Write the point-slope form of the line's equation satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation.

Slope =3, passing through (6 ,2)

What is the point-slope form of the equation of the line?

__?

(Simplify your answer. Use integers or fractions for any numbers in the equation.)

What is the slope-intercept form of the equation of the line?

__?

(Simplify your answer. Use integers or fractions for any numbers in the equation.)

5.

Which value of x, with −4 ≤ x ≤ 4, corresponds to the global (absolute) maximum of the functions f(x) = x 2 (3x 2 − 4x − 36) + c, where c is some given real number?

(a) −2 (b) 2 (c) 3 (d) 0 (e) None of the above.

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14. Find f(x) by solving the initial value problem. f 0 (x) = e x − 2x; f (0) = 2

(a) f (x) = e x − x 2 (b) f (x) = e x − x 2 + 1 (c) f (x) = e x − x 2 + 2 (d) f (x) = e x − x 2 + C (e) None of the above.

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18. Let f(x, y) = 1 + x + y + x 2 + axy + y 2 be a function of two variables. Suppose it is always true that ∂f /∂x = ∂f /∂y . What is the value of a?

(a) Any value of a is possible since mixed partial derivatives are equal. (b) 0 (c) 2 (d) 1 (e) There is no such value of a since the partial derivatives must be different.

Please provide the correct solution for each problem for a thumbs up. thank you!

Fundamental Theorem of Line Integrals Consider the line integral: ∮_C▒〈6xy+6x, 3x^2-3 cos⁡(y) 〉 ∙dr ⃗ where C is the line segment from the origin to (4, 5). Evaluate this integral by rewriting the integral as a standard single variable integral of the parameter t. Without finding a potential function show that the integrand is a gradient field. Find a potential function for the vector field. Use the Fundamental Theorem of Line Integrals to evaluate this integral.

Two chemical factories are discharging toxic waste into a large lake, and the pollution level at a point ? miles from factory A toward factory B is ?(?) = 3?2 − 72? + 576 parts per million for 0 ≤ ? ≤ 50. Find where the pollution is the least

Solve the following constant coefficient linear differential equations using Laplace Transform (LT), Partial Fraction Expansion (PFE), and Inverse Laplace Transform (ILT). You must check answers in the t-domain using the initial conditions.

Note: Complex conjugate roots

y ̈ (t) + 6 ̇y (t) + 13y (t) = 2

use the initial conditions
y(0) = 3, ̇y(0) = 2.

) Let S be the surface (with outward orientation) given by the boundary of the solid hemisphere x 2 + y 2 + z 2 ≤ 4 with z ≥ 0 (i.e., including the disk x 2 + y 2 ≤ 4). Use the Divergence Theorem to compute the flux of the vector field F = xyj + 2yzk over the surface S.

The Cartesian coordinates of a point are given.

(a) (2, −5)

(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) =

(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =

(b) (-2, −2)

(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) =

(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =

(c) (3, 3sqrt3)

(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) =)

(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =

PLZ ANSWER CORRECTLY WILL GIVE THUMBS UP THANK YOU

A spring has a natural length of 50cm. If a 60N force is required to keep the spring compressed 20cm, how much work is done during this compression? How much work is required to compress the spring to a length of 25cm?