Find the Taylor series for f ( x ) centered at the given value of a . (Assume that f has a power series expansion. Do not show that R n ( x ) → 0 . f ( x ) = 2 /x , a = − 4

integration of ((8(x+a))/(x^2+a^2))dx. using trig. sub.

Use Lagrange multipliers to find the absolute extrema of f(x,y) = x² + y² - 2x - 4y + 5 on the region x² + y² <= 9.

            i) A bowl collects 1.67 cm height of water in its first week in a dewdrop. Each week the height of the water increases by 4% more than it did the week before. By how much does it increase in nine weeks, including the first week?

ii) 1000 tickets were sold. Adult tickets cost GHC8.50, children's cost GHC4.50, and a total of GHC7300 was collected. How many tickets of each kind were sold?              

A ferris wheel is 50 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn.

What is the Amplitude in meters=
What is the Midline( y=) in meters=
What is the Period in minuets=
How High are you off of the ground after 2 minutes-in meters=

Round your answers to the three decimal places.

A dock is 1 meter above water. Suppose you stand on the edge of the dock and pull a rope attached to a boat at the constant rate of a 1m/s. Assume the boat remains at water level. At what speed is the boat approaching the dock when it is 10 meters from the dock? 15 meters from the dock? Isn’t it surprising that the boats speed is not constant ?

At 10 meters x’=
At 15 meters x’=

Let L₁ be the line passing through the point P₁=(−1, −2, −4) with direction vector →d=[1, 1, 1]^T, and let L₂ be the line passing through the point P₂=(−2, 4, −1) with the same direction vector.
Find the shortest distance d between these two lines, and find a point Q₁ on L₁ and a point Q₂ on L₂ so that d(Q₁,Q₂) = d. Use the square root symbol '√' where needed to give an exact value for your answer.

Determine the solution to the initial value differential equation;

y′=0.0015y(1100−y), y(0)=32

1. y(x) = ?

2. What is the maximum value of this function. In other words, evaluate: lim x-> inf y(x)

3. Determine x for which y(x) reaches 86% of its maximum value.

Find a possible formula for the trigonometric function whose values are in the following table.

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry along the diagonal is either a 0 or a 1.

Hint 1. Suppose A = {~v1, . . . , ~vn} and B = { ~w1, . . . , ~wn}. If the k th column of [T] A B consists of all zeros, what can you deduce?

Hint 2. Suppose A = {~v1, . . . , ~vn} and B = { ~w1, . . . , ~wn}. If the k th column of [T] A B has a one in the k th entry and all other entries are zero, what can you deduce?

Hint 3. Now construct bases with the properties found in Hint 1 and Hint 2.

Hint 4. Theorem 18 part 5 is your friend.

Hint 5. The proof of the Rank-Nullity Theorem is your best friend.

Let f(x, y) =x^2+ 3y^2−2x−12y+ 13 on the domain A given by the triangular region with vertices (0,0),(0,6), and (2,0).

Find the maximum of f on the boundary of A.

Let PQ be a focal chord of the parabola y² = 4px. Let M be the midpoint of PQ. A perpendicular is drawn from M to the x-axis, meeting the x-axis at S. Also from M, a line segment is drawn that is perpendicular to PQ and that meets the x-axis at T. Show that the length of ST is one-half the focal width of the parabola.

Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (−2, 8) and (2, 2).

Find an equation of the normal line to the parabola y = x² − 8x + 7  that is parallel to the line x − 2y = 2.