Basic Mathematics - Unit Wise 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 Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount A in the account after x years have elapsed.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]

1.  (i) What is even and odd function? Give example of each and write their symmetricity.                                                                                                         (1+1+1+1+1+1)

   (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]

    (ii) Find the domain and range of the following functions.

                    (2)

                  (2)

3.  (i) Sketch the graph of the function f(x)=x². Shifted vertically up to 1 and -2 units and horizontally up to 3 and -2 units.

4. Graph the following functions. Write their symmetricity and specify the interval over which the function is increasing and decreasing.

    a. y = -x³

    b. y = x²                        (2.5+2.5)

1.  a) What do you mean by asymptotes? How many types of asymptotes define each? (1+3)

    b) Find horizontal and vertical asymptotes of the following functions

            Does there other asymptotes exist?                (5+1)

    (ii) Find the  algebraically of the following functions.

            a)         (2.5)

            b)     (2.5)

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

        and draw the curve.

6. Show that  has a continuous extension to x=2.

7. Evaluate the

        (i)         (2.5)

        (ii)         (2.5)

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.

    b) An open top box is to be made by cutting small congruent squares from the corners of square sheet of tin having length 12 inch and is bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?        (8)

5. Find the equations of tangent and normal to the curve x³ + y³ - 9xy = 0 at the point (2, 4).   (3+2)

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?

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

                f(x) = x² - 2 = 0

3.  a) Define Newton's Raphson method with their formula.    (2)

6. What is L' Hospital rule? Using this rule evaluate the following

        a. 

        b.                 (1+2+2)

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].

11. Determine the concavity and find the inflection point of the function

        f(x)=x³-3x²+2

11. State Mean value theorem. Verify the mean value theorem if f(x) = x² + 2x -1 on [0, 1].    (1+4)

2. Define area between two curves

    a) Find area of the region enclosed by the parabola y=2 - x² and the line y = -x.                       

7. Define integration. Evaluate the following integral.

        a. 

        b.                     (1+2+2)

9. Integrate the following

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

    b) Define volume integral. Find the volume of solid generated by revolving the region bounded by the curve y² = x and the line y = 1, x = 4 about the line y = 1.   (1+5)

10. Find the area of the surface generated by revolving the curve  , about x-axis.

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.

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

4.  Find the initial value problem in

9. Solve the following differential equation.

10. Draw a phase line for the equation 

        and use it to sketch solutions to the equation.

2. Find the Tayllor's series generated by  at a=2 where it anywhere, does the series converges to .        (10)

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

5. Determine the convergence or divergence of the series

8. Define integral test and determine the convergence or divergence of the series.  (1+4)

8. Find the derivatives of the function  at  in the direction of .

10. Find the derivative of , at point P(1,1,0) in the direction of .

12. Find 

        (i) Y = 2u³, u = 8x - 1                    (2.5)

        (ii) Y = sinu, u = x - xcosx              (2.5)

11. Find the second order derivative

    of .

12.  a. Find  if .

        b. Find the slope of circle  at the point (3,4).

12. Find the local extreme values of the function