If the original quantity is 15 and the new quantity is 19, estimate the percent change.

For the function y=3x2: (a) Find the average rate of change of y with respect to x over the interval [3,6]. (b) Find the instant

nirvana33 [79]

Answer:

The instantaneous rate of change of $y$ with respect to $x$ at the value $x = 3$ is 18.

Step-by-step explanation:

a) Geometrically speaking, the average rate of change of $y$ with respect to $x$ over the interval by definition of secant line:

$r = \frac{y(b) -y(a)}{b-a}$ (1)

Where:

$a$ , $b$ - Lower and upper bounds of the interval.

$y(a)$ , $y(b)$ - Function exaluated at lower and upper bounds of the interval.

If we know that $y = 3\cdot x^{2}$ , $a = 3$ and $b = 6$ , then the average rate of change of $y$ with respect to $x$ over the interval is:

$r = \frac{3\cdot (6)^{2}-3\cdot (3)^{2}}{6-3}$

$r = 27$

The average rate of change of $y$ with respect to $x$ over the interval $[3,6]$ is 27.

b) The instantaneous rate of change can be determined by the following definition:

$y' = \lim_{h \to 0}\frac{y(x+h)-y(x)}{h}$ (2)

Where:

$h$ - Change rate.

$y(x)$ , $y(x+h)$ - Function evaluated at $x$ and $x+h$ .

If we know that $x = 3$ and $y = 3\cdot x^{2}$ , then the instantaneous rate of change of $y$ with respect to $x$ is:

$y' = \lim_{h \to 0} \frac{3\cdot (x+h)^{2}-3\cdot x^{2}}{h}$

$y' = 3\cdot \lim_{h \to 0} \frac{(x+h)^{2}-x^{2}}{h}$

$y' = 3\cdot \lim_{h \to 0} \frac{2\cdot h\cdot x +h^{2}}{h}$

$y' = 6\cdot \lim_{h \to 0} x +3\cdot \lim_{h \to 0} h$

$y' = 6\cdot x$

$y' = 6\cdot (3)$

$y' = 18$

The instantaneous rate of change of $y$ with respect to $x$ at the value $x = 3$ is 18.

5 0

2 years ago

On Monday, Anya read 11 pages of a book in 1/2 hour. On Tuesday, she read 18 pages in 3/4 hour. Did Anya's reading speed increas

Lerok [7]

The way you should go about solving this really depends on how your teacher taught you...However, here's what I would recommend...
You know that 1/2 an hour is equal to 30 minutes, and 3/4 of an hour is equal to 45 minutes.
Using this you can then solve for how many pages she read per minute by dividing the number of pages read by the number of minutes read:11 pages/ 30 minutes to give you Monday's reading speed,and18 pages/ 45 minutes to give you Tuesday's reading speed.

Next, to calculate a percentage increase you need to do the following:
1. Determine the difference between the speeds (this means you will subtract Monday's reading speed from Tuesday's reading speed.)
2. Next you take that number and divide it by Monday's reading speed.
3. Multiply that answer by 100 to get the percentage.

I'm not going to tell you the speeds, as you should try to attempt to solve it by yourself, and I'm sure you need to show your work. I will however tell you that you should find there was a 3.3% increase from Monday to Tuesday.

If you need more help, let me know!

4 0

2 years ago

WILL GIVE BRAINLEIST SUPER EASY NEED TO TURN IN AT 9:45 PLS HURRY IF YOU HAVE TIME PLS HELP WITH OTHER QUESTIONS PLS THANK YOU

nignag [31]

Answer:

A. 24.5

B. 29.6

Step-by-step explanation:

A.

19 - 13.5 = 5.5

30 - 5.5 = 24.5

24.5 - 5.5 = 19

24.5

B.

43.8 - 36.7 = 7.1

36.7 - 7.1 = 29.6

29.6 - 7.1 = 22.5

29.6

7 0

2 years ago

You go to the store and pick out 3 apples for $0.60 each, 2 bananas for $0.50 each, and 1 quart of yogurt for $4.50. At checkout

Stella [2.4K]

Answer:

$6.21

Step-by-step explanation:

1 apple = $0.60

3 apples = $1.80 (Multiply 0.60 by 3)

2 bananas = $1.00 (Multiply 0.50 by 2)

1 quart of yoghurt = $4.50

Add the prices up first:

1.80 + 1.00 + 4.50 = $7.30

7.30 off 15% = $6.21

5 0

3 years ago

Use the graph to find the unit rate

Diano4ka-milaya [45]

<h3>Answer:</h3>

2.4 potatoes/pound

<h3>Explanation:</h3>

The unit rate is the slope of the line. The slope of the line is ...

... (change in y-value)/(change in x-value)

... = (36 potatoes - 12 potatoes)/(15 pounds - 5 pounds)

... = (24 potatoes)/(10 pounds)

... = 2.4 potatoes/pound

_____

The relationship here is proportional, so the line goes through the point (0, 0) and the unit rate can be figured from any point:

... (12 potates)/(5 pounds) = (24 potatoes)/(10 pounds) = (36 potatoes)/(15 pounds) = 2.4 potatoes/pound

4 0

2 years ago

Read 2 more answers