Hey, anyone good at tangents?
10(iv) Given that 24x+3y+2=0 is the equation of the tangent to the curve at the point (p,q), find p and q. (The curve is y=1/3x³-9x.)
Thank you very much
Here's how i found the possible x-values, maybe you or someone else can figure out which is the right one without calculus.
p and q are just the x and y values (respectively) for which the line and the curve intersect.
now 24x + 3y + 2 = 0
=> 3y = -24x - 2
=> y = -8x - 2/3
now equate the functions:
since y = -8x - 2/3 and y = 1/3x³-9x
then they intersect when
-8x - 2/3 = (1/3)x³ - 9x
=>(1/3)x^3 - x + 2/3 = 0
=> x^3 - 3x + 2 = 0
we see that x = 1 is a root, so x - 1 is a factor. divide by x - 1. we obtain:
x^3 - 3x + 2 = (x - 1)(x^2 + x - 2) = (x - 1)(x + 2)(x - 1) = 0
so x = 1 or x = -2
these are the possible values for p, at one of these points we have the tangent, i just can't find a simple way to figure out which i don't want to use calculus since you're doing AS math and you're not comfortable with it
i can tell you the answer is x = p = 1, and you could just plug that value into either of the other functions to get q, but i used calculus to come to that conclusion
now we found the equation of the line to be:
y = -8x - 2/3
this is a line with slope -8
now all we need to do, is find the points on the cubic graph where the slope is -8. whichever value corresponds with the values we got before is the right one. (Remember, we said the tangent is either at x = 1 or x = -2)
let's find the formula for the slope of the curve at any value of x. we use the derivative for this. that is, dy/dx gives the slope of the tangent line for any value of x.
y = (1/3)x^3 - 9x ..............take the derivative, we get:
=> dy/dx = x^2 - 9
we want x^2 - 9 = -8 ........that is, we want the slope of a point on the cubic to be equal to the slope of the tangent line. so,
x^2 - 9 = -8
=> x^2 - 1 = 0
=> (x + 1)(x - 1) = 0
=> x = -1 or x = 1
now the line only cuts the curve at x = 1 and x = -2. since it does not cut the curve at -1, x = 1 must be our answer.
so x = p = 1
now to find q
we have y = -8x - 2/3, so when x = 1 we have,
y = -8 - 2/3 = -26/3
so (p,q) = (1, -26/3)
Hope you could follow that. if not, wait on one of the smart people to give you an easier way
when i did the second part, i found what points on the curve had the same slope of the line. x = -2 was not such a point, so i discarded it. for the other point, x = -1, the line we were talking about did not even touch the curve (since the line ONLY touches the curve at x = -2 and x = 1 as we found), so that could not be it either. so the only possible answer is x = 1