Math 111 – Calculus I



Math 111 – Calculus I.

Week Number Five Notes

Fall 2003

Limits at Infinity and Horizontal Asymptotes

Previously, we studied functions that grew without bound as x approached some real number a. However, there are instances where we want to study the “long term” behavior of a given function (in particular, to determine whether or not there is a bound on the growth of the function and, if it is bounded, attempt to determine one specific limiting value). We will do this by studying “limits at infinity”. We’ll begin with the following notation.

Definition 5.1: Assume f is defined for all real numbers greater than or equal to some number a. The following notation

indicates that the value of f(x) approaches L as x gets arbitrarily large (where L is (i) a real number, (ii) positive infinity, or (iii) negative infinity). We read this as follows: the limit as x approaches infinity of f(x) equals L.

EXERCISE: Design analogous notation and terminology for the following:

The limit as x approaches minus infinity of f(x) is negative infinity.

Let’s look at some pictures of some graphs below and see if we understand this concept intuitively.

Consider the graphs of the following two functions below.

f(x) = sin(x)/x for x in [-100,100] g(x) = 1/x2 for x in [-2,2]

Answer the following questions. What is

This motivates the definition of a horizontal asymptote. More precisely, the line y = a is a horizontal asymptote of f if f has a real number limit L at either positive or negative infinity.

Hence, in the above examples, the x-axis (y = 0) is a horizontal asymptote of both f and g.

Let’s do two more examples.

Example 5.2: Find all horizontal asymptotes (if any exist) of the following function.

Example 5.3 (an application from chemistry involving mixing solutions – p. 141 Section 2.5 of text – problem 43): A tank contains 5000 L of pure water. Brine containing 30 gallons of salt per liter is pumped into the tank at a rate of 25 liters per minute.

a) Show that the concentration of salt (after t minutes in grams per liter) is given by the following.

b) What is the limiting concentration of brine in the tank (i.e. what is the asymptotic behavior of C(t) as t approaches positive infinity)?

II. Tangent Lines and Instantaneous Velocities Revisited

Now that we have algebraic tools for computing limits, we will revisit some of the problems that we studies earlier in section 2.1 (look back at notes from week number two when we approximated these values using the graphing calculator). Let’s redo these problems using current techniques (and do a couple of other examples as well).

Example 5.5: Find the equation of the tangent line to the parabola y = 1/(x2 + 1) at the point P = (0,1) using empirical techniques.

Example 5.6 (an application from business – section 2.6 of the textbook – problem 25):

The cost (in dollars) of producing x units of a certain commodity is given by

C(x) = 5000 + 10x +.05x2.

a) Find the average rate of change of C with respect to x when the production level changes from x = 100 to x = 101.

b) Find the instantaneous rate of change of C with respect to x when the production level x is 100 units (this is called the marginal cost).

III. The Derivative of a Function f

We are finally ready to define the central concept in the study of differential calculus: THE DERIVATIVE.

Definition 5.7: Assume f is a function of a real variable and a is a member of the domain of f. If the following limit exists,

this limit is called the derivative of f at the number a. It is denoted f’(a).

Note that if we let x = a + h in (*), we obtain the following equivalent limit for f’(a).

Let’s look at a few more examples.

a) Compute f’(2).

b) Given an x > 0, compute f’(x) (this is equivalent to finding a general “formula” for f’(x)).

c) Show that f’(0) does not exist. Is there an appropriate limit you can attempt to compute? What is this limit? Use this to determine a “candidate” for the equation of the tangent line to f at the point (0,0). Does this seem sensible?

Example 5.9(problem 26, p. 150, section 2.6 of textbook): Assume a tank holds 100000 gallons of water that can be drained from the bottom of the tank in an hour. Torricelli’s Law asserts that the volume of water remaining in the tank after t minutes is given by the following function.

Find the rate that the water is flowing out of the tank for any time 0 < t < 60. Specifically calculate these instantaneous flow rates at t = 0, 10, 20,30,40,50, and 60 minutes respectively.

The Derivative as a Function

Let D = {x in the domain of f such that f’(x) exists}. Then, f’(x) is a WELL-DEFINED function with domain D. There are many instances in which it is convenient to think of the derivative as a function of f (rather than just calculating derivatives “value by value”).

ALTERNATIVE NOTATION FOR f’(x): dy/dx, Dx(f(x))

There are a number of reasons why you might want to study the derivative as a FUNCTION. Here are two.

1) The graph of f’ gives you an idea about whether f’ is (i) increasing on an open interval (ii) decreasing on an open interval (iii) has a maximum or minimum on a given open interval(see sketch of f(x) = x2 and tangent lines to f at x = -1,0,and 1 below).

The graphs above indicate (this is NOT a formal proof) that f is decreasing at x = -1 (in fact it is decreasing for all x < 0), f has a minimum value at x = 0 (actually called a local minimum of f – it is in fact f’s global minimum), and f is increasing at x = 1 (in fact it is increasing for all x > 0). The derivative, as a function, provides us with this information (QUESTION: What is f’(x) for f(x) = x2?).

2) There are practical problems in which we want to compute rates of change of the derivative function. For example, to define the instantaneous acceleration of a particle at a time t, we would want to compute the derivative of the velocity function at time t (remember that the velocity function is the derivative of the displacement function at time t).

Let’s look at one more example to end this section.

Example 5.10: Consider the sketch of the following function f below.

Sketch of f

a) Identify all values of x in the domain of f where f is not differentiable.

b) Where is f increasing on its domain? Where is f decreasing? Partition the domain into regions where f is increasing/decreasing.

c) Sketch the graph of f’(x) given f(x).

Non Hand-In Homework Problems Associated with Week #5 Notes

Sections 2.5 through 2.8

Section 2.5: 4,9,18-25,27,29,33,35,44

Section 2.6: 7-10,12,17,25

Section 2.7: 4-10,14,15,17,19,23,26,36

Section 2.8: 1,3,6,7,12,22,24,32

Read Sections 2.9 and 3.1 of the textbook

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