The 1st and 2nd Derivatives & Applications of Derivatives

Question 1

What is f’(x)

Answer 1

Instantaneous rate of change - The gradient of the tangent to the curve at a point

Question 2

if f’(x) >0

Answer 2

gradient is increasing

Question 3

if f’(x) <0

Answer 3

gradient is decreasing

Question 4

if f’(x) =0

Answer 4

gradient is stationary

Question 5

Stationary point

Answer 5

gradient of the tangent to the curve at the point is horizontal

Question 6

Maximum turning point

Answer 6

gradient goes from-ve –>0–>+ve

i.e /-\

Question 7

Minimum turning point

Answer 7

gradient goes from +ve –> 0 –> +ve

i.e _/

Question 8

point of inflection

Answer 8

point on curve where tangent crosses the line

i.e the concavity changes

Question 9

local maximum

Answer 9

Given point (a, f(a))
if f(x) is less than or equal to f(a)

Question 10

local minimum

Answer 10

given the point (a, f(a))
if f(x) is greater than or equal to f(a)

Question 11

how to find stationary points

Answer 11

Let the first derivative =0

Question 12

global maximum

Answer 12

the highest point at the endpoint of a given domain

Question 13

global minimum

Answer 13

lowest point at the other endpoint of a given domain

Question 14

how to find max and min value of a function in a given domain

Answer 14

sub each value of x in the domain into the function

Question 15

Second derivative

Answer 15

the rate of change of the first derivative (the rate of change of the gradient)

Question 16

if f’‘(a)>0

Answer 16

concave upwards and there is a minimum turning point there

Question 17

if f’‘(a)<0

Answer 17

concave downwards and there is a maximum turning point there

Question 18

if y’‘=0

Answer 18

there is an inflection point at a point on the curve AND concavity changes at this point

Question 19

max turning point…

Answer 19

y’=0 and y’‘<0

Question 20

min turning point…

Answer 20

y’=0 and y’‘>0

Question 21

horizontal point of inflection…

Answer 21

y’=0, y’‘=0 and concavity changes

Question 22

when y’‘(a)=0

Answer 22

more work is needed to decide the nature of the point

Question 23

what table to use with f(x)

Answer 23

table of signs –> to find where the function is positive or negative

Question 24

what table to use with f’(x)

Answer 24

table of slopes –> to determine the nature of stationary points

Question 25

what table to use with f’‘(x)

Answer 25

table of concavity –> to find any points of inflection and the concavity of the function

Question 26

point of inflection

Answer 26

concavity changes

Question 27

how to find stationary points

Answer 27

let y’=0

Question 28

how to find global max and min

Answer 28

examine and compare the: turning points, boundaries of the domain (or behaviour for large x) and any discontinuities of f’(x) (when y=0/0)

Question 29

steps to find the nature of stationary points

Answer 29

derive, solve y’=0
if stationary points exist –> sub into original to find coords
easy to differentiate–> second derive –> sign of y’’
- zero –> table of concavities
- positive –> min turning point
- negative –> max turning point
not easy to differentiate –> table of slopes with y’
_/ –> min turning point
/-\ –> max turning point
— –> horizontal point of inflection

Question 30

how to solve max/min qs

Answer 30

the three cs

construct: draw a diagram, assign pronumerals/form equations, note restrictions
calculus: differentiate, find T.Ps then find nature
conclude: evaluate and compare T.Ps with endpoints/discontinuities, state final answer

Question 31

Displacement

Answer 31

Relative position to starting point

Question 32

when displacement is negative

Answer 32

particle is left of origin

Question 33

when displacement is positive

Answer 33

particle is right of origin

Question 34

velocity

Answer 34

rate of change of position with respect to time

Question 35

when velocity is negative

Answer 35

particle is travelling left

Question 36

when velocity is positive

Answer 36

particle is travelling right

Question 37

speed=

Answer 37

the absolute value of velocity

Question 38

acceleration

Answer 38

rate of change of velocity with respect to time

Question 39

when acceleration is negative

Answer 39

acting to the left

Question 40

when acceleration is positive

Answer 40

acting to the right

Question 41

speeding up=

Answer 41

velocity and acceleration acting in the same direction

Question 42

slowing down=

Answer 42

velocity and acceleration acting in opposite direction

Question 43

“constant velocity”=

Answer 43

when a=0

Question 44

“at rest”=

Answer 44

when v=0

Question 45

“at the origin”=

Answer 45

when x=0

Question 46

First derivative of displacement (x dot)

Answer 46

velocity

Question 47

Second derivative of displacements (x double dot)

Answer 47

acceleration

Question 48

second derivative shows the…

Answer 48

concavity

Question 49

To find max/min of displacement

Answer 49

solve f’(t)=0 (velocity=0)

Question 50

To find max/min of velocity

Answer 50

solve f’‘(t)=0 (acceleration=0)