If f(x)=[(x−1)/(1+5x)], there are two points on the graph of
y=f(x) at which the tangent lines are parallel to the line x−2y=2.
Type in the x values of
these points. Type the smaller number first, then the larger number.
Sure, I'll type 'em in -- what are they?
Oh, you mean I have to solve for them first? OK, here goes.
The line x-2y=2 has slope 1/2
f(x) has slope 10/(5x+1)^2 for all x ≠ -1/5
SO, we have to find where the slope = 1/2
10/(5x+1)^2 = 1/2
20 = (5x+1)^2
20 = 25x^2 + 10x + 1
25x^2 + 10x - 19 = 0
x = -1.09 or 0.69
If you sketch the curve, you'll see that this appears correct.
To find the points on the graph of y = f(x) where the tangent lines are parallel to the line x - 2y = 2, we need to find the derivative of f(x) and set it equal to the slope of the given line. Let's go step by step.
1. Find the derivative of f(x):
To find the derivative of f(x), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative is given by (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.
In this case, our function is f(x) = (x - 1) / (1 + 5x). Let's apply the quotient rule:
f'(x) = [ (1 + 5x) * (1) - (x - 1) * (5) ] / (1 + 5x)^2
= [ 1 + 5x - 5x + 5 ] / (1 + 5x)^2
= 6 / (1 + 5x)^2
So, the derivative of f(x) is f'(x) = 6 / (1 + 5x)^2.
2. Determine the slope of the given line:
The given line is x - 2y = 2. To determine its slope, we need to rearrange it in the form y = mx + b, where m represents the slope. Let's solve for y:
x - 2y = 2
-2y = 2 - x
y = (1/2)x - 1
From this equation, we can see that the slope of the given line is m = 1/2.
3. Set the derivative of f(x) equal to the slope of the given line:
We want to find the x-values of the points where the tangent lines are parallel to the given line. That means the slope of the tangent lines should be equal to the slope of the given line.
Setting the derivative equal to the slope, we have:
6 / (1 + 5x)^2 = 1/2
Now, let's solve for x:
6 = (1/2)(1 + 5x)^2
12 = (1 + 5x)^2
Taking the square root of both sides:
√12 = 1 + 5x
√12 - 1 = 5x
(√12 - 1)/5 = x
So, we have found the x-value of one point where the tangent line is parallel to the given line.
4. Find the second x-value:
To find the second x-value, we need to consider the symmetry of the graph of y = f(x). Since f(x) is a rational function, it is symmetric about the vertical line x = -1.
The x-value we found in step 3 was the x-value of the point where the tangent line is parallel to the given line in the positive x-axis region. So, to find the second x-value, we can just negate the x-value we found.
The second x-value is -[(√12 - 1)/5] = -(-√12 + 1)/5 = (√12 - 1)/5.
Therefore, the two x-values where the tangent lines are parallel to the line x - 2y = 2 are (√12 - 1)/5 and -(√12 - 1)/5.
Hope this helps!