Basic Mathematics Model Question

Group A         [2 × 10 = 20]

Attempt any two questions.

1.  (a) In 2000; 100 is invested in a savings account, where it grows by accruing interest that is compounded annually (once a year) at an interest rate of 5:5%. Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount A in the account after x years have elapsed.     [5]

   (b) Define when the function f(x) is odd and even. Also, define when a function f(x) is increasing and decreasing? If y = x² is a given function then determine the interval in which the function is increasing and decreasing and draw the graph of the given function. [1 + 1 + 1 + 2]

2. A rock breaks loose from the top of a tall cliff             [3 + 3 + 4]

    (a) Find average speed during the first 2 sec of fall.

    (b) What is its average speed during the 1sec interval between second 1 and second 2?

    (c) Find the speed of the falling rock at t = 1 and t = 2.

3. (a) Find the positive root of the equation         [3]

                f(x) = x² - 2 = 0

   (b) Find the Taylor series and the Taylor polynomials generated by f(x) = e^x at x = 0.         [3]

   (c) Use the Trapezoidal Rule with n = 4 to estimate . Compare the estimate with the exact value. [4]

Group B

Attempt any 8 questions.        [8 x 5 = 40]

4. Define horizontal asymptote to a curve y = f(x). Find the horizontal asymptote to the curve

        and draw the curve.

5. (a) Find the slope of the curve  at any point  What is the slope at the point x = −1 ?

    (b) Where does the slope equal −1/4?

    (c) What happens to the tangent to the curve at the point (a, 1/a) as a changes?

6. Water runs into a conical tank at the rate 9ft³/minutes . The tank stands point down and has a height of 10ft and a base radius of 5ft. How fast is the water level rising when the water is 6ft deep?

7. Find the absolute maximum and minimum values of  on the interval [−2, 3].

8. Find the area between the curves y = x ² − 2 and y = 2.

9. A pyramid 3m high has a square base that is 3m on a side. The cross section of the pyramid perpendicular to the altitude xm down from the vertex is a square xm on a side. Find the volume of the pyramid.

10. Draw a phase line for the equation 

        and use it to sketch solutions to the equation.

11. Find the second order derivative

    of .

12. Find the local extreme values of the function