2.5 Continuity Worksheet - Classwork for PDF

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This worksheet will prepare you for the quizzes and future exams for this topic in the class....

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(2.5) Continuity 1. Here is the graph of a function f . 4

(a) At what values of x in (−4, 4) do you think f is not continuous? In other words, at what values of x does f have a discontinuity?

3 2 1 1

-4 -3 -2 -1

2

3

4

-1 -2 -3 -4

(b) For each discontinuity that you identified, determine whether it is a jump discontinuity, a removable discontinuity (hole), or an infinite discontinuity. (An infinite discontinuity is one type of essential discontinuity.)

2. The Definition of Continuity. A function f is continuous at x = a if three things are equal:

This definition implies three things:

What makes each type of discontinuity fail the definition of continuity? • If the graph of a function f has a vertical asymptote at x = a, then f is discontinuous at x = a because:

• If the graph of a function f has a jump at x = a, then f is discontinuous at x = a because:

• If the graph of a function f has a hole at x = a, then f is discontinuous at x = a because:

3. A function can also be continuous from one side but not the other! Looking back at #1, do you see a value of x where this happens? Which two quantities from the definition of continuity are equal there?

A function f is continuous on an interval if it is continuous at every number in the interval.

4. Looking again at the graph in #1, on which intervals is f continuous?

If f and g are continuous at some value a and if c is some constant, then the following functions are also continuous at a. f +g

cf f −g

fg

f where g(x) 6= 0 g

5. The following types of functions are continuous on their domains. For each type, describe the domain. (a) polynomials (b) rational functions (c) radical functions (d) sine and cosine (e) tangent (f) exponential functions (g) logarithmic functions 6. Let f be the function defined by ( −x2 + 7 f (x) = 2x + a

for x ≤ 1 for x > 1

For what values of a will f be continuous on all real numbers? (Is a part of the input or the output?)

7. Where are the following functions discontinuous? (a) f (x) =

x2 − x − 2 x−2

(b) g(x) =

1 x2 − 2x + 1

Is the discontinuity in f removable? That is, can you define a new function, h, so that f and h agree everywhere that f is defined and so that h is continuous? Can you do this for g ?

Theorem: If f is continuous at b and lim g(x) = b, then lim f (g(x)) = x→a

=

.

x→a

In other words: Theorem: If g is continuous at a and f is continuous at is continuous at .

, then the composite function f ◦ g

8. Where are the following functions continuous? (a) j(x) =

ln(x) + tan−1 (x) x2 − 1

(b) g(x) =

√ 1 − sin x

(c) f (x) = cos(3x − sin x) 9. Use #8(c) to evaluate lim cos(3x − sin x). x→π

10. In each part, can you sketch a continuous function f (x) defined on [−4, 3] which has all three of the given properties, or is it impossible to do so? (a) • f (−4) = 2 • f (3) = 5 • f has no zeros in [−4, 3]

(b) • f (−4) = 2 • f (3) = −1 • f has no zeros in [−4, 3]

11. One of the above graphs was impossible, and one was not. What made the impossible graph impossible? Besides zero, are there other numbers that it would be impossible to “leave out” of the range, if the function had to be continuous? What numbers are those?

Intermediate Value Theorem. Suppose f is continuous on the closed interval [a, b]. Let N be any number between f (a) and f (b), and assume f (a) 6= f (b). Then there exists a number c between and

such that

.

We can illustrate this with a graph:

12. (Modified from Example 10 in Stewart.) Show that there is a solution to 4x3 + 3x = 6x2 + 2 between 1 and 2. (Hint: use the IVT.)

13. If f (x) = 3ex + 7x3 , show that there is a number c such that f (c) = 4.7.

14. Is the following argument correct or incorrect? Why? In the 2018 Super Bowl, the Eagles beat the Patriots 41 to 33. Since the Eagles had 0 points at the beginning of the game and 41 at the end, the Intermediate Value Theorem says that the Eagles must have had exactly 4 points at some moment in the game....