Expert: Paul Klarreich - 4/8/2008

Question
5. (a) Find the equation of the tangent line to y = 1
x at the generic point (a, 1/a)

(b) Let f (a) be the area of the triangle bounded by this tangent line, the x-
axis, and
the y-axis. Show that f

Answer
Questioner:   gwendolyn
Category:  Calculus
Private:  No
 
Subject:  does this help
Question:  5. (a) Find the equation of the tangent line to y = 1
x at the generic point (a, 1/a)

(b) Let f (a) be the area of the triangle bounded by this tangent line, the x-
axis, and the y-axis. Show that f

Show what?  
..............................
Hi, Gwendolyn,

I'll try to answer, but here is what YOU must do in the future:

1. Read some of the archived questions - click on Browse Past Answers - to see how to write mathematical expressions.

2. Proofread your question before sending it.  Make sure YOU know what the question says.  Maybe show it to someone and ask he can read it out loud to you and it makes sense.

a) Find the equation of the tangent line to y = 1/x  at (a, 1/a)

Slope = dy/dx at  x = a:

dy/dx = -1/x^2.
At x = a,  dy/dx = -1/a^2.

Use the point-slope form:  y - y0 = m(x - x0)

y - 1/a = -1/a^2(x - a)

Simplify a bit:

y - 1/a = -x/a^2 + 1/a

y  = -x/a^2 + 2/a

I think that will do it.


(b) Let f (a) be the area of the triangle bounded by this tangent line, the x-
axis, and the y-axis. Show that f

I'll assume you just want to compute f(a).

Now find the x- and y-intercepts of your tangent line:

y  = -x/a^2 + 2/a

y-intercept:  let  x = 0:

y = 2/a

x-intercept:  let  y = 0:

0  = -x/a^2 + 2/a

x/a^2 = 2/a

x = 2a.

So you have a triangle with  base = 2a, height = 2/a

Area = f(a) = (1/2) base * height

f(a) = (1/2)(2a)(2/a) = 2

That's it.  It's also the first and last time I will attempt to figure out what your question says.