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The drawing below shows a square with side a. A straight line intersects the square and encloses

an area A. The heights x and y on the left and right side (in a distance d from the square) of

the intersecting line can be varied. Assuming that x  y and x; y  a, nd an expression for

the enclosed area A(x; y) with respect to x and y.



Give an example of a function of two variables such that f(0,0) = 0 but f is NOT continuous at (0,0). Explain why the function f is NOT continuous at (0,0).


It's time to tidy up your work desk. you are given 27cm^2 of cardboard to build a rectangular box without a lid to store small electronic components. By using the knowledge of partial derivative, determine the maximum volume of this box




The derivative of a differentiable function ff(xx) is given as


ff′

(xx) = xx + 3

(xx − 2)2 .

a. Find intervals of increase and decrease for ff(xx).

b. Determine values of xx for which relative maxima and minima occurs on the graph of

ff(xx).

c. Find ff′′(xx) and determine intervals of concavity for the graph of ff(xx).

d. For what values of xx do inflection points occur on the graph of ff(xx).



The drawing below shows a square with side a. A straight line intersects the square and encloses

an area A. The heights x and y on the left and right side (in a distance d from the square) of

the intersecting line can be varied. Assuming that x  y and x; y  a, nd an expression for

the enclosed area A(x; y) with respect to x and y.



n the right triangle ABC, AB = 2, BC = 4 and ED is a line parallel to AB. Find the



angle α = angle BAD which minimizes the distance L, where L = AD + ED



A soft drink manufacturer requires services of your software house to automate the process of can designing. The available material for the bottom and top of the can cost 20�ђ�ѡ��њ2, and for the sides the material cost is 10�ђ�ѡ��њ2. The company is considering two design options; one design does not include the lid and is an open top design, and the other design includes the lid and thus is a closed top design. The company wants to take desired volume, and design (open top or closed top) as input and give dimensions and surface area of the can that will result in the minimum cost as output for the specified volume and design. To test your code the company has asked you to use 275�њ3 as volume for both designs. Furthermore, if the budget of the company is $2 million, how many cans of the required design will the company be able to make?


c) Describe geometrical meaning of definite integral with figure.


a) Define differentiation and integration in calculus. Also write down the 

 differences between them.



3. a) Define tangent and normal of a curve with figure. Also find the equation of tangent and normal of the ellipse (x ^ 2)/4 + (y ^ 2)/16 = 1 at the point (- 1, 3) .

b) Explain maximum and minimum value of a function with graphically. Evaluate maximum and minimum value of the function f(x) = x ^ 3 - 3x ^ 2 + 3x + 1