This was a multiple choice question asking which of the following is an equation of the line tangent to the graph of f(x)=x^4+2x^2 at the point where f'(x)=1?
and the answer is y=x-0.122, but I don't understand why.
I tried to take the derivative of f(x) and got 4x^3+4x = 1 and solved for x, but I am not getting the answer. What am I doing wrong and how do I solve this?
Thanks!
you did nothing wrong, you are just faced with an equation very hard to solve.
4x^3 + 4x - 1 =0
I got x = .2367329 by a method called Newton's Method, which is sort of hard to explain in this forum.
substituting back in the original I go y=.1152257
so we have the point (.2367,.1152) and slope of 1
When used in y=mx+b I indeed got
y = x - .1215, they rounded it off to 3 decimals.
I don't know if you are in highschool or college, so I don't know what kind of tools you have to solve equations of that type.
how did you get b=.1215 though?
y=mx+b
.2367 = .1152 + b
b=.2367-.1152
b=.1215
The equation is not difficult to solve at all! You only need to know is a simple trick:
4x^3+4x - 1 = 0 --->
x^3 = -x + 1/4 (1)
The trick is to compare this with the equation for (a+b)^3, which you can write as:
(a+b)^3 = 3 a b(a+b) + a^3 + b^3 (2)
If you compare (2) to (1), you see that if you can find values for a and b such that
3 a b = -1 and a^3 + b^3 = 1/4 (3)
then a + b is a solution to the equation! But finding the numbers a and b such that (3) is satisfied amounts to solving a quadratic equation:
3 a b = -1 --->
a^3 b^3 = -1/27
So, if we put A = a^3 and B = b^3, then we have:
A B = -1/27
A + B = 1/4
A solution is:
A = 1/8 + sqrt[1/64 + 1/27]
B = 1/8 - sqrt[1/64 + 1/27]
And a solution for x is:
x = a + b = A^(1/3) + B^(1/3) =
[sqrt(1/64 + 1/27) + 1/8]^(1/3) -
[sqrt(1/64 + 1/27) - 1/8]^(1/3)
which gives the same result as Reiny gave above.
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(1/64)^x=Sqrt1/8
The equation that you obtained, 4x^3 + 4x = 1, is indeed the correct equation to find the x-coordinate of the point at which the slope is 1. However, solving this equation directly can be challenging and may require more advanced methods.
One approach to solving this equation is by using numerical methods like Newton's Method. Newton's Method is an iterative numerical method that can approximate the roots of a function. In this case, we want to find the value of x at which 4x^3 + 4x = 1.
To use Newton's Method, you initially make a guess for the value of x and then iteratively refine that guess until you reach a desired level of accuracy. The formula for using Newton's Method is:
x_(n+1) = x_n - (f(x_n) / f'(x_n))
Here, f(x) represents the function 4x^3 + 4x - 1, and f'(x) represents its derivative, which is 12x^2 + 4.
Let's say we start with an initial guess of x = 1. Plugging this value into the formula, we have:
x_(1) = 1 - ((4(1)^3 + 4(1) - 1) / (12(1)^2 + 4))
Evaluating this expression, we get x_(1) = 1 - 9/16 = 7/16.
Now, we take this new value of x and substitute it back into the formula. Continuing this process of iteration, we eventually reach a value of x that satisfies the equation to the desired level of accuracy. It turns out that this value is approximately x ≈ 0.2367.
Once you have obtained this value, you can substitute it back into the original function f(x) = x^4 + 2x^2 to find the corresponding y-coordinate. In this case, the value of y is approximately y ≈ 0.1152.
Therefore, the point of tangency on the graph of f(x) = x^4 + 2x^2 is approximately (0.2367, 0.1152). Given that the slope of the tangent line is 1, you can use the point-slope form of a linear equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point, to obtain the equation of the tangent line as y = x - 0.1215 (rounded to 3 decimal places).
I apologize for the earlier confusion in my response. I hope this detailed explanation helps you understand the process of solving this problem. Let me know if you have any further questions!