College Algebra Laboratory I.

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This definition can be turned around into the following fact. This kind of discontinuity in a graph is called a jump discontinuity.

Jump discontinuities occur where the graph has a break in it as this graph does and the values of the function to either side of the break are finite i. This kind of discontinuity is called a removable discontinuity.

Removable discontinuities are those where there is a hole in the graph as there is in this case. A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it.

In other words, a function is continuous if its graph has no holes or breaks in it. Example 2 Determine where the function below is not continuous. So all that we need to is determine where the denominator is zero. A nice consequence of continuity is the following fact. With this fact we can now do limits like the following example.

Example 3 Evaluate the following limit.

Below is a graph of a continuous function that illustrates the Intermediate Value Theorem. Also, as the figure shows the function may take on the value at more than one place. It only says that it exists.

These are important ideas to remember about the Intermediate Value Theorem. A nice use of the Intermediate Value Theorem is to prove the existence of roots of equations as the following example shows.

For the sake of completeness here is a graph showing the root that we just proved existed. Note that we used a computer program to actually find the root and that the Intermediate Value Theorem did not tell us what this value was. If it does, then we can use the Intermediate Value Theorem to prove that the function will take the given value.

62 Chapter 2 Limits and Continuity 6. Power Rule: If r and s are integers, s 0, then lim x→c f x r s Lr s provided that Lr s is a real number. The limit of a rational power of a function is that power of the limit of the func-tion, provided the latter is a real number. chapter P (review) Class Expectations PreCalculus Expectations pdf CHAPTER ONE Fundamentals. Section notes: PCalc Ppdf homework: PCalc P.1 srmvision.com classwork (a) PRECALC CH1A Interval srmvision.com (b) Working w srmvision.com (c Domain of Functions. Limits are the most fundamental ingredient of calculus. Learn how they are defined, how they are found (even under extreme conditions!), and how they relate to continuous functions. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more.

We now have a problem. So, what does this mean for us? Okay, as the previous example has shown, the Intermediate Value Theorem will not always be able to tell us what we want to know.LIMITS AND CONTINUITY 1.

The concept of limit x2 − 4. Examine the behavior of f (x) as x approaches 2. Example Let f (x) = x−2 Solution. Review 1 Limits Continuity Pcalc To AP Calc Essay AP Calculus Review Limits, Continuity, and the Definition of the Derivative Teacher Packet Advanced Placement and AP are registered trademark.

and the Definition of the Derivative Page 11 of 18 π ⎧ ⎪sin x.. lim+ x→ 0 1 2 a (B) 1 a (C) a (D) 2 a (E) Does not exist (D) 1 (E) ∞ ln 2 x = 2x (A) − ∞ (B) −1 (C) 0 ⎧ srmvision.com Dallas.

x. Chapter 1 review. This is the end of the test. When you have completed all the questions and reviewed your answers, press the button below to grade the test. Review Exercises Use the Textbook Edition Upgrade Tool to automatically update assignments from this title to corresponding questions in the newest edition of .

Calculus I Chapter 1 and 2 Test Review Key 1. Evaluate the expression) 3 2 Use the graph as shown to determine the following limits, and discuss the continuity of the function at.

(i) (ii) (iii) (i) 2 (ii) 2 (iii) 2 Calculus I - Chapter 1_2 Test Review srmvision.com Justify your answer. 2 (c) 4 = 4a − 2b 1 = 4a − b 2 pts for finding a in terms of b 1 pt for continuity equation 1 pt for differentiability equation Continued on next page.

1 pt value The function.

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Limits and continuity | Calculus, all content ( edition) | Khan Academy