Could you please help me with this question:
Determine the values of x so that the tangent to the function (y=3/cubic root of x) is parallel to the line x+16y+3=0
Thank you
(1) Get the slope of the line x+16y+3=0
It will help if you rewrite it as
y = -x/16 - 3/16
(2) require that the slope of the function
y = 3/x^(1/3).
which is dy/dx = -1/x^(4/3)
equal the slope value from (1). Solve for x.
Thank you drwls,
But how do I equate the two equations?
I know the answer is 8, but I can't solve it properly.
To find the values of x for which the tangent to the function is parallel to the given line, we need to find the derivative of the function and set it equal to the slope of the line.
Step 1: Find the derivative of the function:
To find the derivative of y = 3/∛x, we can rewrite the function as y = 3x^(-1/3).
Using the power rule, the derivative of y with respect to x is:
dy/dx = -1/3 * 3 * x^(-1/3 - 1)
= -x^(-4/3)
So, the derivative of the function y = 3/∛x is dy/dx = -x^(-4/3).
Step 2: Determine the slope of the line:
The equation of the line is x + 16y + 3 = 0.
Rewrite it in slope-intercept form (y = mx + c):
16y = -x - 3
y = -(1/16)x - 3/16
The slope of the line is the coefficient of x, which is -1/16.
Step 3: Set the derivative equal to the slope of the line and solve for x:
-x^(-4/3) = -1/16
Use cross-multiplication to solve for x:
-16 = -x^(-4/3)
Flip both sides of the equation to make it easier to work with:
16 = x^(-4/3)
To get rid of the negative exponent, take the reciprocal of both sides:
1/16 = x^(4/3)
Now, cube both sides:
(1/16)^3 = (x^(4/3))^3
1/4096 = x^4
Take the fourth root of both sides to solve for x:
x = ±∛(1/4096)
The values of x that satisfy the condition are x = ∛(1/4096) and x = -∛(1/4096).
Therefore, the values of x for which the tangent to the function y = 3/∛x is parallel to the line x + 16y + 3 = 0 are x = ∛(1/4096) and x = -∛(1/4096).
To find the values of x where the tangent to the function is parallel to the given line, we need to use the concept of differentiation. Here's how you can solve it step by step:
1. Start by differentiating the given function y = 3/(cubic root of x) with respect to x.
To differentiate y, we use the power rule:
dy/dx = d/dx(3/(cubic root of x))
= d/dx(3x^(-1/3)) [Rewriting the expression]
= -1/3 * 3x^(-1/3-1) [Applying the power rule]
= -x^(-4/3)
= -1/(x^(4/3))
So, the derivative of y with respect to x is -1/(x^(4/3)).
2. Now, let's find the slope of the given line x + 16y + 3 = 0.
We put it in the slope-intercept form: y = mx + c.
The equation x + 16y + 3 = 0 can be rearranged as:
16y = -x - 3,
y = (-1/16)x - 3/16.
Comparing this equation with y = mx + c, we see that the slope (m) of the line is -1/16.
3. Since we want the tangent to be parallel to the line, the slope of the tangent and the slope of the line must be equal.
Therefore, -1/(x^(4/3)) = -1/16.
4. Solve the equation -1/(x^(4/3)) = -1/16 for x.
To do this, we can cross-multiply:
-1 * 16 = x^(4/3),
-16 = x^(4/3).
5. To simplify further, we can raise both sides of the equation to the power of 3/4:
(-16)^(3/4) = (x^(4/3))^(3/4),
(-16)^(3/4) = x,
x = (-16)^(3/4).
6. Finally, simplify the value of x.
The value of (-16)^(3/4) can be determined by taking the cube root of the absolute value of -16 and then raising it to the power of 3:
(-16)^(3/4) = (|(-16)|^(1/3))^3
= 2^3 [Since (-16)^(1/3) = -2]
= 8.
So, the values of x that satisfy the condition are x = 8.
Therefore, the tangent to the function y = 3/(cubic root of x) is parallel to the line x + 16y + 3 = 0 when x = 8.