Prove that if a diagonal of a parallelogram bisects the angles whose vertices it joins, the parallelogram is a rhombus

How do we interpret a confidence interval?

Let f (x, y) = -x³ - y³ + 9xy - 26. Check that (0,0) and (3,3) are stationary points of f and classify these points as maximum, minimum or saddle point. Obtain the maximum or minimum value of f.

Find the distance between the skew lines given by the following parametric equations:

L1: x=2t y=4t z=6t
L2: x=1-s y=4+s z=-1+s

The production costs, in $, per week of producing x widgets is given by C(x)=65000+4x+〖0.2x〗^2-〖0.00002x〗^3 and the demand function for the widgets is given by p=500-0.5x . Find the Marginal Revenue equation. Find the Marginal Cost equation. Find the Marginal Revenue and Marginal Cost for the firm when it is producing 300 widgets. Based on your numbers, would you advise the company to increase, decrease, or make no change to the level of production? Explain why.

Find the equation (in terms of x and y) of the tangent line to the curve r=2sin5θ at θ=π/3.

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $16 per linear foot to install and the farmer is not willing to spend more than $4000, find the dimensions for the plot that would enclose the most area.

(width, length) =

Given f(x) = 1 x 2 − 1 , f 0 (x) = −2x (x 2 − 1)2 and f 00(x) = 2(3x 2 + 1) (x 2 − 1)3 . (a) [2 marks] Find the x-intercept and the y-intercept of f, if any. (b) [3 marks] Find the horizontal and vertical asymptotes for the graph of y = f(x). (c) [4 marks] Determine the intervals where f is increasing, decreasing, and find the point(s) of relative extrema, if any. (d) [3 marks] Determine the intervals where f is concave up, concave down, and find the inflection point(s), if any. (e) [3 marks] Sketch the graph of f and label all important points.

The temperature in Gavin's oven is a sinusoidal function of time. Gavin sets his oven so that it has a maximum temperature of 320°F and a minimum temperature of 260°. Once the temperature hits 320°, it takes 20 minutes before it is 320° again. Gavin's cake needs to be in the oven for 30 minutes at temperatures at or above 310°. He puts the cake into the oven when it is at 290° and rising. How long will Gavin need to leave the cake in the oven? (Round your answer to the nearest minute.)

____ min

For each of the following mathematical functions (or equations):

(i) Take the first derivative dy/dx,

(ii) Set dy/dx = 0, then solve for x .

(iii) Take the second derivative d(dy/dx)/dx. Is the second derivative positive or negative at x*? Is this a relative minimum point or a relative maximum point? Or neither?

1. 1) Y= 1500 X – (41,000,000 + 500 X + .0005 X2)

2. 2) Y= 12,100,000 + 800X + .004 X2 X

3. 3) Y=(1800-.006X)X

4. 4) Y=1800X-.006X -(12,100,000+800X+.004X )

5. 5) Y= (4) (5000) + 50 (5000/X) + (.5) (X/2)

6. 6) y=x3 –12x2 +36X+8

in the triangle PQR, the point S divides the line PQ in the ratio 1:3, and T divides the line RQ in the ratio 3:2.

PR =a and PQ =b

express the following in terms of a and b

PS , SR, and TQ

Find a basis for the subspace of Pn defined by V={p an element of Pn, such that p(1)=0}. What is the dimension of V?

1. Find the equation of the line with slope of m = −" # and through the point (8,−1).Write your final answer in slope -intercept form. ______________________________

2. Find the equation of the line through the points (2,−7) ??? (−4,−8 ). Write your final answer in slope intercept form. ______________________________

3. Find the equation of the line which passes through the point (10,−3) & is perpendicular to the line 5?+3?=2 ______________________________

Show that ? = {1 − ?, 2 + ?2, 1 + ? − ?2} forms a basis for ?₂(ℝ)

Find the equation (in terms of x and y) of the tangent line to the curve r=5sin5θ at θ=π/3