midterm review

Question 1

at rest, v(t) = ?

Answer 1

v(t) = 0

Question 2

particle moving RIGHT (forward or up)

Answer 2

v(t) > 0 or positive

Question 3

particle moving LEFT (backward or down)

Answer 3

v(t)<0 or negative

Question 4

avg. velocity on [a, b]

Answer 4

(1/b-a) ∫ b to a v(t) dt

or x(b) - x(a) / b - a

Question 5

instantaneous velocity @ time t = a

Answer 5

v(a) = x’(a)

Question 6

acceleration at time t = c

Answer 6

a(c) = v’(c) = x’‘(c)

Question 7

velocity increasing

Answer 7

a(t) > 0 or POS

Question 8

velocity decreasing

Answer 8

a(t) < 0 or NEG

Question 9

speed =

Answer 9

|v(t)|
absolute value of v(t)

Question 10

speed increasing

Answer 10

v(b) and a(b) have SAME signs

both velocity and acceleration are pos/neg

Question 11

speed decreasing

Answer 11

v(b) and a(b) have opposite signs

velocity and acceleration are pos & neg or vice versa

Question 12

total distance traveled on [a, b]

Answer 12

∫b to a of abs. value v(t) dt

b on top of integral, and a on bottom

Question 13

net distance traveled

Answer 13

∫ b to a of v(t) dt

b on top, a on bottom (of integral)

Question 14

position of object at time t=b

Answer 14

x(b) = x(a) + ∫ b to a v(t) dt

b on top, a on bottom (of integral)

Question 15

particle is farthest left or right ?

Answer 15

compare positions (x-values) @ endpoints & @ local min/max

Question 16

velocity at t = b ?

Answer 16

v(b) = v(a) + ∫b to a of a(t) dt

b on top, a on bottom (of integral)

Question 17

moving away or towards origin at t = b ?

Answer 17

AWAY = x(b) & v(b) have SAME signs

TOWARDS = x(b) & v(b) have OPPOSITE signs

Question 18

particle moving away or towards each other

Answer 18

determine each particle’s position and velocity, make sktech and compute.

Question 19

average VALUE of a function, f(x) on [a, b]

Answer 19

1/b -a ∫b to a of f(x) dx

b on top, a on bottom (of integral)

Question 20

average RATE of f(x) on [a, b]

Answer 20

f(b) - f(a) / b - a

or

1/b-a ∫b to a of f PRIME of x dx

b on botom, a on top of integral

Question 21

slope of position = ?

Answer 21

velocity

so x’(t) = v(t)

Question 22

slope of velocity = ?

Answer 22

acceleration
so v’(t) = a(t)

Question 23

during what intervals is the particle moving left ?

Answer 23

left -> v(t) < 0

Question 24

during what intervals is the particle moving right ?

Answer 24

right -> v(t) > 0

Question 25

during what intervals is the particle's acceleration positive ?

Answer 25

a(t) > 0

Question 26

during what intervals is the particle's acceleration negative ?

Answer 26

a(t) < 0

Question 27

when is speed increasing ?

Answer 27

velocity and slope of velocity (or acceleration) have SAME signs

Question 28

when is speed decreasing ?

Answer 28

velocity and slope of velocity (acceleration) have OPPOSITE signs

Question 29

critical point

Answer 29

where the derivative of the function changes signs

Question 30

relative max = ? (think about f'(x) )

Answer 30

when f'(x) changes from POS (inc.) to NEG (dec.)

Question 31

relative min = ? (think about f'(x) )

Answer 31

when f'(x) changes from NEG (dec.) to POS (inc.)

Question 32

when is f increasing ? (given f(x) & f'(x))

Answer 32

where f'(x) is POS or f' > 0

Question 33

when is f decreasing ? (given f(x) & f'(x))

Answer 33

where f'(x) is NEG or f' < 0

Question 34

when is f concave up (given f ''(x) )

Answer 34

f''> 0

Question 35

when is f concave down (given f''(x) )

Answer 35

f'' < 0

Question 36

where is a POI ? (using f'')

Answer 36

x values that f'' changes signs at