Let f(x) = x^6( x − 8 )^7 / (x^2 +2)^2

Mathematics

1 answer:

shutvik [7]3 years ago

5 0

f(x) = x⁶ (x - 8)⁷ / (x² + 2)²

then taking the logarithm of both sides lets us expand the right side as

ln(f(x)) = ln(x⁶ (x - 8)⁷ / (x² + 2)²)

ln(f(x)) = ln(x⁶) + ln((x - 8)⁷) - ln((x² + 2)²)

using the property ln(ab) = ln(a) + ln(b).

We can also use the property ln(aⁿ) = n ln(a), so that

ln(f(x)) = 6 ln(x) + 7 ln(x - 8) - 2 ln(x² + 2)

Then differentiating both sides using the chain rule gives

f'(x)/f(x) = 6 x'/x + 7 (x - 8)'/(x - 8) - 2 (x² + 2)'/(x² + 2)

f'(x)/f(x) = 6 (1)/x + 7 (1)/(x - 8) - 2 (2x)/(x² + 2)

f'(x)/f(x) = 6/x + 7/(x - 8) - 4x/(x² + 2)

Solve for f'(x) :

f'(x) = (6/x + 7/(x - 8) - 4x/(x² + 2)) f(x)

and replace f(x) with the given function :

f'(x) = (6/x + 7/(x - 8) - 4x/(x² + 2)) • x⁶ (x - 8)⁷ / (x² + 2)²

We can expand the product :

f'(x) = 6 x⁵ (x - 8)⁷ / (x² + 2)²+ 7x⁶ (x - 8)⁶ / (x² + 2)² - 4x⁷ (x - 8)⁷ / (x² + 2)³

then simplify by factoring :

f'(x) = x⁵ (x - 8)⁶/(x² + 2)³ • (6 (x - 8) (x² + 2) + 7x (x² + 2) - 4x² (x - 8))

Simplify the rest :

f'(x) = x⁵ (x - 8)⁶ (9x³ - 16x² + 26x - 96)/(x² + 2)³

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Use the given graph to determine the limit, if it exists.

Marina CMI [18]

The limit is the value of the function when you approach the closest possible to the x value without getting to that x value.

The closed point (2,1) is not used to calculate the limit. The closed point (2,1) represents the value of the function at x = 1, but the limit is not the value of the function.

Use the horizontal line that comes from the left and ends at the open point (2,3) to find the limit of f(x) when x approaches to 2 from the left.

That limit is 3, because you can get as close as you want to 3 when you approach x = 2 from the left.

Use the horizontal line that starts at the open point (2-2) and goes to the right, to find the limit of f(x) when x approached to 2 from the right.

That limit is -2 because you can get as close to -2 when you approach x = 2 from the right.

It is important that you know that the definition of limits do not include the value of the function at the point where you want to state the limit.

This result, limit of f(x) when x approaches 2 from the left = 3 and limit of f(x) when x approaches 2 from the right = -2, means that the limit of the function when x approaches 2 does not exist.

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5 0

4 years ago

Two forces with magnitudes of 150 and 100 pounds act on an object at angles of 30° and 120°, respectively. Find the direction an

lyudmila [28]

<h2>Answer:</h2>

$Magnitude=180.27 \ lbf \\ \\ Direction=63.69 \ degrees$

<h2>Step-by-step explanation:</h2>

We have two forces as follows:

<u>First force:</u>

Magnitude: 150 pounds

Angle: 30°

<u>First force:</u>

Magnitude: 100 pounds

Angle: 120°

So the components can be found as follows:

$F_{1x}=150cos(30)=129.90 \ lbf\\F_{1y}=150sin(30) = 75 \ lbf \\ \\ F_{2x}=100cos(120)=-50 \ lbf\\F_{2y}=100sin(120) = 86.60 \ lbf$

So the components of the resultant force can be found by adding each component of the individual forces as follows:

$R_{x}=\Sigma F_{x} \\ R_{y}=\Sigma F_{y} \\ \\ R_{x}=129.90-50=79.90 \ lbf \\ R_{y}=75+86.60=161.6 \ lbf$

Finally, the magnitude and direction of the resultant force is:

$Magnitude \rightarrow R=\sqrt{R_{x}^2+R_{y}^2}=\sqrt{79.90^2+161.6^2}=180.27 \ lbf \\ \\ Direction \rightarrow \theta=tan^{-1}(\frac{R_{y}}{R_{x}})=tan^{-1}(\frac{161.6}{79.90})=63.69 \ degrees$