`f(x) = 1/sqrt(x), x + 2y - 6 = 0` Find an equation of the line that is tangent to the graph of f and parallel to the given line.

Textbook Question

Chapter 2, 2.1 - Problem 37 - Calculus of a Single Variable (10th Edition, Ron Larson).
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dh246 | (Level 1) Adjunct Educator

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Since the tangent line to function f(x) = 1 / sqrt(x) has to be parallel to the line x + 2y - 6 = 0, both needs to have the same slope.
Equation of the straight line can be rewritten in slope-intercept form to : y = -x/2 + 3. Hence the desired slope of our line is -1/2.

And derivative of f(x) will provide the slope of the line tangent to f(x)
d/dx(f(x)) = -1/2 * (x) ^ (-3/2)
setting it equal to -1/2
-1/2 * (x) ^ (-3/2) = -1/2 or x = 1 provides one of the conditions

Corresponding y coordinate can be found from the f(x) = 1/sqrt(x) equation, y = 1/sqrt(1) = 1
The slope of the desired line is equal to -1/2, and we now have a point on the desired line (1, 1), as well the point of tangency

Using the point-slope, equation to desired line can be obtained as follows. Given slope m = -1/2 and a point on the desired line,(x0, y0), the equation of the line is given by
(y - y0) = m(x - x0)
y - 1 = -1/2(x - 1 )
y = -1/2x + 3/2
2y + x - 3 = 0

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hkj1385 | (Level 1) Assistant Educator

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The given line is :- 

x + 2y - 6 = 0

or, y = -(1/2)x + 3 (the line is represented in slope intercept form)

Thus, the slope of the line = -(1/2)

Now, the tangent to the curve f(x) = 1/(x^(1/2)) is parallel to the above line

Thus, the slope of the tangent = slope of the line = -(1/2).......(1)

The given function is:-

f(x) = 1/(x^(1/2))

differentiating both sides w.r.t 'x' we get

f'(x) = -(1/2)*[1/{x^(3/2)}]

Now, slope of the tangent  = -(1/2)

Thus, -(1/2)*[1/{x^(3/2)}] = -(1/2)

or, x = 1

Putting the value of x =1 in the given equation of curve, we get  

f(1) = y = 1

Hence the tangent passes through the point (1,1)

Thus, equation of the tangent at the point (1,1) and having slope = -(1/2) is :-

y - 1 = (-1/2)*(x - 1)

or, 2y - 2 = -x + 1

or,  2y + x - 3 = 0 is the equation of the tangent to the  given curve at (1,1)

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