Need help with differentiation + intergration

Just got this question and im stuck :frown:

much appreciated if people know how to do it.

4. The curve C with equation y = f(x) is such that

dy/dx = 3√x + 12/√x , x > 0.

(a) Show that, when x = 8, the exact value of dy/dx is 9√2.

The curve C passes through the point (4, 30).

(b) Using integration, find f(x).

Reply 1

kikzen

14

a)x=8

then dy/dx = 3rt(8) + 12rt(8)/8 [rationalising the denominator]
=3*2*rt(2)+12*2*rt(2)/8
=rt(2)[6+3]
=9rt(2)

b) dy/dx = 3√x + 12/√x , x > 0.
remember, rt(x) = x^(1/2); 1/rt(x) = x^(-1/2)
so
intemagrating,
y = 3(2/3)x^3/2 + 24x^(1/2) + c

and the curve goes through (4, 30)
so plug those in
30 = 2(8) + 24(2) + c
c = -34

y = 3(2/3)x^3/2 + 24x^(1/2) -34?

along those lines. i may have messed up some arithmetic or something though

Reply 2

Simba

16

Fenchurch

Just got this question and im stuck :frown:

much appreciated if people know how to do it.

4. The curve C with equation y = f(x) is such that
dy/dx = 3√x + 12/√x , x > 0.
(a) Show that, when x = 8, the exact value of dy/dx is 9√2.
The curve C passes through the point (4, 30).
(b) Using integration, find f(x).

a) At x = 8, dy/dx = 3√8 + 12/√8
= 6√2 + (12√8)/8
= 6√2 + 3√2
= 9√2

b) y = ∫(3x^1/2 + 12x^-1/2)dx
= 2x^3/2 + 24x^1/2 + c

Sub in x = 4 and y = 30:

30 = 16 + 12 + c

c = 2

Therefore y = 2x^3/2 + 24x^1/2 + 2

Hope this helps,

~~Simba

Reply 3

Fenchurch

OP

Cheers Simba!!!

One thing though, how can you be in year 11 if you got your gcse marks, or you did french early as well as maths?

Reply 4

Fenchurch

OP

quick question, il be back to ask for more soon no doubt but

if u got

x²=y²+7
x=2y-2

as a simultaneous equation

getting it o

2y-x=2
y²-x²=-7

i was wondering what step you would use next to make the y or x equal, not sure bout wether you square one then leave the other

Reply 5

mikeyT

2

Fenchurch

quick question, il be back to ask for more soon no doubt but

if u got

x²=y²+7
x=2y-2

as a simultaneous equation

getting it o

2y-x=2
y²-x²=-7

i was wondering what step you would use next to make the y or x equal, not sure bout wether you square one then leave the other

When you have quadratic terms you should get two answers.
Just start at the beginning and don't bother putting all the variables on one side.
Use x=2y-2 and sub this into the other equation
(2y-2)²=y²+7
Multiply out and rearrange
3y²-8y-3=0
(3y+1)(y-3)=0
So y=-1/3 and y=3
Then x=-8/3 when y=-1/3
and x=4 when y=3

Need help with differentiation + intergration

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