Answer 55

(1/ulna)(du/dx)

Question 56

Δx =

Answer 56

dx=(x+Δx)-(x); x2-x1

Question 57

Δy=

Answer 57

f(x+Δx)-f(x); y2-y1

Question 58

dy=

Answer 58

f’(x)(dx)

Question 59

diff bw Δy and dy

Answer 59

Δy is the exact change