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University of Dar Es Salaam MT 261 Tutorial 1

This document contains 21 calculus problems involving functions of multiple variables, partial derivatives, limits, and optimization using Lagrange multipliers. The problems cover topics like finding partial derivatives of functions, limits of multivariable functions, determining rates of change, using partial derivatives to find stationary points, and applying Lagrange multipliers to optimization problems.

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Gilbert Furia
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250 views4 pages

University of Dar Es Salaam MT 261 Tutorial 1

This document contains 21 calculus problems involving functions of multiple variables, partial derivatives, limits, and optimization using Lagrange multipliers. The problems cover topics like finding partial derivatives of functions, limits of multivariable functions, determining rates of change, using partial derivatives to find stationary points, and applying Lagrange multipliers to optimization problems.

Uploaded by

Gilbert Furia
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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UNIVERSITY OF DAR ES SALAAM MT 261 TUTORIAL 1


MT261: Calculus of Several variables for non-majors Instructions: Answer ALL questions in this tutorial sheet.

1. What are the domain and range of the function f (x, y) = xy ? y2 (1)

x2

2. Discuss the continuity of the following functions in the whole plane (a)
x2 y x4 +y 2

if(x, y) = (0, 0) (2) if(x, y) = (0, 0)

f (x, y) =

0 (b)
x3 x2 y+xy 2 y 3 x2 +y 2

if(x, y) = (0, 0) (3) if(x, y) = (0, 0)

h(x, y) =

0 k is a real number. (c) Find x4 y 4 , (x,y)(0,0) x2 y 2 lim 3. Show that x3 + y 3 7 = , and 2 + y2 (x,y)(1,2) x 5 lim

if it exist

(4)

x3 + y 3 =0 (x,y)(0,0) x2 + y 2 lim

(5)

2 4. (a) f (x, y, z) = e2x cos(z) + e3y sin(z). Find the rst order partial derivatives of f (b) f (x, y) = sin(xy 2 ) Find the second order partial derivatives of f
z t z u

5. Let x2 ey , x = sin(t), and y = t3 . Find

6. Let z = x ln y, x = u2 + v 2 and y = u2 v 2 . Find 7. If V = f (x2 + y 2 ) show that x V y V = 0 y x

and

z v

8. A cylinder has dimensions r = 5cm, h = 10cm. Find the approximate increase in volume when r increases by 0.2cm and h decreases by 0.1cm 9. If
ws3 . d4

Find the percentage increase in y, when w increases 2%, s decreases by 3%

and d increases by 1%. 10. If


V , R

V = 250volts and R = 50, nd the change in I resulting from an increase of

1volt in V and increase of 0.5 in R. 11. The two sides forming the right-angel of a right-angled triangle are denoted by a and b. The hypotenuse is h. If there are possible errors of + 0.5% in measuring a and b, nd the maximum possible error in calculating (i) the area of the triangle and (ii) The length of h. 12. The total surface area of S of a cone of base radius r and perpendicular height h is given by S = r2 + r r2 + h2 . If r and h are each increasing at the rate of 0.25cm/sec, nd the rate at which S is increasing at the instant when r = 2cm and h = 4cm. 13. Find the limits

3 (i) lim
(x,y)(2,1)

ln(xy 1)

(ii) ey sin(x) (x,y)(0,0) x lim 14. If z=f x y , show that x z z +y x y

15. if is a function of independent variables x, y, z which are changed to independent variables u, v, w by the transformation x = u
vw u

,y=

uw v

z=

uv , w

show that

+v +w =x +y +z u v w x y z

16. If x = f (x, y) with u = x2 y 2 and v = xy, nd x x y y , , , u v u v 17. If z = 2x2 + 3xy + 4y 2 and u = x2 + y 2 , v = x + 2y, determine x x y y , , , , and u v u v z z , u v

18. Find the stationary values, if any, of the function z = x3 6xy + y 3 . 19. Determine stationary values of z = x3 3x + xy 2 and their nature. 20. Determine the relative dimensions of a rectangular box, without a top and having a specic volume, if the least amount of material is to be used in its manufacture. 21. Use the Lagrange multiplier to nd the shortest distance from the origin to the plane Ax + By + Cz = D.

by mwl. Mahera*****************END***************************************

4 c e f g a b c a b

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