MTL 100 (Calculus)
Tutorial Sheet No. 6
Applications of differential calculus of functions of several variables
1. Expand the following functions into Taylor’s series around (0, 0) :
1
(i) 1−x−y−xy
(ii) tan−1
x−y
1+xy
(iii) ln(1 − x) ln(1 − y)
(iv) ln 1−x−y+xy
1−x−y
(v) ex cos y.
2. Expand sin x sin y in the powers of (x − π/4) and (y − π/4). Find the terms of the first and
second order terms and also the remainder R2 .
3. Let f (x, y) = 41 xy 3 − yx3 + 12 x2 y 2 − 2x + 3y − 4. Find the increment gained by the function
when the independent variables are changed from the values x = 1, y = 2 to x = 1+h, y =
2 + k. compute f (1.02, 2.03) up to the second order(included) terms.
4. Using Taylor’s formula, find quadratic and cubic approximations ex sin y at origin. Estimate the errors in approximations if |x| ≤ 0.1; |y| ≤ 0.1.
5. Find the maximum value of f (x, y, z) = x2 + 2y − z 2 subject to the constraints g(x, y, z) ≡
2x − y = 0 and h(x, y, z) ≡ y + z = 0.
6. Find the maximum value of w = xyz along all points lying on the intersection of the two
planes x + y + z = 40 and z = x + y.
7. If a, b, c are positive numbers find the extreme value that f (x, y, z) = ax + by + cz can take
on the sphere x2 + y 2 + z 2 = 1.
8. Find the local Max/Min of z = x2 + y 2 subject to the condition stated:
(i) 3x2 +xy +3y 2 = 1
(ii) xy = 14
(iii) x2 +4xy +y 2 = 1
(iv) 2x2 +xy +2y 2 = 1
9. Find the foot of the perpendicular from P(6,2,3) to the plane z = 5x − y + 2, by minimizing
the distance from P to (x, y, z) where (x, y, z) is in the plane.
10. Use the method of Lagrange’s multipliers to locate all the local maximas and minimas.
Also find absolute maximum and absolute minimum values of the function
(i) f (x, y, z) = x2 + y 2 where x4 + y 4 = 1
(ii) f (x, y, z) = xy where 2x2 + y 2 = 1
(iii) f (x, y, z) = x + y + z where x2 + y 2 = 1 and y 2 + z 2 = 1.
11. Find the critical points of f (x, y) and determine its relative extrema, if any, in the following
cases:
(i) f (x, y) = x4 + y 4 − 3xy
(ii) f (x, y) = (x2 + y 2 )e4x+2x
(iii) f (x, y) = 2 sin(x + 2y) + 3 cos(2x − y)
(iv) f (x, y) =
2
−x
.
1+x2 +y 2
12. Show that f (x, y) = (x4 + y 4 + 1)−1 has an absolute maximum at (0, 0).
13. Show that x2 + y 2 +
2
x
+
2
y
has relative minimum at (1, 1).
14. Find for what values of (x, y, z) has a relative maximum/ minimum (if there is any):
(i) x2 + y 2 + z 2 = 3
(ii) x2 + y 2 = 2z
(iii) x2 − y 2 = 2z
3
2
(iv)x2 + xy + y 2 + ax + ay
15. Find the absolute maximum and absolute minimum of f on the regions described by the
(in)equalities:
(i) f (x, y) = xy; x2 + 2y 2 ≤ 1
(ii) f (x, y) = 8xy + y; 0 ≤ y ≤ 15 − x; 0 ≤ x ≤ 5
(iii) f (x, y) = − 1+xx2 +y2 ; |x| ≤ 1, |y| ≤ 2
(iv) f (x, y) = x2 + 2xy − 4x + 8y; 0 ≤ x ≤ 1, 1 ≤ y ≤ 2
(v) f (x, y) = x + y; x2 + y 2 + z 2 = 1
(vi) f (x, y, z) = xyz; x2 + y 2 = 1, y = z.
16. Find the greatest and the least values of the function f (x, y) = x2 y(4 − x − y) on and
inside the triangle bounded by the straight lines x = 0, y = 0, x + y = 6.
17. Split a positive number into the sum three positive numbers whose product is maximum
possible.
18. In a given sphere of radius R, inscribe a rectangular parallelepiped of maximum volume.
19. Of all rectangular parallelepipeds having a given diagonal find the one with maximum
volume.
n 2
20. Prove the inequality: ( x1 +x2 +···+x
) ≤
n
21. On the ellipse
x2
4
+
y2
9
x21 +x22 +···+x2n
.
n
= 1 find the nearest and farthest points from the straight line
3x + y − 9 = 0.
22. Find the regular triangular pyramid of a given volume having the least sum of edges.
23. Prove that the products of sines of the angles of a triangle is a maximum when the triangle
is equilateral.
24. Find the shortest distance between the curves x2 + 2xy + 5y 2 − 16y = 0 and x + y − 8 = 0.
-END2