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3 years ago

12

Find a linearization function at a suitably chosen integer near a at which the given function and its derivative are easy to eva

luate

1 answer:

baherus [9]3 years ago

5 0

Here it is: 2+3+4+5+6

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All digits in a dropdown number are different, and one of its digits is the average of all its digits. It has at least two digit

Svetradugi [14.3K]

1. Start the search among 2-digit numbers. A dropdown number (DDN) with 2 digits is a number $ab$ such that

$\dfrac{a+b}2 = a \implies a + b = 2a \implies b = a$

or

$\dfrac{a+b}2 = b \implies a+b = 2b \implies a=b$

but both of these solutions violate the requirement that the digits are distinct, so there are no 2-digit DDNs.

A 3-digit DDN $abc$ is such that

$\dfrac{a+b+c}3 = a \implies a+b+c = 3a \implies b+c = 2a$

or $a+c=2b$ if the average is $b$ , or $a+b=2c$ if the average is $c$ . The smallest possible value for $a$ is 1 since we require 3 digits. Then $b+c=2$ , and we can pick $b=0$ and $c=2$ to get the smallest DDN, 102.

2. In a 4-digit DDN $abcd$ , we have

$\dfrac{a+b+c+d}4 = a \implies a + b + c + d = 4a \implies b+c+d=3a$

or $a+c+d=3b$ or $a+b+d=3c$ or $a+b+c=3d$ .

We're free to fix $a=1$ and $b=0$ to try to get the smallest DDN. This leaves us with $c+d=3$ or $c+d=-1$ or $1+d=3c$ or $c=3d$ .

The first two cases are impossible - the only choices for $c,d$ such that $c+d=3$ are 1 and 2, and the sum of two positive integers must be positive. The smallest possible value of $c$ is 2; this leaves us with $1+d=6$ or $2=3d$ , but the latter case is impossible because 3 does not divide 2. So $d=5$ , and the <em>smallest</em> 4-digit DDN is 1025.

To find the largest DDN, start with the largest possible values for $a$ and $b$ . Let $a=9$ and $b=8$ . Then $c+d=19$ or $c+d=15$ or $17+d=3c$ or $17+c=3d$ . At most, we can have $c+d=13$ with 7 and 6, so the first two cases are impossible. If we maximize $c=7$ , then either $17+d=21\implies d=4$ or $24=3d\implies d=8$ (which we don't want). So the <em>largest</em> 3-digit DDN is 9874.

3. I don't have an analytical solution to this, but using brute force (program) the total count is 112.

4. It is possible; consider 1249 and 1250, with digital averages

$\dfrac{1+2+4+9}4=4 \text{ and } \dfrac{1+2+5+0}4=2$

which happens to be the smallest pair. (Also found with brute force.)

3 0

2 years ago

Read 2 more answers

Can someone help me find X of this triangle? thanks :)

tino4ka555 [31]

Since it is an equilateral triangle, all of the sides are the same length.

So, x=14 :)

Hope this helps!

7 0

3 years ago

Read 2 more answers

How does a nuclear power plant produce electricity?

alexdok [17]

I believe the answer is A.Quickly moving neutrons coming out of the reaction are slowed down by water. The water heats up and turns into steam. The steam turns the turbine and produces electricity.

5 0

3 years ago

An animal shelter spends $3.50 per day to care for each bird and $6.50 per day to care for each cat. Brooklyn noticed that the s

Vikki [24]

Solution: Let $x$ be the number of birds and $y$ be the number of cats.

$x+y=29 \implies y=29-x$

$3.50 x +6.50 y= 146.50$

$3.50 x+6.50(29-x)=146.50$

$3.50x+188.5-6.5x=146.50$

$3x=188.5-146.50$

$3x=42$

$x=\frac{42}{3} =14$

Therefore, there were 14 birds at the shelter on Friday

8 0

4 years ago

I need help ASAP please help

alukav5142 [94]

Answer:

The origin is (0,0)

So: Its in the middle of the graph.

Step-by-step explanation:

5 0

3 years ago