1. Using differentials, find the approximate value of each of the following: (a) 17 81 1 4 (b) ( ) 1 33 5 2. Show that the function given by log ( ) x f x x = has maximum at x = e. 3. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ? 4. Find the equation of the normal to curve x2 = 4y which passes through the point (1, 2). 5. Show that the normal at any point q to the curve x = a cosq + a q sinq, y = a sinq aq cosq is at a constant distance from the origin. 6. Find the intervals in which the function f given by 4sin 2 cos ( ) 2 cos x x x x f x x = + is (i) increasing (ii) decreasing. 7. Find the intervals in which the function f given by 3 3
Question 1:Using differentials, find the approximate value of each of the following: (a) 17 81 1 æ 4 è ç ö ø ÷ (b) ( ) 1 33 5 −
Question 2:Show that the function given by log ( ) x f x x = has maximum at x = e.
Question 3:The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?
Question 4:Find the equation of the normal to curve x2 = 4y which passes through the point (1, 2).
Question 5:Show that the normal at any point q to the curve x = a cosq + a q sinq, y = a sinq – aq cosq is at a constant distance from the origin.
Question 6:Find the intervals in which the function f given by 4sin 2 cos ( ) 2 cos x x x x f x x − − = + is (i) increasing (ii) decreasing.
Question 7:Find the intervals in which the function f given by 3 3 1 f (x) x , x 0 x = + ¹ is (i) increasing (ii) decreasing.
Question 8:Find the maximum area of an isosceles triangle inscribed in the ellipse 2 2 2 2 1 x y a b + = with its vertex at one end of the major axis.
Question 9:A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
Question 10:The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
Question 11:A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Question 12:A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is 2 2 3 (a3 + b3 )2 .
Question 13:Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has (i) local maxima (ii) local minima (iii) point of inflexion
Question 14:Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x Î [0, p]
Question 15:Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4 3 r .
Question 16:Let f be a function defined on [a, b] such that f ¢(x) > 0, for all x Î (a, b). Then prove that f is an increasing function on (a, b).
Question 17:Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R 3 . Also find the maximum volume.
Question 18:Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle a is one-third that of the cone and the greatest volume of cylinder is 4 3 2 tan 27 ph a . Choose the correct answer in the questions from 19 to 24
Question 19:A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h
Question 20:The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is (A) 22 7 (B) 6 7 (C) 7 6 (D) 6 7
Question 21:The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is (A) 1 (B) 2 (C) 3 (D) 1 2
Question 22:The normal at the point (1,1) on the curve 2y + x2 = 3 is (A) x + y = 0 (B) x – y = 0 (C) x + y +1 = 0 (D) x – y = 1
Question 23:The normal to the curve x2 = 4y passing (1,2) is (A) x + y = 3 (B) x – y = 3 (C) x + y = 1 (D) x – y = 1
Question 24:The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are (A) 8 4, 3 æ ± ö ç ÷ è ø (B) 8 4, 3 − (C) 3 4, 8 æ ± ö ç ÷ è ø (D) 3 4, 8 æ± ö ç ÷ è ø
