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2 years ago

John estimates the value of his car over time. The equation for the line of best fit is approximated as

Mathematics

2 answers:

Nookie1986 [14]2 years ago

7 0

Answer:

For the given equation of the line of best fit, the values that complete the table are as follows: a=14.8, b=0.1, c=9, d=1.1.

Step-by-step explanation:

An equation for a line of best fit follows the same guidelines as a linear function where 'y' represents the total (value), 'x' represents time (years), -2.9 is the rate, and 17.7 is the starting value. The table indicates that for any year, there is a given value, but what we are solving for is the predicted value. The residual is the different between the given and predicted values. So, for 'a', we need to solve for the 'y' in our equation by replacing 'x' with '1', multiplying by -2.9 and adding 17.7. This gives us 14.8. For 'b', we simply need to subtract the given and predicted values to get a residual of 0.1. For 'c', we again solve for 'y' by replacing 'x' with '3' in our given equation to get 9. And, for 'd' we subtract the given value of 5 and the predicted value of 6.1 to get 1.1.

alexira [117]2 years ago

6 0

Answer:

Step-by-step explanation:

a = 14.8

b = 0.1

c = 9

d = -1.1

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Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to

nikklg [1K]

Answer:

Part 1)

See Below.

Part 2)

$\displaystyle (-0.179, -0.178) \cup (-0.010, 0.012)$

Step-by-step explanation:

Part 1)

The linear approximation L for a function f at the point x = a is given by:

$\displaystyle L \approx f'(a)(x-a) + f(a)$

We want to verify that the expression:

$1-36x$

Is the linear approximation for the function:

$\displaystyle f(x) = \frac{1}{(1+9x)^4}$

At x = 0.

So, find f'(x). We can use the chain rule:

$\displaystyle f'(x) = -4(1+9x)^{-4-1}\cdot (9)$

Simplify. Hence:

$\displaystyle f'(x) = -\frac{36}{(1+9x)^{5}}$

Then the slope of the linear approximation at x = 0 will be:

$\displaystyle f'(1) = -\frac{36}{(1+9(0))^5} = -36$

And the value of the function at x = 0 is:

$\displaystyle f(0) = \frac{1}{(1+9(0))^4} = 1$

Thus, the linear approximation will be:

$\displaystyle L = (-36)(x-(0)) + 1 = 1 - 36x$

Hence verified.

Part B)

We want to determine the values of x for which the linear approximation L is accurate to within 0.1.

In other words:

$\displaystyle \left| f(x) - L(x) \right | \leq 0.1$

By definition:

$\displaystyle -0.1\leq f(x) - L(x) \leq 0.1$

Therefore:

$\displaystyle -0.1 \leq \left(\frac{1}{(1+9x)^4} \right) - (1-36x) \leq 0.1$

We can solve this by using a graphing calculator. Please refer to the graph shown below.

We can see that the inequality is true (i.e. the graph is between y = 0.1 and y = -0.1) for x values between -0.179 and -0.178 as well as -0.010 and 0.012.

In interval notation:

$\displaystyle (-0.179, -0.178) \cup (-0.010, 0.012)$

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2 years ago

Refer to table 28-6. what is the u-2 measure of labor underutilization?

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2 years ago

300 reduced by twice a number is 146. what is the number?

Bogdan [553]

300 - 2x = 146, where x is the number

Subtract 300 from both sides.

-2x = -154

Divide both sides by -2.

x = 77

4 0

2 years ago

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An equation was created for the line of best fit from actual enrollment data. It was used to predict the dance studio enrollment

stealth61 [152]

Answer: First option is correct.

Step-by-step explanation:

Enrollment month Actual Predicted Residual

January 500 8 4

February 400 15 -1

March 550 15 -1

April 13 12 -1

May 16 17 -1

June 14 15 -1

Since we know that

Residual value = Actual value - Predicted value

Sum of residuals is given by

$4-1-1+1-1-1=1$

since we can see that sum of residual is more than 0.

So, it can't be a good fit .

Hence, No, the equation is not a good fit because the sum of the residuals is a large number.

Therefore, First option is correct.

6 0

2 years ago

Find the equation of the line which passes through (−2, 3) and the point of intersection of the lines x + 2y=0 and 2x − y − 12=0

Alik [6]

Answer: y = 0.794*x + 4.588

Step-by-step explanation:

A linear relationship can be written as:

y = a*x + b

where a is the slope and b is the y-axis intercept.

For a line that passes through the points (x1, y1) and (x2, y2), the slope can be written as:

a = (y2 - y1)/(x2 - x1).

In this case the points are:

(-2, 3) and the intersection of the lines:

x + 2y = 0

2x - y - 12 = 0

To find the intersection of those lines, we can first isolate one variable in one side of each equality, i will isolate the variable y.

y = -x/2

y = 2x - 12

Now we can write:

-x/2 = 2x - 12

Solving this we can find the value of x at which both lines intersect.

2x + x/2 = 12

(5/2)*x = 12

x = 12*(2/5) = 4.8

Now we evaluate one of the lines in that point and get:

y = -4.8/2 = -2.4

Then these lines intersect at the point (4.8, -2.4)

Now we can find the slope of our equation.

a = (-2.4 - 3)/(4.8 - (-2)) = 0.794

then we have:

y = 0.794*x + b

And we know that when x = -2, y = 3

then:

3 = 0.794*-2 + b

3 + 1.588 = b = 4.588

Then the equation is:

y = 0.794*x + 4.588

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2 years ago

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