Basic Mathematics 2079

Section A

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

1. (i) If a function f(x) is defined by 

Evaluate f(-3), f(-1), and f(0) and sketch the graph. [1+1+1+2]
(ii) Define different types of discontinuity at a point. At what points  the function becomes  continuous of the function  f(x) = (x-2)/(x²-7x+10)? [3+2]

2. Define Gradient vector and directional derivative.[2+3+2+3]

Find the direction in which f(x, y) = 
(i) Increases and decreases most rapidly at the point (1,1).
(ii) What is the direction of zero change in f at (1,1)? 
(iii) Derivative of f(x, y) at the point (1,1) in the direction v = 3i - 4j. 

3. Define the concavity of the function. [2+3+3+2]

The graph of the function f(x) = x⁴ - 4x³ + 10 then
(i) Find the intervals on which f is increasing and on which f is decreasing. 
(ii) Find where the graph of f is concave up and where it is concave down. 
(iii) Find the local maximum or local minimum value of the function if they exist. 

Section B

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

4. (i) Find the domain and range of the function 

. [2]
(ii) Draw the graph of the function y = x² shifted up by 1 unit, down by 2 units, also shift 3 units to the left, and 2 units  right with new position of the function. [3]

5. Define horizontal and vertical asymptotes.[2+3]

 Find the appropriate asymptotes for the function f(x) = x - √(x² + 16). 

6. Find the area of the region between the x-axis and the graph of

 f(x) = x³ - x² - 2x.  1 ≤ x ≤ 2.                          [5]

7. Define  arc-length of a curve. Find the length of the curve y = (x/2)^(2/3) from x = 0 to x = 2. [1+4]

8. Evaluate the following integral: [3+2]

9. What is a first-order linear differential equation? [1+4]

Solve the initial value problem:

 . 

10. Determine whether the following series are convergent or divergent:[5]

11. What is the chain rule for a function w = f(x, y)?[1+3+1]

To  use this rule ,find the derivative of w = xy w.r.t t along the path x = cost, y = sint. Also, find the derivative of w at t = π/2. 

12. Find dy/dx for the following:[3+2]
(i) y² - x² = cos(xy) 
(ii) x² = 9/y²