BE Computer Engineering (Pokhara University) Calculus I (PU, MTH 110) Question Paper 2079

(a) Define the limit of a function $f(x)$ as $x \to a$ using the $\varepsilon$–$\delta$ definition. Using this definition, prove that $\lim_{x \to 3}(2x - 1) = 5$. (7)

(b) Examine the continuity of the function

at $x = 2$. If it is discontinuous, classify the type of discontinuity and state how (if possible) it may be removed. (7)

(a) State Rolle's theorem and the Mean Value Theorem for derivatives. Verify the Mean Value Theorem for the function $f(x) = x^3 - 3x$ on the interval $[-2, 2]$ and find all values of $c$ that satisfy the conclusion. (7)

(b) A rectangular box with an open top is to be constructed from a square sheet so that its volume is $32\,\text{cm}^3$. Find the dimensions of the box that minimize the amount of material used, justifying that your answer corresponds to a minimum using the second-derivative test. (7)

(a) Define convergence of an infinite series $\sum_{n=1}^{\infty} a_n$. State and prove the integral test for the convergence of a series of positive terms. (6)

(b) Test the following series for convergence or divergence, naming the test you apply in each case:

(i) $\displaystyle\sum_{n=1}^{\infty} \frac{n}{2^n}$

(ii) $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n \ln n}$

(iii) $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}$ (6)

(a) Explain how a point is represented in polar coordinates and derive the relations connecting polar coordinates $(r, \theta)$ with Cartesian coordinates $(x, y)$. Convert the Cartesian equation $x^2 + y^2 = 4x$ into polar form and identify the curve. (5)

(b) Sketch the cardioid $r = 1 + \cos\theta$ and find the area of the region enclosed by it. (5)

Evaluate the following limits:

(a) $\displaystyle\lim_{x \to 0} \frac{\sin 3x}{\tan 5x}$

(b) $\displaystyle\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x}$

(c) $\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$ using L'Hôpital's rule.

(a) Find $\dfrac{dy}{dx}$ if $x^y = y^x$. (4)

(b) If $y = e^{ax}\sin bx$, show that $y_2 - 2a\,y_1 + (a^2 + b^2)y = 0$, where $y_1$ and $y_2$ denote the first and second derivatives of $y$ with respect to $x$. (3)

Evaluate the following integrals:

(a) $\displaystyle\int x^2 e^{x}\,dx$ (using integration by parts)

(b) $\displaystyle\int \frac{2x + 3}{(x - 1)(x + 2)}\,dx$ (using partial fractions)

(c) $\displaystyle\int \frac{dx}{\sqrt{4 - x^2}}$

(a) Evaluate $\displaystyle\int_{0}^{\pi/2} \sin^4 x \cos^2 x \, dx$. (4)

(b) Find the area of the region bounded by the curve $y = x^2$ and the line $y = x + 2$. (3)

(a) Define an improper integral of the first kind. Evaluate $\displaystyle\int_{1}^{\infty} \frac{dx}{x^2}$ and state whether it converges or diverges. (3)

(b) Determine whether the integral $\displaystyle\int_{0}^{1} \frac{dx}{\sqrt{x}}$ converges, and if so, find its value. (3)

(a) Find the radius and interval of convergence of the power series $\displaystyle\sum_{n=1}^{\infty} \frac{(x - 2)^n}{n \, 3^n}$. (4)

(b) Obtain the Maclaurin series expansion of $f(x) = \cos x$ up to the term in $x^4$. (2)

Air is being pumped into a spherical balloon so that its volume increases at the rate of $100\,\text{cm}^3/\text{s}$. At the instant when the radius of the balloon is $5\,\text{cm}$, find the rate at which (a) the radius and (b) the surface area of the balloon are increasing. $\left(V = \frac{4}{3}\pi r^3,\; S = 4\pi r^2\right)$

(a) Find the slope of the tangent to the polar curve $r = 2 + 2\sin\theta$ at $\theta = \dfrac{\pi}{6}$. (3)

(b) Find the length of the arc of the curve $r = e^{\theta}$ from $\theta = 0$ to $\theta = \pi$. (3)