Complete a thorough curve analysis of the function ?(?) = ?4 − 12?³ + 48?² − 64?.

Follow the Algorithm for Curve Sketching as outlined in Section 4.5 of your textbook. You are expected to clearly demonstrate all aspects of the curve analysis, summarize your results clearly, and draw (by hand) an accurate sketch of the function with all key points labelled on the sketch. Show all algebraic steps to your process clearly. Annotate your work to demonstrate what information is gained from each part of the process. State conclusions clearly.

This task does ask you to demonstrate clearly how the function equation, the first derivative, and the second derivative can all be used to fully understand all important characteristics of the graph.

The Vector Field f(x, y) = (2x + 2y^2)i + (4xy - 6y^2)j has exactly one potential function f (x, y) that satisfies f(0, 0). Find this potential function , then find the value of this potential function at the point (1, 1).

Compute the directional derivative of T(x,y,z)=200e^(-x^2-3y^2-9z^2) at the point (2,-1,2) in the direction toward the point (3,-3,3). What is the maximum rate of change of T at the point (2,-1,2)?

I have solve this by mathematica...

1) Determine the distance between the point (-2, 6) and the line 3x -4y -10 = 0
2) Determine the distance between the point (4, -1) and the line 3x -4y +12 = 0
3) Determine the distance between two parallel lines which equations are  3x -4y +6 = 0 and 6x -8y +9 = 0

What is the power series representation of ln(1-2x)?

What is its radius of convergence?

Find the equation of the tangent line to the curve at the given point using implicit differentiation.

Cardioid

(x² + y² + y)² = x² + y²  at  (−1, 0)

Consider the scalar functions

f(x,y,z)g(x,y,z)=x^2+y^2+z^2,

g(x,y,z)=xy+xz+yz,

and=h(x,y,z)=√xyz

Which of the three vector fields ∇f∇f, ∇g∇g and ∇h∇h are conservative?

1. You invested $ 10,000 in a mutual fund, and after 10 years, the original investment has doubled. Find use   

a). - Find the yearly percentage growth rate

b).- The rate of the account growing per year

2. Consider the rational function f(x)= ( x-4/ -4x - 16)

a). The equation of the vertical asymptotes

b). The equation of the horizontal asymptotes

c). The derivative of the function

d). Critical point if any(s)

e). Do you have any Extrema points, Explain why?

f). Sketch the graph label the asymptotes and intercepts

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is given by a(t) = 60t ft/s2 , at which time the fuel is exhausted and it becomes a freely “falling” body and falls to the ground.

(a) Determine the position function s(t) for all times t > 0.

(b) What is the maximum height achieved by the rocket? (

c) At what value of t does the rocket land?

Find the arc length of the curve on the given interval.

x= lnt , y = t + 1, 1 ≤ t ≤ 2

Optimization:

a) You are attempting to use 1000 yards of fencing to make one rectangular pasture. What’s the largest pasture you can make, and what are the lengths of the sides of the fencing? (First, write a function A(x) that expresses the total area in terms of the width x.)

b) Using the same amount of fencing (1000 yards) to make two rectangular pastures of equal area, what are the largest pastures you can make, and what are the lengths of the sides of the fencing?

c) Using the same amount of fencing to make n pastures?

d) Write an equation for A(x) based on n pastures. Then take the derivative and check your answer.

1. Consider the following curve f(x)= X^3 - 5x^2 +7x-5 Find the coordinates of the minimum and the maximum.

2. The curve y= x^3 + ax^2 + bx + c has a relative max at x=-3 and a relative minimum at x= 1. Find the values of a and b.

3. Find the equation of the perpendicular line to the curve x^2 + 2xy - 2y^2 + x=2 at the point (-4,1)

4. Find the slant asymptote f(x)= (4x^2 - 2x +5 / 2x - 1)