Page 265 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes dv dv dr dr dv dv
t. Using chain rule, we can write or .
dt dr dt dt dt dr
dv dv 2
Note that 5. Also 4 r , which can be determined if r is known. To find r, we note
dt dr
4 3
that volume becomes 15 ´ 5 = 75 cubic feet, after 15 seconds, therefore, we have r 75 or
3
225 1/3
r .
4

dr 5 4 2/3
Thus, = 0.058 ft/sec.
dt 4 225

Rule 5.

Inverse function Rule
If y = f(x) and x = g(y) are inverse functions which are differentiable, then we can write g[f(x)] = x.
Differentiating both sides w.r.t. x we have
g¢[f(x)] × f¢(x) = 1

1
or g¢(y) =
f ( )
x

dx 1
or = .
dy dy
dx

Example

Find the equation of a tangent at the point (2, 3) to the rectangular hyperbola xy = 6. Show that
(2, 3) is middle point of the segment of tangent line intercepted between the two axes. What are
its intercepts on the two axes?
Solution:

6 dy 6 6
We can write y or = = -1.5
x dx x 2 4

Equation of tangent is (y – 3) = –1.5(x – 2) or y = 6 - 1.5x.
The point of intersection of the tangent with y-axis is obtained by substituting x = 0 in the above
equation. This point is (0, 6). Similarly (4, 0) is a point of intersection of the tangent with x-axis.

Since (2, 3) is the middle point of the line joining the points (0, 6) and (4, 0), hence the result.
Intercepts of the tangent on x and y axes, are 4 and 6 respectively.

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