Basic Mathematics 2080

Section A

Long Answer Questions
Attempt any TWO questions. [2 × 10 = 20]

1.(a) Define gradient of a vector function f(x, y, z) and find the derivative of

 f(x, y, z) = x³ - xy² - z at point P₀(1,1,0) in the direction of 𝑉⃗ = 2𝑖⃗ - 𝑗⃗ + 6𝑘⃗. [5]

(b) Define Volume of a solid and find the volume of the solid generated by revolving the region bounded by y = √x and the lines y = 1, x = 4 about the line y = 1. [5]

2. Evaluate:
(a)     (2.5)

(b)     (2.5)
Solve the differential equation   [5]

3.(a) State Rolle's Theorem and show that x³ + 3x + 1 = 0 has exactly one real solution. [5]
(b) Find the area of the region enclosed by the parabola y = 2 - x² and the line y = -x. [5]

Section B

Short Answer Questions
Attempt any EIGHT questions. [8 × 5 = 40]

4. Define the absolute value function and sketch the graph of  absolute value. [5]

5. Find the limit of [5]

6. State the integral test and apply it to test the convergence of the series  . [5]

7. Find the Taylor Series generated by f(x) = 1/x at a = 2. Where, if anywhere, does the series converge to 1/x? [5]

8. Define implicit differentiation and find the slope of the circle x² + y² = 25 at the point (3, -4). [5]

9. Define partial derivative and find the value of at the point (4, -5) if 

f(x, y) = x² + 3xy + y - 1. [5]

10. Evaluate:[5]
(a) ∫ (sin 2x cos x + cos 2x sin x) dx 
(b) 

11. Determine the concavity of y = 3 + sin x on [0, 2π]. [5]

12. Test for convergence of the series. [5]