BE Computer Engineering (Pokhara University) Calculus I (PU, MTH 110) Question Paper 2078

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(a) State the precise (ε–δ) definition of the limit of a function. Using this definition, prove that $\lim_{x\to 3}(4x-5)=7$. (5)

(b) Examine the continuity of the function

at $x=2$. If $f$ is discontinuous, classify the discontinuity and state how it may be removed. (5)

(c) Using the definition of the derivative (first principles), find $\dfrac{dy}{dx}$ for $y=\sqrt{2x+1}$, and hence evaluate the slope of the tangent at $x=4$. (6)

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(a) State Rolle's Theorem and the Lagrange Mean Value Theorem. Verify the Mean Value Theorem for $f(x)=x^3-3x$ on the interval $[-2,2]$ and find all values of $c$ that satisfy the conclusion. (6)

(b) A closed right-circular cylindrical can is to hold $500\,\text{cm}^3$ of liquid. Find the dimensions (radius and height) that minimise the total surface area of the can. (6)

(c) Evaluate $\displaystyle\lim_{x\to 0}\frac{e^x-1-x}{x^2}$ using L'Hôpital's rule, justifying each application. (4)

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(a) Derive the reduction formula for $I_n=\displaystyle\int \sin^n x\,dx$ in terms of $I_{n-2}$, and use it to evaluate $\displaystyle\int_0^{\pi/2}\sin^4 x\,dx$. (8)

(b) Evaluate $\displaystyle\int \frac{2x+3}{(x-1)(x^2+1)}\,dx$ using the method of partial fractions. (8)

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(a) State the integral test for the convergence of an infinite series. Using it, determine whether the series $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n(\ln n)^2}$ converges or diverges. (6)

(b) Find the interval of convergence and the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\frac{(x-2)^n}{n\,3^n}$. (6)

Evaluate the following limits:

(a) $\displaystyle\lim_{x\to 0}\frac{\tan x-\sin x}{x^3}$ (4)

(b) $\displaystyle\lim_{x\to \infty}\left(1+\frac{3}{x}\right)^{2x}$ (3)

(a) If $y=x^x$, find $\dfrac{dy}{dx}$ using logarithmic differentiation. (3)

(b) Find $\dfrac{dy}{dx}$ for the implicit relation $x^3+y^3=6xy$ at the point $(3,3)$. (4)

The radius of a spherical balloon is increasing at a rate of $2\,\text{cm/s}$. At the instant when the radius is $10\,\text{cm}$, find the rate at which (a) the volume and (b) the surface area of the balloon are increasing.

Evaluate the following integrals:

(a) $\displaystyle\int x^2 e^{x}\,dx$ using integration by parts. (4)

(b) $\displaystyle\int \frac{dx}{\sqrt{9-4x^2}}$. (3)

(a) Using the property $\displaystyle\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx$, evaluate $\displaystyle\int_0^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$. (4)

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

(a) Evaluate the improper integral $\displaystyle\int_1^{\infty}\frac{dx}{x^2}$ and state whether it converges or diverges. (3)

(b) Test the convergence of $\displaystyle\int_0^{1}\frac{dx}{\sqrt{1-x}}$, evaluating it where it converges. (4)

(a) Determine whether the sequence $a_n=\dfrac{3n^2+2n}{n^2+1}$ converges, and if so find its limit. (3)

(b) Test the convergence of the series $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^n}$ using the ratio test. (4)

(a) Sketch the cardioid $r=1+\cos\theta$ and identify its symmetry. (2)

(b) Find the area enclosed by one loop of the curve $r=2\sin 2\theta$. (4)