Page 94 - Mathematics Class - XI

dy
6. Similarly, take another point (– 4, 3) on the circle. Verify that at (– 4, 3) = tan a, where a is the angle
dx
made by the tangent to the circle at the point (– 4, 3) with the positive direction of x-axis (Fig. (a)).

7. Take other sheet with the graph of (x + 3) + y = 25 and take the point (1, 3) on it. Repeat the above process
2
2
dy
using set square and wires as shown in Fig. (b), i.e., verify at (1, 3) = tan q
dx
8. Now take the third sheet, showing the graph of the curve xy = 9. Take the point (3, 3) on it. Place one
perpendicular side of set square along the line y = x and a wire along the other side touching the curve
at the point (3, 3) and find the angle made by the wire with the positive direction of x-axis as shown in
dy
Fig. (c). Let it be q. Verify that at (3, 3) = tan q.
dx

OBSERVATION
dy −4
1. For the curve x + y = 25, at the point (4, 3) = = tan q,
2
2
dx 3
Value of q = 2.2 (approx)
dy
So, at the point (4, 3) = tan q
dx
dy 4 4
2. For the curve x + y = 25, at the point (– 4, 3) = , tan a =
2
2
dx 3 3
dy
So, at the point (– 4, 3) = tan a.
dx
dy −4 −4
3. For the curve (x + 3) + y = 25, at (1, 3) = , value of q = 2.2 (approx.), tan q = .
2
2
dx 3 3
dy −4
So, at the point (1, 3) = .
dx 3
dy 3p
4. For the curve xy = 9, at (3, 3) = –1, q = , tan q = –1.
dx 4

CONCLUSION
This activity shows that derivative at a point P (a, b) is equal to slope of tangent at P (a, b),

dy
i.e., at P(a, b) = tan q.
dx Knowledge Booster

APPLICATION ● If dy = 0, then the tangent to curve y = f(x) at the point (x, y) is
dx
With this activity we can verify the result that parallel to the x-axis.
the slope of the tangent at a point is equal to
the value of the derivative at that point for ● If dy =∞, dx = ,0 then the tangent to the curve y = f(x) at the point
dy
dx
other curves. (x, y) is parallel to the y-axis.
dy
● If = tan q > 0, then the tangent to the curve y = f(x) at the point
dx
(x, y) makes an acute angle with positive x-axis and vice versa.

92