School of Mathematics and Statistics
Carleton University
Math. 69.104
ASSIGNMENT 1
SOLUTIONS
Due: October 19, 1999 (or Oct.18,1999 Mr. Dubé),

1. [10] Let a > b be two real numbers. Using any method, evaluate the following limit:

   

   .

   Solution There are two cases, namely, a=0 and .

   Case 1 If then we can use L'Hospital's Rule. Indeed, in this case,

   

   Case 2 On the other hand, if a =0, then we rationalize the numerator (since we cannot use L'Hospital's Rule here since the ``derived limit" does not exist ...) to find,

   

   since -b = |b| when b < 0, by definition. Now as the last limit does not exist as we have different left- and right-hand limits at x = 0, namely, and respectively. Hence the limit does not exist if a=0. Combining these results we get that

   

2. [5]a)

   Let Find the derivative of f, i.e., , using any method whatsoever.

   Solution The Box method is easiest. Let . Then, by the Chain Rule,

   

   But

   

   It follows that

   

   [5]b)

   Evaluate to 3 significant digits.

   Solution 0.605

3. [5]a)

   Without finding its derivative, determine the number of real roots of the derived polynomial function where

   

   Solution Since this is a polynomial of degree 4 (in disguise), its derivative must be of degree 3 (since its leading coefficient is 1) and so, by the Fundamental Theorem of Algebra, it can have at most three roots. That there are 3 real roots follows from part (b) below.

   [5]b)

   Determine intervals containing the roots in (a).

   Solution Use Rolle's Theorem applied to the function f(x). Note that f(1) =0 and f(2) = 0. Hence it's derivative, somewhere in the open interval (1,2). A similar argument applied to the intervals (2,3) and (3,4) shows that somewhere in these intervals too. Hence the three roots belong to the intervals (1,2), (2, 3), (3,4).

   Hint: Use Rolle's Theorem.

4. Find the derivatives of the following functions using any method.

   [5] a)

   ,

   Solution Apply the natural logarithm to both sides and use implicit differentiation along with the Product Rule. We get,

   

   and so

   

   Solving for we get

   

   [5] b)

   

   Solution Use the Box method, for example. Then

   

5. Evaluate the following limits (Show all work).

   [5] a)

   .

   Solution L'Hospital's Rule is best.

   

   [5] b)

   .

   Solution Note that , by definition of the principal part of the Arctangent function. Thus the required limit must be 0.

6. [10] It is known that barometric pressure p changes with altitude h according to the law

   

   where denotes normal pressure (given at the outset) and c is a constant. At an altitude of 5540 meters the pressure is found to be equal to one-half the value of normal pressure. Find the rate of change of the barometric pressure with respect to altitude. Hints: Verify that the Law is really an exponential law ``in disguise". Now proceed as in the derivation of the Law of Radioactive Decay and find c given that At an altitude of 5540 meters the pressure is found to be equal to one-half the value of normal pressure. Your final answer should look like

   

   where you need to find c explicitly.

   Solution Apply the exponential function to both sides to find

   

   or, equivalently,

   

   We are given that when h = 5540 then . Using this in the last display gives

   

   which, when solved for c, gives . It follows that

   

Total: /60

• About this document ...