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Grace [21]
2 years ago
15

Can someone please help me with continuity? I would like to check my work, I’m a bit confused on the differentiability part

Mathematics
1 answer:
nordsb [41]2 years ago
6 0

For <em>f</em> to be <u>continuous</u> at some point <em>x</em> = <em>c</em>, you require that

• <em>f(c)</em> exists and is finite, and

• the limits of <em>f</em> as <em>x</em> approaches <em>c</em> from either side match, and their value must be <em>f(c)</em>

<em />

For <em>f</em> to be <u>differentiable</u> at <em>x</em> = <em>c</em>, you require that

• <em>f</em> is continuous at <em>c</em> (i.e. the above conditions are all met), and

• the derivative <em>f '</em> is continuous at <em>c</em>

Without having the graph of <em>f '</em>, you can assess whether the last condition is met by some function by mentally tracing a tangent line to the graph as you get closer to <em>c</em>. If the slope of the tangent doesn't change, then the function is differentiable. This is usually accompanied by jumps or sharp corners in the graph.

<em />

Some examples:

• every polynomial is both continuous and differentiable

• the absolute value function |<em>x</em>| is continuous but not differentiable at <em>x</em> = 0

• 1/<em>x</em> is neither continuous nor differentiable at <em>x</em> = 0

(a) Neither continuous nor differentiable

Why? <em>f</em> has a vertical asymptote at <em>x</em> = -2 and <em>f</em> (-2) does not exist. <em>f</em> is not continuous, and therefore not differentiable.

(b) Neither continuous nor differentiable

Why? From the left, <em>f</em> is approaching 2.5, while from the right, it's approaching 5.

(c) Neither continuous nor differentiable

Why? The limits from either side of <em>x</em> = 2 match and are equal to 0.5, but <em>f</em> (2) itself does not exist (which is indicated by the point being a hollow circle).

If the circle was instead filled in, so that <em>f</em> (2) = 0.5, then <em>f</em> would be continuous there, but still not differentiable. Notice the sharp corner. Or, using the tangent-line analysis, the slope of the tangent to the left of <em>x</em> = 2 is negative, but to the right it would be positive.

(d) Both continuous and differentiable

Why? <em>f</em> doesn't have any special features at <em>x</em> = 3 that would suggest it's not continuous nor differentiable.

(e) Neither continuous nor differentiable

Why? Similar reasoning as in (a). There's another vertical asymptote, but the graph shows <em>f</em> (4) = 1. However, the limits from either side of <em>x</em> = 4 are positive infinity, not 1.

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Joshua's recipe for chili requires 2 3/4 ​ pounds of beef. Sondra's recipe for chili requires 2 2/3 ​ pounds of beef. Which of t
Lemur [1.5K]

Answer:

Step-by-step explanation:

Your question isn't complete but let me help out based on the information given.

Joshua's recipe for chili requires 2 3/4 ​ pounds of beef. Sondra's recipe for chili requires 2 2/3 ​ pounds of beef.

The pounds of beef required by both of them will be:

= 2 3/4 + 2 2/3

= 2 9/12 + 2 8/12

= 4 17/12

= 4 + 1 5/12

= 5 5/12

The difference in the pounds of beef required by both of them will be:

= 2 3/4 - 2 2/3

= 2 9/12 - 2 8/12

= 1/12

Also, we.can also infer that Joshua recipe for chili requires more pounds of beef than that of Sondra

4 0
3 years ago
write about which method you prefer to use to divide by 5 counting up, counting back on a number line, or dividing by 10, and th
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Answer:

In First method : counting up, counting back on a number line,

If we want the quotient after dividing the number by 5 then we count how many 5 we get from 0 to the dividend.

For example :  \frac{30}{5}

Since, from 0 to 30 there are six 5's obtained. ( because 5 × 6 = 30 )

Thus,  \frac{30}{5}=6

In Second Method : dividing by 10, and then doubling the quotient.

First we divide the number by 10 then multiply the quotient by 2.

For Example: \frac{30}{5}

Since, \frac{30}{10}=3

2\times 3 = 6

Thus, \frac{30}{5}=6

Now, when we compare the above methods then we conclude that for the smaller numbers first method is appropriate because for small numbers we can easily count total 5's from 0. While for large numbers Second method is appropriate because it is hard to count the total 5's for the large number.


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3 years ago
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Answer:

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2 ( n + 5 ) = -2

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Now take 10 to the right side.

2n + 10 -10 = -2 - 10

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Then divide both sides by 2.

\frac{2n}{2} =\frac{-12}{2}

And now you'll the value of n.

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