7/28/2019 MVT Tangent and Normal, Rate of Measure
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Math 12th
MVT, Tangent and Normal, Rate of Measure
1. Verify Rolle's Theorem for the following function:
( ) ( ) 30,3 2 = xxxxf [Ans. C = 1]
2. Usin Rolle's theorem, find the points on the curve [ ]2,2,2 =
xxy where the
tangent is parallel to the axisx [Ans. (0, 0)]
3. Verify Lagrange's mean value theorem for the following
function:
( ) ( ) ( ) ( ) [ ]4,0321 inxxxxf = [Ans.3
22 =C ]
4. Find the equation of the tangent line to the curve4
sin0,cos1
=== atyx .
[Ans. ( ) ]2
1112
2
1
4
=
xy
5. Find a point on the parabola ( ) 22= xy where the tangent is
parallel to the chora
joining the points (2, 0) and (4, 4). [Ans. (3, 1)
6. Find the equation of the tangent to the curve.
++=
4
35,
2
1int765
2 potheatxxy .
[ ]03344. =+ yxAns
7. Show that the curves kxyandyx == 2 at right angles if 18 2
=K
8. Using differentials, find the approximate value of .26 [Ans.
(5.1)
9. Find the point on the curve xy 82 = for which the abscissa
and ordinate change at
the some rate.
[Ans. (2, 4)]
10. Using differentials, find the approximate value of 37.0 .
[Ans. 0.608]
11. Using differentials, find the approximate value of .48.0
[Ans. 0.693]
12. A balloon which always remains spherical is being inflated
by pumping in gas at
the rate of 900 cubic cm/ sec. Find the rate at which the radius
of the balloon is
increasing when the radius of the balloon is 15 cm.
[Ans. .sec/1
cm
]
13. If the tangent to the curve ( )611
3++= Patbaxxy is parallel to the line ,5=xy find
the value ofa and b. [Ans. 5,2 == ba ]
14. A point source of light along a straight road is at a height
of 'a' metres. A boy 'b
metres in height is walking along the road. How fast is his
shadow increasing if he
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is walking away from the light at the rate of 'C' metres per
minute?
[Ans. ]min/ utemba
bc
15. Find the equation of tangent to the curve .4
2cos,3sin=== attytx
[Ans. ]02223 =+ xy
16. At what points will the tangent to the curve 2136152 23 +=
xxxy be parallel to
?axisx Also, find the equations of tangents to the curve at
those points.
( )
( )
=
=
06;6,3
,07;7,2.
y
yAns
17. Find the points on the curve 3xy = at which the slope of the
tangent is equal to y
coordinate of the point. ( )
27
1,
3
1;0,0.Ans
***
