Lachlan1996
given that f(x) = ax^3 + bx^2, f(2)=-4 and f'(3)=99 find f(x), f(3) and f'(2)
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Lachlan1996
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i cant find a way to isolate a or b to solve
tjones89
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have you found the derivative?
Lachlan1996
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the derivative is 3ax^2 + 2bx but that doesnt help solve it
mathslover
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f(x) = ax^3 + bx^2
f(2) = 8a + 4b = -4
f(3) = 27a + 9b = 99
mathslover
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8a + 4b = -4
27a+9b=99
from equation 1 , we have : a = \(\LARGE\frac{-4-4b}{8}=\frac{-1-b}{2}\)
put this value in the second equation.
mathslover
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Can you now solve for a, b ? @Lachlan1996
Lachlan1996
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wait wait, hold on youve lost me here.
mathslover
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lol, take your time. Sorry :)
Lachlan1996
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uhhh i got 27a + 6b =99 but that is the gradient as in the derivative
Lachlan1996
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so i dont think you can simultaneously solve it
Lachlan1996
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because you are simultaneously solving a derivative and a point
EulerGroupie
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Yes! Now use the f(2)=-4 in the original equation to get your second equation with the two unknowns... a and b
EulerGroupie
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Two equations with two unknowns should be solvable. Pick your favorite method for solving systems... @mathslover is using substitution.
Lachlan1996
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but he is solving a f' (x) with a f (x)
EulerGroupie
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a and b will be the same in the original and the derivative
Lachlan1996
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ah yes i see
mathslover
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f(2) = 8a + 4b = -4
f(3) = 27a + 9b = 99
a = (-1-b)/2
put this in f(3) equation.
27(-1-b)/2 + 9b = 99
(-27-27b+18b)=198
-27 -9b = 198
-9b = 198+27 = 225
b = -225/9
b = -25
You can solve for a now.
Lachlan1996
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in the above equation isnt it meant to be 6a not 9a though?
Lachlan1996
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wait no i mean 6b not 9b
EulerGroupie
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Its actually a really cool way to come up with a system... one eq from the original and one from the derivative. Slick, huh?!
correct... I got 6b rather than 9b as well
EulerGroupie
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27a+6b=99
8a+ 4b=-4
Lachlan1996
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uhh okie doke so a=0.5b-0.5
Lachlan1996
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however substituting that into 27a + 6b = 99 i got 8.333333 as being b
EulerGroupie
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I got a=7, b=-15
Lachlan1996
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oh no made an error forgot to subtract 6b
mathslover
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OH MY FAULT , sorry :(
Lachlan1996
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no, no it was a simple mistake, i just wasnt sure if you had made a mistake or were doing something else
Lachlan1996
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this seems like alot of work just to start off the question, do you think this would be in an exam?
mathslover
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Depends but I do think, Yep! :)
EulerGroupie
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For calculus, it doesn't seem out of bounds for a test.
Lachlan1996
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ahh god, in a year 11 maths exam? i think im buggered then
mathslover
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Whether it will come or not in exam but I think you must be PREPARED
Lachlan1996
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mmm yes my exam is in 5 weeks.
EulerGroupie
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Creating a small system and solving it with substitution is fairly standard in calculus.
mathslover
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You have much time for it, go through these type of questions and ask your problems here...
Lachlan1996
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okie doke, thanks very much for the help mate, its much appreciated. thankyou to you both
EulerGroupie
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You're welcome.
mathslover
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Welcome! :)
Lachlan1996
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uhhhh small problem
Lachlan1996
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we got the answer wrong? its meant to be y=7x^3 -15x^2
mathslover
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Try to do it again, I think calculation mistake?
EulerGroupie
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yes... a=7, b= -15
Lachlan1996
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ah yes, last step of my working didnt put the -ve in the 15
mathslover
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hmn, Take care of these small mistakes .
EulerGroupie
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whew! you scared me ;)
Lachlan1996
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will do, doesnt help when you get frustrated but. maths is not easy for me
sauravshakya
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I think it is f'(3)=99 not f(3)=99