TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2079 B.S.
Candidates are required to give their answers in their own words as far as practicable.
Group A
Long Answer QuestionsAttempt any Two questions.
- 1(a) If a function is defined by $f(x)=\{1+x \text{ if } x \le -1, x^2 \text{ if } x > -1\}$, evaluate $f(-3)$, $f(-1)$ and $f(0)$ and sketch the graph. (b) Prove that $\lim_{x \to 0} \frac{|x|}{x}$ does not exist.
- 2(a) Sketch the curve $y=x^2+1$ with the guidelines of sketching. (b) If $z=xy^2+y^3$, $x=\sin t$, $y=\cos t$, find $dz/dt$ at $t=0$.
- 3(a) Estimate the area between the curve $y=x^2$ and the lines $x=0$ and $x=1$, using rectangle method, with four sub intervals. (b) A particle moves along a line so its velocity $v=t^2-2t+10$. i) Find displacement and distance traveled during $1 \le t \le 4$.
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4(a) Define initial value problem. Solve: $y''+y'-6y=0, y(0)=1, y'(0)=0$. (b) Find Taylor's series expansion for $\cos x$ at $x=0$.
- 5(a) Ground temp is $20$°C and height $2$km is $10$°C. Express $T$ as a function of height $h$. (b) Draw graph and find/explain slope. (c) Temperature at $2$km height?
- 6Find the equation of tangent at $(1,3)$ to the curve $y=2x^2+1$.
- 7State Rolle's theorem and verify for $f(x)=x^2-9, x \in [-3,3]$.
- 8Starting with $x_1=1$, find the third approximation $x_3$ to the root of $x^3-x-5=0$.
- 9Show that the integral $\int_{0}^{3} \frac{dx}{x-1}$ diverges.
- 10Use Trapezoidal rule to approximate the integral of $\frac{1}{x} dx$ from $1$ to $2$ with $n=5$.
- 11Find derivative of $\vec{r}(t)=t^2\vec{i} - te^{-t}\vec{j} + \sin 2t\vec{k}$ and find unit tangent vector at $t=0$.
- 12What is a sequence? Is the sequence convergent?