How do you find the equation of the line tangent to #f(x)= 1/x#, at (1/2,2)?

Question

Ezra Adams · Accepted Answer

#y-2=-4(x-1/2)#
#y=-4x+4# #f'(x)=-1/x^2# #f'(1/2)=-1/(1/4) = -4# #y-2=-4(x-1/2)# #y=-4x+4#

Charles Anctil · Answer

#4x+y=4#

If #color(white)("XXX")f(x)=1/x=x^(-1)# then #color(white)("XXX")f'(x)=(-1)(x^(-2)) = -1/x^2#

At #(color(red)(1/2),2)# the slope of the tangent line is #color(white)("XXX")f'(color(red)(1/2))=-1/((color(red)(1/2))^2) = color(green)(-4)#

The slope-point form of the tangent line using a slope of #color(green)(m=-4)# and the point #(color(red)(barx),color(blue)(bary)) =(color(red)(1/2),color(blue)(2))# is #color(white)("XXX")y-color(blue)(bary)=color(green)(m)(x-color(red)(barx))#

or, with the given values: #color(white)("XXX")y-color(blue)(2=color(green)(-4)(x-color(red)(1/2))# #rarr# #color(white)("XXX")y-2 = -4x +2# in standard form: #color(white)("XXX")4x+y=4#

graph{(1/x-y)(4x+y-4)=0 [-3.973, 4.8, -1.268, 3.117]}

Aurora Alvarado · Answer

undefined To find the equation of the line tangent to the function f(x) = 1/x at the point (1/2, 2), we can use the concept of differentiation.

First, we need to find the derivative of the function f(x). The derivative of 1/x can be found using the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1). Applying this rule, the derivative of 1/x is -1/x^2.

Next, we substitute the x-coordinate of the given point (1/2, 2) into the derivative to find the slope of the tangent line. Plugging in x = 1/2 into the derivative -1/x^2, we get -1/(1/2)^2 = -1/(1/4) = -4.

Now that we have the slope of the tangent line, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Substituting the values x1 = 1/2, y1 = 2, and m = -4, we get y - 2 = -4(x - 1/2).

Simplifying the equation, we have y - 2 = -4x + 2. Rearranging the terms, we get the equation of the line tangent to f(x) = 1/x at (1/2, 2) as y = -4x + 4.