a) Find the equation of the normal line at the point (−2, 1 − 3) to the ellipsoid x² /4 + y² + z² / 9 = 3

b) Find a plane through P (2, 1, 1) and perpendicular to the line of intersection of the planes: 2x+y−z = 3 and x+2y+z = 2.

can you find a linear homogeneous constant-coefficient DE and such that x.e^(-2x).sinx is a solution.

Write the answer in a.y''' + by'' +cy'+dy=0

Maximize 250 + 7x − 5y subject to:

2x + 3y ≤ 90

2x+ y≤62

x+ y≥20

x ≥ 0, y ≥ 0

(27) Give an example of an infinite incidence geometry (i.e. an incidence geometry with an infinite number of points and an infinite number of lines).

How do I find the missing sides of an isosceles trapezoid?

Find the Radius and center of the sphere. 2x² +2y² + 2z² + x + y + z = 9

For f(x)=9x+5 and g(x)=x² find the following composite functions and state the domain of each

a) f o g b) g o f c) f o f d) g o g

a. the domain of f o g is

b. (g o f )(x)=

a. the domain of g o f is

c. ( f o f )(x)=

the domain of f o f is

d. ( g o g)(x)=

the domain of g o g is

y''+4y'+4y=2e^{-2x} Find general solution.

The population of Egypt in 2002 was 73,312,600 and was 78,887,000 in 2006. Assume the population of Egypt grew exponentially over this period.

1.   Determine the 4-year growth factor and percent change.

2. Determine the 1-year growth factor and percent change.

3. Define a function that gives the population of Egypt in terms of the number of years that have elapsed since 2002.

4. Re-write your function from part(c) so that it gives the population of Egypt in terms of the number of decades that have elapsed since 20002.

5. Assuming Egypt’s population continued to grow according to this model, how long will it take for the population of Egypt to double?

We got for x > 0 given a differential equation y’ = 1-y/x, with start value y(2)= 2

1. Find the Taylor polynomial of first and second degree for y(x) at x =2.
2. Show that y(x) =x/2 +2/x solves the given equation.

For the function

a) f(x)=x^3-9x^2+23x-15

b)f(x)=(x+3)^2(2x+1)(x-1)

c)f(x)=-(x^2-6x+9)(x^2-x-6)

Find:

1) the zeros

2) the y-intercept

3) left-right end behavior

4) the sketch of the graph

T/F and why(true/false)?

(Part a) If the partial derivatives f_x(x,y), f_y(x,y) exists for all values x,y ,then f(x,y) is continuous.

(Part b) Suppose f(x,y) is a function which is differentiable and nonnegative everywhere, and is zero at the origin

,f(x,y) ≥ 0, f(0,0) = 0.Then the gradient vector is zero at the origin is,∇f(0,0) = (0,0).

(Part c) If F(x,y) is continuous on the entire plane, then there is a function f:R→R satisfying

f′(x) =F(x,f(x)), f(0) = 0 for all x∈R.

solve using series solutions

(x^+1)y''+xy'-y=0

Consider a solid object with the base in the first quadrant bounded by y(x) = 1-( x^2/16) , x-axis, and y-axis. If the cross section that perpendicular to the x-axis is in the form of square, determine the volume of this object!

A computer aided drafting (CAD) instructor at a community college wants to use his rapid prototype machine to produce demo items for a seminar on engineering technology. He plans to bring two different types, a Golden Ruler and a Fibonacci Gauge. He will bring at least 20 of each. he has 45 cubic inches of filler and 95 cubic inches of material available to make the demos, but only has 86 hours of free time to run the machine. he is able to save time by running the demos in batches. Use the information in the table to determine how many of each demo item he needs to bring while minimizing the total cost. Set up but do not solve.