a)f(u,v) fuction is provide f(6,-2)=2020, f_u(6,-2)=2, f_v(6,-2)=3 equations. g(x,y,z)=f(3yz+x²,2x+2y²-z²) so,

find tangent plane of g(x,y,z)=2020 at the point (0,1,2).

b)Find the tangent line ,which is parallel to question a) tangent plane, of r(t)=<t²+1,2t+7,4t-t²>(-∞<t<∞)

1.) Let f′(x) = 3x^2 − 8x. Find a particular solution that satisfies the differential equation and the initial condition f(1) = 12.

2.) An object moving on a line has a velocity given by v(t) = 3t^2 −4t+6. At time t = 1 the object’s

position is s(1) = 2. Find s(t), the object’s position at any time t.

(a) Determine the inverse Laplace transform of F(s) =(2s−1)/s^2 −4s + 6
(b) Solve the initial value problem using the method of Laplace transform. d^2y/dx^2 −7dy/dx + 10y = 0, y(0) = 0, dy/dx(0) = −3.
(c) Solve the initial value problem:
1/4(d^2y/dx^2)+dy/dx+4y = 0, y(0) = −1/2,dy/dx(0) = −1.

using these axioms ( finite affine plane ) :

AA1 : there exists at least 4 distinct points , no there of which are collinear .

AA2 : there exists at least one line with n (n>1) points on it .

AA3 : Given two distinct points , there is exactly one line incident with both of them .

AA4 : Given a line l and a point p not on l , there is exactly one line through p that does not intersect l .

prove : in an affine plane of order n , each line contains exactly n points .

Find the solutions of the equation

4x^2 + 3 = 2x

Among all triangles with a perimeter of 2s=9 units, find the dimension of the triangle with the maximum area. (Hint: Heron's Formula for the area of a triangle may be useful - A=Sqrt(s(s-a)(s-b)(s-c).  

Use the Laplace Transform method to solve the following differential equation problem: y 00(t) − y(t) = t + sin(t), y(0) = 0, y0 (0) = 1

Please show partial fraction steps to calculate coeffiecients

Consider the following differential equations:

?³?/??³ + 9 ?²?/??² + 20 ??/?? + 12? = 15

?(0) = ?̇(0) = ?̈(0) = 0

a) Find the solutions to the equation using Laplace transform.
b) Use the Final Value Theorem to determine ?(?) as ?→∞ from?(?).

Note: Dots denote differentiation with respect to time.

Given that h(x) = x.sinx . Find the root of the function h(x) = 1, where x is between [0, 2] using substitution method.

Two towns had a population of 12,000 in 1990. By 2000, the population of town A had increased by  13⁢ % while the population of town B had decreased by  13⁢ %. Assume these growth and decay rates continued.

a. Write two exponential population models A(T) and B(T) for towns A and B, respectively, where T is the number of decades since 1990.

A(T)=

12000*e0.12*T

B(T)=

12000*e−0.14*T

b. Write two new exponential models a(t) and b(t) for towns A and B, where t is the number of years since 1990.

Give the answers in the form C⋅at. Round the growth factor to four decimal places.

a(t)=

12000*1.13t

b(t)=

12000*0.87t

c. Find A(2), B(2), a(20), and b(20) and explain what you have found.

Round your answers to the nearest integer.

A(2)=

B(2)=

a(20)=

b(20)=

Each of these values represent the population

2 years after 199020 decades after 19902 decades after 1990

Q­: In the questions below, nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. In how many ways can this be done if:

1. Ben is to be in the picture?
2. Both Ed and Gail are in the picture?
3. Neither Ed nor Fran are in the picture?
4. Dot is on the left end and Ed is on the right end?
5. Hal or Ida (but not both) are in the picture.
6. Ed and Gail are in the picture, standing next to each other?
7. Ann and Ben are in the picture, but not standing next to each other?

Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $25,000 at a rate of 8%/year compounded monthly. Her bank is now charging 11.3%/year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of 3 yr for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank? (Round your answers to the nearest cent.)

interest paid to manufacturer?

interest paid to bank?

savings?

1-: ?(?) = ln (3 − √2? + 1)   ? ′(0) =?

2-Passing through the point x = 1 and ? = 2?
Perpendicular to the straight line tangent to 3 + 5? - 2 parabola
what is the equation of the normal?

Find the mass of the solid, moment with respect to yz plane, and the center of mass if the solid region in the first octant is bounded by the coordinate planes and the plane x+y+z=2. The density of the solid is 6x.

Problem: Prove that every polynomial having real coefficients and odd degree has a real root

This is a problem from a chapter 5.4 'applications of connectedness' in a book 'Principles of Topology(by Croom)'

So you should prove by using the connectedness concept in Topology, maybe.