Write the equation of the tangent line to the graph of ?(?) = (2)/(3−?) at the point where x = 4. You must use the limit definition for any derivatives and show each process by step. Use proper notation.

Find the domain for each function please explain. f(x)=10x^2 + x f(x)= -2/x^2

Discuss a real time application of first order differential equation [Answer should include literature review and real applications maximum 200 words].

a) Given a vector field à = zỹ +(3y + 2)2 î in cartesian coordinates, determine whether it is solenoidal (V · À = 0), conservative (D x X = 0)

I Div x A (Cylinderical Coordinates)

ii) Calculate integral A*dl , where the contour C is the unit circle (r=1) traversed in anticlockwise direction

Find the value of k such that the graph of y = f(x) has no vertical asymptote given by x = 2 where

f(x) = (4x 3 − 4x 2 + kx + 14)/ 4x 2 − 12x + 8 .

Then find all the intercepts, asymptotes, local extreme values, points of inflection, monotonicity intervals, concavity intervals. Finally sketch the graph of y = f(x).

Find the general solution for differential equation

x^3y'''-(3x^2)y''+6xy'-6y=0, y(1)=2, y'(1)=1, y''(1)=-4  

1) Determine the angle between vectors:

U = <2, -3, 4> and V= <-1, 3, -2>

2) determine the distance between line and point

P: -2x+3y-4z =2

L: 3x – 5y+z =1

3) Determine the distance between the line L and the point A given by

L; (x-1)/2 = (y+2)/5 = (z-3)/4 and A (1, -1,1)

4) Find an equation of the line given by the points A, B and C.

A (2, -1,0), B (-2,4,-1) and C ( 3,-4,1)

5) Determine whether the lines are parallel, perpendicular or neither.

(x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 = (z-2)/6

6) A) Find the line intersection of vector planes given by the equations

-2x+3y-z+4=0 and 3x-2y+z=-2

B) Given U = <2, -3, 4> and V= <-1, 3, -2>

Find

a. U . V

b. U x V

7) Find the angle between the planes:

3x -5y+7z -4=0 and 3x -2y+5z +3 =0

Find the area under the graph of f over the interval

left bracket 7 comma 9 right bracket[7,9].

​f(x)equals=StartSet Start 2 By 2 Matrix 1st Row 1st Column 4 x plus 9 comma 2nd Column for x less than or equals 8 2nd Row 1st Column 81 minus 5 x comma 2nd Column for x greater than 8 EndMatrix

The area is

nothing.

2)

Find the area of the region bounded by the graphs of the given equations.

yequals=x plus 6x+6​,

yequals=x squaredx2

The area is

nothing.

​(Type an integer or a simplified​ fraction.)

How do you tell if a graph of a rational function has a hole in the graph or a vertical asymptote? Give an example of each case.

[system of linear Differential Equations] Use matrix methods to solve the follow initial -value problem,

u (t) = 2u (t) + 2v (t) + 4

V (t) = u (t) + 3v (t) – 1

u (0) = 2

v (0) = -1

[ find, u (t) and v (t) ].

Write the equation of the tangent line to the curve : y³ + 2x²y - 8y = x³ + 35

when x = -2

(a)

C: line segment from (0, 0) to

(7, 4)

counterclockwise around the triangle with vertices (0, 0),

(8, 0),

and

(0, 4)

A metal rod at 38°C is placed in a room at a constant temperature of 0°C.

(a) If after 20 minutes the temperature of the rod is 20°C, find the temperature function T(t) that
models the temperature T of the rod at time t. Assume Newton's Law of Cooling.
Note: You must state the differential equation that models this situation and include
how to solve this DE as part of your solution.

(b)Determine the time it will take for the bar to reach 10°C.

(c)Determine the temperature of the metal rod after 15 minutes.

find the maclaurin series for f and its radius of convergence.

(1) f(x) = 10^x
(2) f(x) = e^x^2
(3) f(x) = (1-x)^-5
(4) f(x) = ln(1+x^2)