assignment (eum113) (24 sept)

7/30/2019 Assignment (EUM113) (24 Sept)

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EUM113/3_SemI (2012/2013 )

The intention of this assignment is to assess your understanding on Calculus of OneVariable

Your assignment must be handwritten

Please submit the hard copy of the assignment to Dr. Loh (Room 3.09 PPKM) ON or BEFOREON or BEFORE 5 October 2013 , 5.00 pm.

Please note that LATE submissions are not considered .


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EUM113/3_SemI (2012/2013)

Assignment 1

1) The rate of growth of bacteria in a culture is found to be satisfying the equation

= 0.5 where N is the number of bacteria and t is the time in days. If initially

= 200 , find N when t=6.

2) We have a cable that weighs 2 lbs/ft attached to a bucket filled with coal that weighs800 lbs. The bucket is initially at the bottom of a 500 ft mine shaft. Answer each of the following about this.

(a) Determine the amount of work required to lift the bucket to the midpoint of the shaft.

(b) Determine the amount of work required to lift the bucket from the midpointof the shaft to the top of the shaft.

(c) Determine the amount of work required to lift the bucket all the way up theshaft.

3) (a) Verify the mean value theorem for integral and find all values of c in that interval.

(i) ( ) = 1, [1, ] (ii) ( ) = , [ , ] (b) Use the Simpsons 1/3 rule to approximate 21 by taking n=4.

4) Determine all the numbers c which satisfy the conclusions of the Mean Value

Theorem for derivative for the following function.

(a) ( ) = 3+ 2 2 on [ 1,2 ] (b) ( ) = + 1 on [0,3 ]


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5) Suppose that we know that ( ) is continuous and differentiable on [6,15]. Lets alsosuppose that we know that (6) = 2 and that we know that ( ) 10 . What isthe largest possible value for

(15 )?

6) Evaluate the following limits

(a) lim 23 +26 +8 (b) lim 1 5 4 1 (c) lim 5 +757 (d) lim sec tan

7) Verify the theorem lim ( ) ( ) = lim ( ) lim ( ) where all limitsexist for the function given below:

lim ( ) =lim 1sin and lim ( ) =lim cos 8) Determine the value of a that will make the function below continuous at = 0 .

( ) = < 0 = 02 > 0 9) A function

( ) is defined as

( ) = 0 < 12 = 1+ 1 1 < 2 Find the point or points of discontinuity of the function ( ) at = 1 .

10) If

( ) = 1 > 10 = 11 < 1

Show that the function ( ) is continuous but not differentiable at = 1 .11) Find the Maclaurin series for the following functions:

(a) e (b) cos (c) ln(1 + ) (d) cosh


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12) If ( ) = ln find its Taylor series at = 1 . Hence find ln 1.1 correct to six decimalpoints.

13) Find the Taylor series expansion about = 12

for the function

( ) = sin .

assignment (eum113) (24 sept)

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