Tribhuvan University
Institute of Science and Technology
TU Exam Year: 2079
Candidates are required to give their answers in their own words as far as practicable.
SECTION A
Attempt all questions.
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1.(a) If a function is defined by \( f(x) = \begin{cases} 1 + x, & x \leq -1 \\ x^2, & x > -1 \end{cases} \), 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 \( \frac{dz}{dt} \) 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 that its velocity \( v \) at time \( t \) is \( v = t^2 - 2t + 10 \).
(i) Find the displacement of the particle during the time period \( 1 \leq t \leq 4 \).
(ii) Find the distance traveled during this time period. -
4.(a) Define initial value problem. Solve: \( y'' + y' - 6y = 0 \), \( y(0) = 1 \), \( y'(0) = 0 \).
(b) Find the Taylor's series expansion for \( \cos x \) at \( x = 0 \). -
5.(a) Dry air is moving upward. If the ground temperature is 20° and the temperature at a height of 2 km is 10°C, express the temperature \( T \) in °C as a function of the height \( h \) (in kilometers), assuming that a linear model is appropriate.
(b) Draw the graph of the function and find the slope. Hence, give the meaning of the slope.
(c) What is the temperature at a height of 2 km? -
6.Find the equation of tangent at \( (1,3) \) to the curve \( y = 2x^2 + 1 \).
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7.State Rolle's theorem and verify the theorem for \( f(x) = x^2 - 9 \), \( x \in [-3, 3] \).
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8.Starting with \( x_1 = 1 \), find the third approximation \( x_3 \) to the root of the equation \( x^3 - x - 5 = 0 \).
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9.Show that the integral \( \int_0^3 \frac{dx}{x-1} \) diverges.
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10.Use Trapezoidal rule to approximate the integral \( \int_1^2 \frac{dx}{x} \) with \( n = 5 \).
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11.Find the derivative of \( \mathbf{r}(t) = t^2 \mathbf{i} - t e^{-t} \mathbf{j} + \sin 2t \mathbf{k} \) and find the unit tangent vector at \( t = 0 \).
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12.What is a sequence? Is the sequence \( a_n = \frac{n}{5+n} \) convergent?