QUESTION 1

Mathematics

1 answer:

Kitty [74]3 years ago

3 0

It's difficult to make out what the force and displacement vectors are supposed to be, so I'll generalize.

Let θ be the angle between the force vector F and the displacement vector r. The work W done by F in the direction of r is

W = F • r cos(θ)

The cosine of the angle between the vectors can be obtained from the dot product identity,

a • b = ||a|| ||b|| cos(θ) ==> cos(θ) = (a • b) / (||a|| ||b||)

so that

W = (F • r)² / (||F|| ||r||)

For instance, if F = 3i + j + k and r = 7i - 7j - k (which is my closest guess to the given vectors' components), then the work done by F along r is

W = ((3i + j + k) • (7i - 7j - k))² / (√(3² + 1² + 1²) √(7² + (-7)² + (-1)²))

==> W ≈ 5.12 J

(assuming F and r are measured in Newtons (N) and meters (m), respectively).

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$\displaystyle\int\frac{4x^2-6}{(x+5)(x-2)(3x-1)}\,\mathrm dx$

You have a rational expression whose numerator's degree is smaller than the denominator's. This tells you you should consider a partial fraction decomposition. We want to rewrite the integrand in the form

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You can use the "cover-up" method here to easily solve for $a,b,c$ . It involves fixing a value of $x$ to make 2 of the 3 terms on the right side disappear and leaving a simple algebraic equation to solve for the remaining one.