Find the equation of the tangent to the curve y = 2 - sqrt(x) perpendicular to the straight line y + 4x - 4 = 0
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I don't know the answer. If I knew it, I wouldn't post it here asking what is the answer.
To find the equation of the tangent to the curve y = 2 - sqrt(x) that is perpendicular to the straight line y + 4x - 4 = 0, we can follow these steps:
1. Find the slope of the given straight line: The given equation of the straight line is y + 4x - 4 = 0, which can be rewritten as y = -4x + 4. The slope-intercept form of a line is y = mx + c, where m is the slope of the line. Therefore, the slope of the given line is -4.
2. Find the derivative of the curve: The equation of the curve is y = 2 - sqrt(x). To find the slope of the tangent to the curve, we need to find the derivative dy/dx. Taking the derivative of y with respect to x, we get:
dy/dx = d/dx(2 - sqrt(x))
= -1/2 * 1/sqrt(x)
= -1/(2√x)
3. Find the slope of the tangent: The slope of the tangent to the curve is equal to the derivative, so the slope of the tangent is -1/(2√x).
4. Determine the slope of a line perpendicular to the given straight line: The slope of the tangent to the curve needs to be perpendicular to the given line, which has a slope of -4. The product of the slopes of two perpendicular lines is -1, so we can find the slope of a line perpendicular to the given line by taking the negative reciprocal of its slope. Thus, the slope of a line perpendicular to y + 4x - 4 = 0 is 1/4.
5. Equate the slopes and solve for x: Set the slope of the tangent to the curve, which we found from step 3, equal to the slope of a line perpendicular to the given line, which we found from step 4:
-1/(2√x) = 1/4
6. Solve the equation for x: Cross-multiply the equation from step 5 to get:
-4√x = 2
Then, divide both sides by -4 to solve for √x:
√x = -1/2
Square both sides of the equation to get:
x = 1/4
7. Find the corresponding y-coordinate: Substitute x = 1/4 into the equation of the curve y = 2 - sqrt(x):
y = 2 - √(1/4)
y = 2 - 1/2
y = 3/2
So the point of tangency on the curve y = 2 - sqrt(x) is (1/4, 3/2).
8. Find the equation of the tangent line: To find the equation of the tangent line, we use the point-slope form. We have the point (1/4, 3/2) on the tangent line, and the slope of the tangent is -1/(2√x), which we found from step 3. Substituting these values into the point-slope form equation, we get:
y - (3/2) = (-1/(2√(1/4))) * (x - (1/4))
Simplifying the equation, we get:
y - (3/2) = (-1/(2 * 1/2)) * (x - (1/4))
y - (3/2) = -2 * (x - (1/4))
y - (3/2) = -2x + 1/2
Finally, rearranging the equation, we get the equation of the tangent line:
y = -2x + 5/2