Find the absolute max and min of f(x)= e^-x sin(x) on the interval [0, 2pi]

Find the absolute max and min of f(x)= (x^2) / (x^3 +1) when x is greater or equal to 0

. True or false?

(1) There is at most one pivot in any row.

(2) There is at most one pivot in any column.

(3) There is at least one pivot in any row.

(4) There is at least one pivot in any column.

(5) There cannot be more free variables than pivot variables.

(6) There is a linear system that has exactly two solutions.

(7) The weights c1, ..., cp ? R in a linear combination c1v1 + ... + cpvp cannot all be zero.

(8) Given nonzero vectors u, v in R n , span{u, v} contains the line through u and the origin. Hint: can you describe this line as a set of vectors?

(9) Asking whether the linear system corresponding to a1 a2 a3 b is consistent, is the same as asking whether b is a linear combination of a1, a2, a3.

(10) If the augmented matrix of a linear system has two identical rows, the linear system is necessarily inconsistent

Suppose a colony of mushrooms triples in size every 10 days. If there are 10 mushrooms to start, how many days until there are 1000 mushrooms? (A) 41.9181 days (B) 56.4091 days (C) 37.9010 days (D) 29.8974 days (E) 49.3955 days

Use the substitution x = e^t to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. (Use yp for dy/dt and ypp for d²y/dt².)

x²y'' − 3xy' + 13y = 2 + 3x

1. For the equation x²+y²−2x−4y−11=0, do the following.

(a) Find the center (h,k) and radius r of the circle.

(b) Graph the circle.

(c) Find the intercepts, if any.

ln(cos6x) Calculate the first four terms with the maclaurin series?

Solve the following system. If the system's equations are dependent or if there is no solution, state this.
3x - 4y - z = 18
4x + 2y + z = 16
7x - 2y + 3z = 19

a) Let f(x) = −x^4 − 4x^3 . (i) Find the intervals of increase/decrease of f. (ii) Find the local extrema of f (values and locations). (iii) Determine the intervals of concavity. (iv) Find the location of the inflection points of f. (v) Sketch the graph of f. (You can choose your own scale for the graph)

b) A farmer wants to fence in an area of 6 km2 in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. Find the dimensions of the rectangular field that minimize the amount of fence used.

Express the area of the region bounded by the graph of f(x) = 2 + √ x and the x-axis in the interval [3, 8] as a limit of sums using right endpoints. Do not evaluate the limit.

Let f(x) = −x^4 − 4x^3.

(i) Find the intervals of increase/decrease of f.

(ii) Find the local extrema of f (values and locations).

(iii) Determine the intervals of concavity.

(iv) Find the location of the inflection points of f.

(v) Sketch the graph of f. (You can choose your own scale for the graph)

Separation of Variables:

Let v(t) be the velocity, in knots per hour, of a vessel at time t, t = 0. After the second engine is turned on, its velocity satisfies the differential equation ??/??=3?+22, with initial condition v (0) = 2.
First use separation of variables to find an expression for v in terms of t. Secondly, f the vessel cruising speed is 25 knots per hour, how long will the vessel take to reach this speed?

The number? H(t) of veterans from foreign wars who are homeless or at risk of becoming homeless can be approximated by the exponential? function,

H(t)=65(1.65)^t where t is the number of years since 2000.

?a) In what year were there 14,000 veterans who were homeless or at risk of becoming? homeless?

?b) What is the doubling time of homelessness among? veterans?

A room has volume 100m³, and initially contains normal air. Every minute, 2m³ of gas is pumped in, which is 1.9m³ air, and 0.1m³ xenon. Every minute, 2m³ of well-mixed gas is pumped out. What volume of xenon will be in the room after 1 hour?

Factor the polynomial as a product of linear factors: x⁴-5x³+12x²-2x-20