Question

A curve has a gradient function of ax+b where a and b are constants. Given that the curve has a minimum value of y of 1 when x=1/3 and a gradient of 4 when x=1, find the equation of the curve.

Answer 1

@myininaya can you help me with this ?

Answer 2

Do you know how to integrate?

Answer 3

a bit.

Answer 4

if you know the the curve has the slope formula ax+b
then you can find the general form for a curve with that slope by integrating the slope.

Answer 5

if you dont mind, can u show me the step by drawing? it make me understand more easily

Answer 6

integrate (ax+b)?

Answer 7

ax^2/2a + bx/b

Answer 8

that's now how you integrate ax+b

Answer 9

a and b are constants

how do you integrate x? x^2/2
to integrate 1? you say what?

Answer 10

\[\int\limits_{}^{}(ax+b) dx=a \int\limits_{}^{}x dx+b \int\limits_{}^{}1 dx\]

Answer 11

oh ok. i understand that. and then?

Answer 12

what did you get after integrating (ax+b)?

Answer 13

hmm.. idk

Answer 14

have you ever seen this before:
\[\int\limits_{}^{}x^n dx =\frac{x^{n+1}}{n+1}+C \text{ where } n \neq -1 \]
\[\int\limits_{}^{}k dx=kx+C \text{ where } k \text{ is a constant }\]

Answer 15

only the upper one. the second one , nope

Answer 16

so you don't know how to integrate 5 but x^1?

Answer 17

\[\int\limits_{}^{}f dx=F+C \text{ where F'=f}\]
Like do you know how to integrate 5?

Answer 18

Like think of a function where it's derivative is 5

Answer 19

aahh, ok...

Answer 20

ok what is it?

Answer 21

if you can't integrate 5 then I don't know if I can help you with your problem because I have to go to bed soon.
I'm getting you example of how to to integrate a constant so you can integrate the constant in your problem.


Another word for integrate is find the antiderivative
Can you find the antiderivative of 5?

Answer 22

Do you know what function gives you the derivative 5?

Answer 23

no.

Answer 24

Do you know anything about derivatives?

Answer 25

dy/dx?

Answer 26

What can you put in this blank so that the below statement is true.
\[\frac{d}{dx} ( \text{ ________________) } =5\]

Answer 27

5x

Answer 28

or even 5x+c where c is a constant
\[\frac{d}{dx}(5x+c)=5 =>\int\limits_{}^{} 5 dx=5x+c\]
So the derivative of 5x+c is 5
so the antiderivative of 5 is 5x+c

Answer 29

do you see how to integrate a constant now?

Answer 30

yes,i can see it now

Answer 31

\[\int\limits_{}^{}(ax+b) dx=?\]

Answer 32

remember a and b are just constants

Answer 33

a∫x^2 +c? ax^2 +c? both wrong?

Answer 34

a∫x^2/2 + b∫x +c

Answer 35

here is another example:
\[\int\limits_{}^{}(5x+1)dx=5\frac{x^2}{2}+x+c\]
here is another another example:
\[\int\limits_{}^{} (\pi x-7)dx=\pi \frac{x^2}{2}-7x+c\]
now you try again to integrate (ax+b)

Answer 36

or did you mean to say earlier it was ax^2/2+bx+c?

Answer 37

yes

Answer 38

ok that is what our curve looks like
y=ax^2+bx+c
has the slope ax+b

great
now we just need to find a and b and c

Answer 39

it tells you what point is on the curve which is what?

Answer 40

minimum point?

Answer 41

yep and what is that point

Answer 42

idk, maybe (1/3, 1)

Answer 43

why is it maybe
that is what it says

so you can find a relationship between a,b, and c using y=ax^2/2+bx+c
using the fact you know (1/3,1) is on the curve.

Answer 44

you can replace x with 1/3 and y=1
we will come back in use whatever equation you get from that.

Answer 45

you are also given the gradient is ax+b
and they tell you some things about the gradient
they tell you the gradient is 4 when x=1
so what does that tell you about a relationship just between a and b?

Answer 46

a is gradient. b is y intercept. ?

Answer 47

no
it says the gradient is ax+b

Answer 48

that is how we were about to find the general form for the curve which is y=ax^2/2+bx+c

Answer 49

I want you to tell me a relationship between a and b given the gradient is 4 when x=1

Answer 50

what equation does that give you in terms of a and b?

Answer 51

gradient(x)=ax+b
you are given gradient(1)=4
use that to give me a relation between a and b.

Answer 52

you can do this replace gradient(x) with 4 and replace x with 1

Answer 53

m=ax+b
m=4
4=a(1)+b
4=a+b
4-b=a

Answer 54

great so we have 4=a+b
or you can express it as 4-b=a

now you are also given something else about the gradient
it says you have a min at x=1/3
what does that mean the gradient is at x=1/3?