UNIVERSITY OF THE PUNJAB
B.S. 4 Years Program / First Semester - Spring 2024
Roll N
Paper: MS-152 A
Time: 3 Hrs .
pect: Calculus and Analytical Geometry
THE ANSWERS MUST BE ATTEMPTED ON THE ANSWER SHEET PROVIDED
Q.1. Solve the following:
\( (6 \times 5=30) \)
(?) Evaluate the limit \( \lim _{x \rightarrow+\infty} \frac{\sqrt{3 x^{4}+x}}{x^{2}-8} \).
(iii) Let \( y=x^{5} \sec \left(\frac{1}{x}\right) \). Find \( \frac{d y}{d x} \).
(iii) Confirm that the formula \( (5+\Delta x)^{3} \approx 125+75 \Delta x \) is the local linear approximation \( f(x)=(4+x)^{3} \) at \( x_{0}=1 \), where \( \Delta x=x-1 \).
(iv) Estimate the area between the graph of the function \( f(x)=\frac{1}{x} \) and the interve \( [a, b]=[1,2] \). Estimate the specified area using \( n=5 \) rectangles.
(i) Evaluate the integral \( \int_{0}^{\sqrt{\pi}} 5 x \cos \left(x^{2}\right) d x \).
(vi) Determine wether the points \( P_{1}(6,9,7), P_{2}(9,2,0) \), and \( P_{3}(0,-5,-3) \) lie on the same line.
Solve the following.
\[
(3 \times 10=30)
\]
Q.2. Find: (a) the intervals on which \( f(x)=\frac{x}{x^{2}+2} \) is increasing, (b) the intervals on which \( f \) is decreasing, (c) the open intervals on which \( f \) is concave up, (d) the open intervals on which \( f \) is concave down, and (e) the \( x \)-coordinates of all inflection points.
2.8. Evaluate the integral \( \int_{0}^{3} \frac{x^{3}}{\left(3+x^{2}\right)^{5 / 2}} d x \).
2.4. Find the distance between the skew lines \( L_{1}: x=3-t, y=4+4 t, z=1+2 t \) an \( L_{2}: x=t, y=3, z=2 t \).
