What can you say about a solution of the equation y' = - y2 just by looking at the differential equation? The function y must be decreasing (or equal to 0) on any interval on which it is defined. The function y must be increasing (or equal to 0) on any interval on which it is defined.

Answers

Answer 1

Answer:

Answer:

The function y must be decreasing (or equal to 0) on any interval on which it is defined.

Step-by-step explanation:

The derivative of a function gives us the rate at which that function is changing. In this case, -y^2, yields a negative value for every possible value of y, thus, the rate of change is always negative and the function y is decreasing (or equal to 0) on any interval on which it is defined.

Answer 2

Answer:

Final answer:

The differential equation y' = - $y^2$ implies that y is either decreasing or constant wherever it is defined, because the derivative y' is non-positive.

Explanation:

By examining the differential equation y' = - $y^2$ , we can infer some characteristics about the solutions without solving it. If y is a solution to this equation, then y' represents the derivative of y with respect to x. This derivative tells us about the rate of change of the function y.

Since the right side of the equation is - $y^2$ , and a square of a real number is always non-negative, multiplying by -1 makes it non-positive. This implies that the derivative y' is either less than or equal to zero. Therefore, wherever the function y is defined, it must be either decreasing or constant (equal to zero). If y is positive, y will decrease because of the negative sign in front of the square. If y is negative, squaring it results in a positive number, but the negative sign still ensures that the rate of change is non-positive.

Conclusion: the function y is decreasing or remains constant on any interval it is defined; it cannot be increasing.

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A line passes through the points ( - 9, - 24) and (8, – 9).Calculate the slope of the line. Write your answer as a fraction.
(If the slope is undefined, enter "DNE")

Answers

Answer:

15 / 17

Step-by-step explanation:

slope = (y2 - y1) / (x2 - x1)

= (-9 + 24) / (8 + 9)

= 15 / 17

Consider the following functions. f(x) = x − 3, g(x) = x2 Find (f + g)(x). Find the domain of (f + g)(x). (Enter your answer using interval notation.) Find (f − g)(x). Find the domain of (f − g)(x). (Enter your answer using interval notation.) Find (fg)(x). Find the domain of (fg)(x). (Enter your answer using interval notation.) Find f g (x). Find the domain of f g (x). (Enter your answer using interval notation.)

Answers

Answer:

$(f+g)(x)=x-3+x^2$ ; Domain = (-∞, ∞)

$(f-g)(x)=x-3-x^2$ ; Domain = (-∞, ∞)

$(fg)(x)=x^3-3x^2$ ; Domain = (-∞, ∞)

$((f)/(g))(x)=(x-3)/(x^2)$ ; Domain = (-∞,0)∪(0, ∞)

Step-by-step explanation:

The given functions are

$f(x)=x-3$

$g(x)=x^2$

$(f+g)(x)=f(x)+g(x)$

Substitute the values of the given functions.

$(f+g)(x)=(x-3)+x^2$

$(f+g)(x)=x-3+x^2$

The function $(f+g)(x)=x-3+x^2$ is a polynomial which is defined for all real values x.

Domain of (f+g)(x) = (-∞, ∞)

$(f-g)(x)=f(x)-g(x)$

Substitute the values of the given functions.

$(f-g)(x)=(x-3)-x^2$

$(f-g)(x)=x-3-x^2$

The function $(f-g)(x)=x-3-x^2$ is a polynomial which is defined for all real values x.

Domain of (f-g)(x) = (-∞, ∞)

$(fg)(x)=f(x)g(x)$

Substitute the values of the given functions.

$(fg)(x)=(x-3)x^2$

$(fg)(x)=x^3-3x^2$

The function $(fg)(x)=x^3-3x^2$ is a polynomial which is defined for all real values x.

Domain of (fg)(x) = (-∞, ∞)

$((f)/(g))(x)=(f(x))/(g(x))$

Substitute the values of the given functions.

