Coordinates and Graphs

[fdragon]

In the figure, line l is tangent to the graph of y= 1/x^2 at point P, with coordinates (w, 1/w^2), where w> 0 . Point Q has coordinates (w, 0). Line l crosses the x-axis at the point R, with coordinates (k,0)

a... Find the value of k when w=3

b... For all w > 0, find k in terms of w

c... Suppose that w is increasing at the constant rate of 7 units per second. When w=5, what is the rate of change of k with respect to time?

d... Suppose that w is increasting at the constant rate of 7 units per second. When w=5, what is the rate of change of the area of (Triangle)PQR with respect to time? Determine whether the area is increasing or decreasing at this instant.



Totally don't have a clue... lol.

 

[pka]

Quite frankly it appears as if you have put no effort into this your own homework.
All that I will do for you, until we see some of your work, is the tell you that the equation of the line l is:
\(\displaystyle y - \frac{1}{{w^2 }} = \frac{{ - 2}}{{w^3 }}\left( {x - w} \right).\)

 

[soroban]

Hello, fdragon!

"Don't have a clue" ??

You're in a Calculus class and know nothing about:

. . \(\displaystyle \begin{Bmatrix}\text{slopes} \\ \text{equations of lines} \\ \text{intercepts} \\ \text{area of a triangle}\end{Bmatrix}\;\;\) precalculus stuff

The only Calculus is taking a derivative.

In the figure, line \(\displaystyle L\) is tangent to the graph of \(\displaystyle y\:=\:\frac{1}{x^2}\)
. . at point \(\displaystyle P\left(w,\,\frac{1}{w^2\right)\), where \(\displaystyle w\,>\,0\)
Point \(\displaystyle Q\) has coordinates \(\displaystyle (w,\,0).\)
Line \(\displaystyle L\) crosses the x-axis at the point \(\displaystyle R(k,\,0)\).

(a) Find the value of \(\displaystyle k\) when \(\displaystyle w\,=\,3\)

(b) For all \(\displaystyle w\,>\,0\), find \(\displaystyle k\) in terms of \(\displaystyle w\)

(c) Suppose that \(\displaystyle w\) is increasing at the constant rate of 7 units per second.
When \(\displaystyle w\,=\,5\), what is the rate of change of \(\displaystyle k\) with respect to time?

(d) Suppose that \(\displaystyle w\) is increasing at the constant rate of 7 units per second.
When \(\displaystyle w\,=\,5\), what is the rate of change of the area of \(\displaystyle \Delta PQR\) with respect to time?
Determine whether the area is increasing or decreasing at this instant.

(b) We have: \(\displaystyle \:y \:=\:x^{-2}\)
Then: \(\displaystyle \:y'\:=\:-2x^{-3} \:=\:-\frac{2}{x^3}\)
At \(\displaystyle x \,=\,w:\;\;y'\,=\,-\frac{2}{w^3}\)

Tangent \(\displaystyle L\) has point \(\displaystyle P\left(w,\,\frac{1}{w^2\right)\) and slope \(\displaystyle m\,=\,-\frac{2}{w^3}\)
. . Its equation is: \(\displaystyle \:y\,-\,\frac{1}{w^2}\:=\:-\frac{2}{w^3}(x\,-\,w)\;\;\Rightarrow\;\;y \:=\:-\frac{2}{w^2}x \,+\,\frac{3}{w^2}\)

For the x-intercept, let \(\displaystyle y\,=\,0:\;\;0 \:=\:-\frac{2}{w^3}x\,+\,\frac{3}{w^2}\;\;\Rightarrow\;\;x\:=\:\frac{3}{2}w\)

. . Therefore: \(\displaystyle \:\fbox{k\:=\:\frac{3}{2}{w}}\)


(a) When \(\displaystyle w\,=\,3:\;k\:=\:\frac{3}{2}(3)\;\;\Rightarrow\;\;\fbox{k\:=\:\frac{9}{2}}\)


(c) We have: \(\displaystyle \:k \:=\:\frac{3}{2}w\)
Differentiate with respect to time: \(\displaystyle \:\frac{dk}{dt}\:=\:\frac{3}{2}\left(\frac{dw}{dt}\right)\)

We are told that \(\displaystyle \frac{dw}{dt}\,=\,7\)
. . Therefore: \(\displaystyle \:\frac{dk}{dt}\:=\:\frac{3}{2}(7)\;\;\fbox{\frac{dk}{dt}\:=\:\frac{21}{2}}\)


(d) The base of the triangle is: \(\displaystyle \:QR\:=\:\frac{3}{2}w\,-\,w\:=\:\frac{1}{2}w\)
. . .The height of the triangle is: \(\displaystyle \Q\:=\:\frac{1}{w^2}\)
The area of the triangle is: \(\displaystyle \L\:A \:=\:\frac{1}{2}\left(\frac{1}{2}w\right)\left(\frac{1}{w^2}\right) \:=\:\frac{1}{4w}\:=\:\frac{1}{4}w^{-1}\)

Then: \(\displaystyle \L\:\frac{dA}{dt} \:=\:-\frac{1}{4}w^{-2}\left(\frac{dw}{dt}\right)\:=\:-\frac{1}{4w^2}\left(\frac{dw}{dt}\right)\)

We are told that: \(\displaystyle \,w\,=\,5,\;\frac{dw}{dt}\,=\,7\)

I'll let you finish it . . .