Question

Find the coordinate(s) of the points on the curve x^3 − 3x^2 where the tangent:

a) is parallel to the x-axis

b) is parallel to the line y = −3x + 2.

Answer 1

(a) parallel to the x axis means the same gradient.. so gradient is defined as
(y2-y1)/(x2-x1), ur y value would be the same as its just a horizontal line, so ur gradient have to be 0

Answer 2

differentiate x^3 -3x^2.. u get 3x^2 - 6x = 0
solve for x..
3x^2= 6x
x^2=2
x= + root 2 n negative root 2

Answer 3

once u get ur x values then sub in to ur original equation to get ur y coordinate

Answer 4

but the answers in the back of the book don't have the x values you stated

Answer 5

for (b), differentiate the line y = -3x + 2, u get -3
then 3x^2 - 6x = -3
solve for x...
3(x^2 -2x +1 )=0
3(x-1)(x-1)=0
x has to be equal to negative 1.. hten sub in to ur equation and get ur y coordinate

Answer 6

whats the answer at the back?

Answer 7

(0,0) and (2,-4)

Answer 8

3x(x-2) = 0

Answer 9

sorry i calculated wrongly

Answer 10

when u reached 3x^2 - 6x = 0, u factorize the common factor out
3x(x-2)=0
so u get x =2 and x = 0

Answer 11

but u get what i am doing
?

Answer 12

some, well I got the x=0 or x=2 after factorizing the common factor, just as u did.

Answer 13

but may I ask, why do we have to solve for x?

Answer 14

u have to solve for x cause u want ur x coordinate o.O? then u can sub ur x coordinate back to the equation x^3 - 3x^2 to get ur y coordinate

Answer 15

yeh and what is the purpose of it then*? I know it's to find the x and y coordinate and the point ultimately.

Answer 16

dnt really get what u mean by whats the purpose.. that point tells u that the gradient is 0

Answer 17

don't worry.

Answer 18

sry :(

Answer 19

it's fine, it's just me, cause I get quite confused sometimes and I like to or at least try to understand things properly.

Answer 20

in a logical order* if that makes sense.

Answer 21

a) parallel to the x-axis = gradient equal to zero
gradient = zero when the function is at a maximum or a minimum
to find the max/min of the function, differentiate the function
so dy/dx= 3x^2 - 6x
make it equals to zero
0= 3x^2-6x
0=x(3x-6)
so x= 0 or 3x-6=0
for that we get x=0 or x=2

so sub the x value into the equation y=x^3-3x^2
u get y=0 when x=0 coordinate = (0,0)
y=-4 when x=2 coordinate = (2,-4)

Answer 22

b) parallel to the line y=-3x+2
gradient must be = -3 to be parallel
dy/dx is the gradient function of the function y

so to find the coordinate:

dy/dx= 3x^2-6x
m(gradient)=-3

so -3=3x^2-6x (gradient of line equals to the gradient of the tangent of that point)
solve that we get x=1
( the x value for that coordinate)

so sub x=1 into the function y=x^3-6x
we get y=-2

so coordinate = (1,-2)