Mathematics I (Calculus) 2077

Group A(10 x 3 = 30)

Attempt any THREE questions.

1(a) If f(x) = x2 then find .

1(b) Dry air is moving upward. If the ground temperature is 20⁰ and the temperature at a height of 1km is 10⁰ C, express the temperature T in ⁰C as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b)Draw the graph of the function in part(a). What does the slope represent? (c) What is the temperature at a height of 2km?(5)

1(c). Find the equation of the tangent to the parabola y = x² + x + 1 at (0, 1)

2(a)A farmer has 2000 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimentions of the field that has the largest area?[5]

2(b)Sketch the curve[5]

3(a)Show that the converges  and diverges .[2]

(b) If f(x, y) = xy/(x² + y²), does f(x, y) exist, as (x, y) → (0, 0)?[3]

3(c) A particle moves in a straight line and has acceleration given by a(t) = 6t² + 1. Its initial velocity is 4m/sec and its initial displacement is s(0) = 5cm. Find its position function s(t).[5]

4. (a) Evaluate[5]

4(b) Find the Maclaurin's series for cos x and prove that it represents cos x for all x.[5]

Group B(10 x 5 = 50)

Attempt any TEN questions.

5. If f(x) = x² - 1, g(x) = 2x + 1, find fog and gof and domain of fog.

6. Define continuity of a function at a point x = a. Show that the function f(x) = is  continuous on the interval[1, -1].

7. State Rolle's theorem and verify the Rolle's theorem for f(x) = x³ - x² - 6x + 2 in [0, 3].

8. Find the third approximation x₃ to the root of the equation f(x) = x³ - 2x - 7, setting x1 = 2.

9. Find the derivatives of r(t) = (1 + t²)i - te^-tj + sin 2tk and find the unit tangent vector at t=0.

10. Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x².

11. Solve: y" + y' = 0, y(0) = 5, y(π/4) = 3

12. Show that the series  converges.

13. Find a vector perpendicular to the plane that passes through the points:p(1, 4, 6), Q(-2, 5, -1) and R(1. -1, 1)

14. Find the partial derivative of f(x, y) = x³ + 2x³y³ - 3y² + x + y, at (2,1).

15. Find the local maximum and minimum values, saddle points of f(x,y) = x⁴ + y⁴ - 4xy + 1.