$((f)/(g))(x)=(x-3)/(x^2)$

The function $((f)/(g))(x)=(x-3)/(x^2)$ is a rational function which is defined for all real values x except 0.

Domain of (f/g)(x) = (-∞,0)∪(0, ∞)

$(f + g)(x) = x^2 + x - 3$ , domain: all real numbers.

$(f - g)(x) = -x^2 + x - 3$ , domain: all real numbers.

$(fg)(x) = x^3 - 3x^2$ , domain: all real numbers.

$f(g(x)) = x^2 - 3$ , domain: all real numbers.

To find (f + g)(x), we need to add the functions f(x) and g(x).

The function f(x) = x - 3 and the function $g(x) = x^2.$

So, $(f + g)(x) = f(x) + g(x) = (x - 3) + (x^2).$

Expanding this equation, we get $(f + g)(x) = x^2 + x - 3.$

To find the domain of (f + g)(x), we need to consider the domain of the individual functions f(x) and g(x).

Since both f(x) = x - 3 and $g(x) = x^2$ are defined for all real numbers, the domain of (f + g)(x) is also all real numbers.

To find (f - g)(x), we need to subtract the function g(x) from f(x).

So, $(f - g)(x) = f(x) - g(x) = (x - 3) - (x^2).$

Expanding this equation, we get $(f - g)(x) = -x^2 + x - 3.$

The domain of (f - g)(x) is also all real numbers, since both f(x) and g(x) are defined for all real numbers.

To find (fg)(x), we need to multiply the functions f(x) and g(x).

So, $(fg)(x) = f(x) * g(x) = (x - 3) * (x^2).$

Expanding this equation, we get $(fg)(x) = x^3 - 3x^2.$

The domain of (fg)(x) is all real numbers, since both f(x) and g(x) are defined for all real numbers.

To find f(g(x)), we need to substitute g(x) into the function f(x).

So, $f(g(x)) = f(x^2) = x^2 - 3.$

The domain of f(g(x)) is also all real numbers, as $g(x) = x^2$ is defined for all real numbers, and f(x) = x - 3 is defined for all real numbers.

In summary:

- $(f + g)(x) = x^2 + x - 3$ , domain: all real numbers.

- $(f - g)(x) = -x^2 + x - 3$ , domain: all real numbers.

- $(fg)(x) = x^3 - 3x^2$ , domain: all real numbers.

- $f(g(x)) = x^2 - 3$ , domain: all real numbers.

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5) if john climbs 30 steps up, then 15 steps down. how many steps did he take?

Answers

Answer:45 steps

Step-by-step explanation:all u have to do is add all that he climbed which is 30+ 15

Answer 45

Step-by-step explanation:

30 + 15 = 45

You have a piggy bank containing a total of 106 coins in dimes and quarters. If the piggy bank contains $17.05, how many dimes are there in the piggy bank? There are how many dimes in the piggy bank.

Answers

Answer:

63 dimes

Step-by-step explanation:

Set up a system of equations where d is the number of dimes and q is the number of quarters:

d + q = 106

0.1d + 0.25q = 17.05

Solve by elimination by multiplying the top equation by -0.25 to eliminate q:

-0.25d - 0.25q = -26.5

0.1d + 0.25q = 17.05

Add together and solve for d:

-0.15d = -9.45

d = 63

So, there were 63 dimes in the piggy bank

Laura is stuck in aisle six at the supermarket trying to decide which jar of peanut butter to buy.She can buy a 16-ounce jar for $2.59 or a 24-ounce jar for $3.29. Which jar has a lower unit price,
per ounce?

Answers

Answer: 24 ounce jar

Step-by-step explanation:

Unit price of 16 ounce jar

= 2.59 / 16

= 0.161875

Unit price of 24 ounce jar

= 3.29 / 24

= 0.137083

Translate the following into algebraic expressions: The first class has a kids in it, the second has b kids in it, and the third class has c kids in it. The kids from all three classes are divided equally between two buses. How many kids are in each bus?