Calculus II BS 3 Semester/Term PU — University of the Punjab 2023

UNIVERSITY OF THE PUNJAB

B.S. 4 Years Program / Third Semester - Fall 2023

Paper: Calculus-li

Course Code: CAL-212

Roll No.

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Time: 3 Hre. Marks: 60

THE ANSWERS MUST BE ATTEMPTED ON THE ANSWER SHEET PROVIDED

Q.1. Solve the following:

(6x5=30)

(i) Find two positive numbers x and y whose sum is 30 and are such that xy? is as

large as possible.

(i) For what value of x in the interval -1 ≤ x S 4, is the graph of the function

f(x) = 2x? - steepest? What is the slope of tangent at this point?

(iii) A business manager determines that f months after production begins on a

next product, number of units produced will be p million per month, where

P= (41. Find p'(t) & p"(t). What happen to the production in the long run

as t → oo?

(iv)

Find the area of the region enclosed by the line y 4x and the curve

y＝ギ+3x。

(r) Evaluate the definite integral f* (× + ½) dx.

(vi) After t hours on job, a factory worker can produce 100te-0.5f units per

hour. How many units does the worker produce during the first 3 hours?

Solve the following:

(3x10=30)

Q. Exaluate Exaluate the integrals:

(a) f(Inx)?dx

(b) J·+2 dx.

Q. 3 A company produces x units of commodity A and y units of commodity B. All the

units can be sold for p=100-x dollars per unit of 4 and q=100-y dollars per unit of

B. The cost in dollars of producing these units is given by the function C(x, y) =

x? + xy + y?. What should be x and y to maximize the profit?