Todays Agenda Turn in Written HW A Upcoming Homework Section 1.5: - - PowerPoint PPT Presentation

Today’s Agenda

• Turn in Written HW A
• Upcoming Homework
• Section 1.5: Continuity
• Section 1.6: Limits involving infinity

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 1 / 10

Upcoming Homework

• WeBWorK #4: Section 1.5 due 9/4/2015
• WeBWorK #5: Section 1.6 due 9/9/2015
• Written HW B: Problems TBA, due 9/11/2015
• WeBWorK #6: Sections 2.1 and 2.2, due 9/14/2015

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 2 / 10

Section 1.5

The Intermediate Value Theorem

Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c ∈ (a, b) such that f (c) = N. The hypotheses of the Intermediate Value Theorem are critical: f must be continuous on the closed interval [a, b], otherwise the conclusion of the theorem does not hold! (Can you think of any examples of what goes wrong when the hypotheses are not satisfied? If f is not continuous on all

• f [a, b]? If the interval is open instead of closed?)

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 3 / 10

Section 1.5

An illustration of the Intermediate Value Theorem:

Image courtesy: http://tutorial.math.lamar.edu Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 4 / 10

Section 1.5

Example 1.5.1

Suppose f is continuous on [1, 5] and the only solutions of the equation f (x) = 6 are x = 1 and x = 4. If f (2) = 8, explain why f (3) > 6.

Example 1.5.2

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. (a) cos x = x, on (0, 1) (b)

3

√x = 1 − x, on (0, 1)

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 5 / 10

Section 1.6

Definition 1.6.1

The notation lim

x→a f (x) = ∞

means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side of a) but not equal to a.

Definition 1.6.2

The vertical line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following statements is true: limx→a f (x) = ∞ limx→a+ f (x) = ∞ limx→a− f (x) = ∞ limx→a f (x) = −∞ limx→a+ f (x) = −∞ limx→a− f (x) = −∞

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 6 / 10

Section 1.6

Definition 1.6.3

Let f be a function defined on some interval (a, ∞). Then lim

x→∞ f (x) = L

means that the values of f (x) can be made as close to L as we like by taking x sufficiently large.

Definition 1.6.4

The horizontal line y = L is called a horizontal asymptote of the curve y = f (x) if either lim

x→∞ f (x) = L

• r

lim

x→−∞ f (x) = L.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 7 / 10

Section 1.6

Comment Some textbooks use stricter definitions than ours does. They require lim

x→a f (x) = ∞

• r

lim

x→a f (x) = −∞

in order to call x = a a vertical asymptote (i.e., it is not enough for the limit to be only one-sided in order to be a vertical asymptote). Similarly, some textbooks require both lim

x→∞ f (x) = L

and lim

x→−∞ f (x) = L

in order for y = L to be called a horizontal asymptote.

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 8 / 10

Section 1.6

Example 1.6.5

In the theory of special relativity, the mass of a particle with velocity v is m = m0

• 1 − v2/c2 ,

where m0 is the mass of the particle at rest and c is the speed of light. What happens as v → c−? (Why can’t we take the limit as v → c+?)

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 9 / 10

Section 1.6

Example 1.6.6

A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt t minutes later (in grams per liter) is C(t) = 30t 200 + t . What happens to the concentration as t → ∞? Certainly we cannot continue to pump brine into the tank forever. What does this limit mean practically?

Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 2 September 2015 10 / 10