Evaluate the line integral ∫CF⋅d r where F=〈4sinx,cosy,xz〉 and C is the path given by r(t)=(t^3,3t^2,3t) for 0≤t≤1 ∫CF⋅d r=

this as a whole question 1, answer all parts please

a)Show that the derivative of f(x) = 6+4x^2 is f(x)'=8x by using the definition of the derivative as the limit of a difference quotient.

b)If the area A = s^2 of an expanding square is increasing at the constant rate of 4 square inches per second, how fast is the length s of the sides increasing when the area is 16 square inches?

c)Find the intervals where the graph of y = x^3-5x^2+2x+4 is concave up and concave down, and find all the inflection points.

d)Find all the relative maximum and/or relative minimum values and points of F(x) = (x^4/3)-2x^2

e)Find all the relative maximum and/or relative minimum values and points of F(x)=x^4-4x on the closed interval [0,4]

f)A particle moves along the x-axis with an acceleration given by a(t)=4t + 7, where t is measured in seconds and s (position) is measured in meters. If the initial position is given by s(0) = 4 and the initial velocity is given by v(0) = 7 then find the position of the particle at t seconds.

g)Find the maximum value of xy if it is required that 7x + 1y = 62

Explain how to use derivatives to find the top of a hill or the bottom of a valley on a graph?

What does it even mean for a derivative to not exist and why is this involved? And what does that have to do with a derivative of zero?

The rate of growth of the population of rabbits in China is proportional to the current rabbit population. The population after t years is R(t). Write the differential equation for which R(t) is a solution. Your equation should involve an unknown constant

Initially, there are 100 rabbits but the population is increasing at a rate of 20 per year. Use this information to find the unknown constant in part That is, write the differential equation (without an unknown constant) for which R(t) is a solution

Find the population of rabbits in year t. That is, find R(t

Find the time t in which there are 1, 000 rabbits

What is the partial fraction decomposition of 5x^2/((x+1)(x^2+3x+2)(x^2+4))

proof a circle is divided into n congruent arcs (n ?? 3), the tangents drawn at the endpoints of these arcs form a regular polygon.

Test the series for convergence or divergence.

∞∑n=1(−1)nn4n

Identify bn.

A 600​-room hotel can rent every one of its rooms at $90 per room. For each​ $1 increase in​ rent,

3 fewer rooms are rented. Each rented room costs the hotel​ $10 to service per day. How much should the hotel charge for each room to maximize its daily​ profit? What is the maximum daily​ profit?

Differentiate.

1a) p(x) = 4th root of (2x-3/x^3)

b) q(x)= (2 sinx)/(1-cosx)

c)r(x)= sin(csc^3(x^4))

d) U(x)= ((cube root of x^2) sinx -2x-3)/sq. rt. x

A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.5 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. Find the work done. (Use 9.8 m/s2 for g.) Show how to approximate the required work by a Riemann sum. (Let x be the height in meters above the ground. Enter xi* as xi.).

Sketch the region enclosed by the given curves and find its area.

a) y=4/x,y=4x,y=(1/4)x,x>0

b) x=y²−4y,x=2y−y²

Someone explain and show how finding a subspace works and knowing how it is one with a matrix example.

In the figure below you see an ellipse which is enclosing a rectangle

The equation of ellipse is given by

x^2/4 + y^2 = 1

Find the length(L) and width (W) of the the rectangle which will maximize its area,( A). What is max(A) ?

Note: Cant upload the figures but i think the equation of ellipse is enough. the rectangle should be fitting inside the ellipse.

1. Sugary drink tax/soda taxand Mammography are different levels of prevention, please categorize them from the three options below and explain why.
   1. Primary level of prevention.
   2. Secondary level of prevention.
   3. Tertiary level of prevention.

2. Explain (Sugary drink tax/soda tax):

   Explain (Mammography):

The temperature , u(x,t), in a metal rod of length L satisfies
          del u/ del t = k del squared u / del x squared limit 0 less than or equal to, x less than or equal to L , t greater than or equal to 0

The ends of the rod at x=0 and x=L , are maintained at a constant temperature T not 0 , so that the boundary conditions are
                u(0, t) =0    u(L, t) = 0
The initial temperature distribution is
      u(x,0) = 4 sin 2 pi x/ L - 6sin (3 pi x/L) +12 sin (5 pi x / L)
Find the temperature, u(x, t).