For the following find: (Show detailed working)

1) general solution of the associated homogns DE

2)  form of the particular solution associated with the undetermined coefficients method. Do not evaluate coefficients

a) (d² y / dx²) - dy/dx = x - 3

b) (d² y / dx²) + 2 dy/dx + y = (1+x)e^-x + x²

^c) (d² y / dx²) + y = sin(2x)

1. Explain how the first derivative test of a function determines where the function is increasing and decreasing.
2. Explain how to apply the second derivative test.
3. What is an inflection point?
4. Why are special methods such as L'Hopital's Rules, needed to evaluate indeterminate forms?

At noon, ship A is 130 km west of ship B. Ship A is sailing east at 30 km/h and ship B is sailing north at 15 km/h. How fast is the distance between the ships changing at 4:00 PM?

Show a composite shape that you could divide up multiple ways to find its area. (illustrate at 3 different ways to divide it up and label dimensions of each division)

for u=<2,1,-3> and v=<1,0,2>,

a) find the angle between u and v

b) use the formula proj_a b=((a•b)/|b|^2)b to find vector projection of v onto u

c) sketch vectors u, v, and the projection found with only using arrows

d) find the area of the parallelogram determined by u and v

The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs is given by the following function.

C(x) = 2000 + 2x − 0.0001x²    (0 ≤ x ≤ 6000)

a. What is the actual cost incurred in producing the 991st and the 1971st disc?

b. What is the marginal cost when x = 990 and 1970?

Williams Commuter Air Service realizes a monthly revenue represented by the following function, where R(x) is measured in dollars and the price charged per passenger is x dollars.

R(x) = 9,600x − 120x²

a. Find the marginal revenue R'(x).

b. Compute the following values.

-R'(39)

-R'(40)

-R'(41)

c. Based on the results of part (b), what price (in dollars) should the airline charge in order to maximize their revenue?

Can somebody explain to me Derivative of Parametric Equations? May I know its importance and applications in real life? i need it for my essay. Thank you

PS. I can't find it on the internet

The scores on a standardized test are normally distributed with a mean of 95 and standard deviation of 20. What test score is 0.5 standard deviations above the mean?

Use Pascal’s triangle to find the possible value(s) of nn if nC3=(n3)=20.

Find the horizontal and vertical tangents to the graph of the cardioid ? = 1 − cos?, 0 ≤ ? ≤ 2?

If f is a polynomial of degree 3 or more how would you use real zeros of f to determine the open intervals over which f(x) > 0 or f(x) < 0. How and where does the sign of f change.

Find the exact location of all the relative and absolute extrema of the function. HINT [See Example 1.] (Order your answers from smallest to largest x.)

g(x) = 2x³ − 6x + 3 with domain [−2, 2]