Determine the coordinates of the point on the graph of f(x)=sqrt(2x+1) where the tangent line is perpendicular to the line 3x+y+4=0
To find the coordinates of the point on the graph of f(x) = sqrt(2x + 1) where the tangent line is perpendicular to the line 3x + y + 4 = 0, we need to follow these steps:
Step 1: Find the derivative of f(x) using the chain rule.
The derivative of f(x) = sqrt(2x + 1) is given by:
f'(x) = (1/2) / (sqrt(2x + 1))
Step 2: Find the slope of the tangent line to the graph of f(x) at a given point (x, f(x)).
The slope of the tangent line is equal to the derivative evaluated at x.
Step 3: Find the slope of the line 3x + y + 4 = 0.
The given line is in the form y = mx + b, where m is the slope of the line.
Step 4: Find the negative reciprocal of the slope of the line to get the slope of a line perpendicular to it.
Step 5: Solve the equation f'(x) = -1/m, where m is the slope of the line.
Step 6: Substitute the value of x obtained from step 5 into f(x) to find the corresponding y value.
Step 7: The coordinates of the point on the graph of f(x) are (x, y) obtained from step 6.
Let's proceed with these steps.
To determine the coordinates of the point on the graph of f(x) = sqrt(2x + 1) where the tangent line is perpendicular to the line 3x + y + 4 = 0, we need to follow the steps below:
Step 1: Find the derivative of f(x)
The derivative of f(x) will give us the slope of the tangent line at any given point on the graph of f(x).
Using the power rule, we can find the derivative of f(x) = sqrt(2x + 1) as follows:
f'(x) = d/dx(sqrt(2x + 1))
To simplify this expression, we can rewrite sqrt(2x + 1) as (2x + 1)^(1/2):
f'(x) = d/dx((2x + 1)^(1/2))
Applying the power rule, we get:
f'(x) = (1/2)(2x + 1)^(-1/2) * (d/dx)(2x + 1)
The derivative of (2x + 1) with respect to x is simply 2, so:
f'(x) = (1/2)(2x + 1)^(-1/2) * 2
Simplifying further, we have:
f'(x) = (2/(2x + 1)^(1/2)
Step 2: Find the slope of the line 3x + y + 4 = 0
We need to find the slope of the line 3x + y + 4 = 0 in order to determine the slope of the perpendicular line.
Rearrange the equation to the form y = mx + b:
y = -3x - 4
Comparing with the slope-intercept form y = mx + b, we see that the slope (m) of the line is -3.
Step 3: Solve for x
Since the tangent line is perpendicular to the line 3x + y + 4 = 0, the slope of the tangent line will be the negative reciprocal of the slope of the given line.
Therefore, the slope of the tangent line is 1/3.
The tangent line's slope, f'(x), is equal to 2 / (2x + 1)^(1/2).
Setting the two slopes equal to each other, we get:
1/3 = 2 / (2x + 1)^(1/2)
To solve for x, we need to square both sides of the equation:
(1/3)^2 = (2 / (2x + 1)^(1/2))^2
1/9 = 4 / (2x + 1)
Cross-multiplying, we have:
1 * (2x + 1) = 4 * 9
2x + 1 = 36
Solving for x, we subtract 1 from both sides:
2x = 36 - 1
2x = 35
Dividing both sides by 2:
x = 35/2
Therefore, the x-coordinate of the point on the graph of f(x) = sqrt(2x + 1) where the tangent line is perpendicular to the line 3x + y + 4 = 0 is x = 35/2.
Step 4: Find the y-coordinate
Substitute the value of x into f(x) to find the corresponding y-coordinate.
f(x) = sqrt(2x + 1)
f(35/2) = sqrt(2 * (35/2) + 1)
f(35/2) = sqrt(35 + 1)
f(35/2) = sqrt(36)
f(35/2) = 6
Therefore, the y-coordinate of the point on the graph of f(x) = sqrt(2x + 1) where the tangent line is perpendicular to the line 3x + y + 4 = 0 is y = 6.
Hence, the coordinates of the point are (x, y) = (35/2, 6).