Question

given the function g(x) = -(1553/25920)*x^4-(3301/8640)*x^3+(3151/480)*x^2+(3181/240)*x-5747/40 . Find the tangent line that touches the curve twice.

Answer 1

\[g(x)=-(1553/25920)x^4-(3301/8640)x^3+(3151/480)x^2+(3181/240)x-5747/40\]

Answer 2

If we assume a single tangent line touches the curve at the points (a,b) and (c,d)
Then the two points should satisfy g(x) from which we can get two equations.
We can also set g'(a) = g'(c) for the third equation
and the slope of the line (d-b)/(c-a) = g'(a) (or g'(c)) for the fourth equation.
Four equations and four unknowns.

Answer 3

b = g(a)
d = g(c)
g'(a) = g'(c)
(d-b)/(c-a) = g'(a)

$ unknowns: a, b, c and d and four equations.

Answer 4

$ should be 4.

Answer 5

equation 1 is \[g(a)=-(1553/25920)a^4-(3301/8640)a^3+(3151/480)a^2+(3181/240)a-5747/40\]
equation 2 is\[g(c)=-(1553/25920)c^4-(3301/8640)c^3+(3151/480)c^2+(3181/240)c-5747/40\]
correct so far?

Answer 6

i am on a really slow computer so please be patient :/ ur help will be greatly appreciated!!

Answer 7

Are these numerical analysis / computer simulation type of problems?
The coefficients look awful and solving the 4 simultaneous equations involving 4th power will be a real challenge.

Answer 8

For the above two equations I will put b on the left of first equation and and d on the left of the second.

Answer 9

it's computer simulated, so i just need to type in those equations and the computer will do it for me. the program i am using is mapple.

Answer 10

okay. cool.

Answer 11

equation 1 is
b=-(1553/25920)a^4 -(3301/8640)a^3 + (3151/480)a^2 + (3181/240)a - 5747/40
equation 2 is
d=-(1553/25920)c^4 -(3301/8640)c^3 + (3151/480)c^2 + (3181/240)c - 5747/40
like this?

Answer 12

yes. a,b,c and d are the unknowns we need for the computer to solve.
We just need two more equations.
So it is time for g'(x)

Answer 13

g(x) = -(1553/6480)*x^3 -(3301/2880)*x^2 + (3151/240)*x + 3181/240

Answer 14

g'(x) = -(1553/6480)*x^3 - (3301/2880)*x^2 + (3151/240)*x + 3181/240

Answer 15

I am not checking your math and so I am assuming you are doing it right. It might be better to double check. Easy to make mistake with so many terms and awful coefficients. Or you can solve the problem and see if the answer matches. If not then you can go look for errors. But the concept is correct.

Answer 16

If equating g'(a) to g'(c) is a nightmare, I'd instead use the following for the third and fourth equations:
(d-b)/(c-a) = g'(a)
(d-b)/(c-a) = g'(c)

Answer 17

i used the computer to solve it so dont worry :). so from this do i sub a in for x and then c in for x, and show them equal to each other like this?
g'(a) = -(1553/6480)*a^3 - (3301/2880)*a^2 + (3151/240)*a + 3181/240
g'(c) = -(1553/6480)c^3 - (3301/2880)*c^2 + (3151/240)*c + 3181/240
Therefore
-(1553/6480)*a^3 - (3301/2880)*a^2 + (3151/240)*a + 3181/240 = -(1553/6480)*c^3 - (3301/2880)*c^2 + (3151/240)*c + 3181/240
like this?

Answer 18

see my reply above. Replace g'(a) and g'(c) on the left with (d-b)/(c-a)

Answer 19

awwwww i now see where u are going with this :)))

Answer 20

or i think i do xD

Answer 21

ok 1 sec, ill try and do that now

Answer 22

so do i know tell the computer to solve for the variables given these 4 equations?

Answer 23

Yes. Once you find (a,b) and (c,d) you can find the equation for the tangent line knowing two points on the line:
Slope m = (d-b) / (c-a)
y - b = m(x - a)

Answer 24

Refer to the Mathematica attachment for a plot and solution.