Functions : Tangent, Increasing or Decreasing and Area under a Curve

2. Let f be a function defined on the closed interval -3≤x≤4 with f(0) = 3. The graph of f', the derivative of f, consists of one line segment and a semicircle.

a) On what intervals, if any, is f increasing? Justify your answer.

b) Find the x-coordinate of each point of inflection of the graph of f on the open interval -3 < x < 4. Justify your answer.

c) Find an equation for the line tangent to the graph of f at the point (0,3).

d) Find f(-3) and f(4). Show the work that leads to your answers.

3. The region R, is bounded by the graphs of x = 5/3 y and the curve C given by x = (1+y^2)^(1/2), and the x-axis. The line and the curve, C, intersect at point P.

a) Find the coordinates of point P and the value of dx/dy, yes this is typed correctly, for the curve C at point P.

b) Set up and evaluate an integral expression with respect to y that gives the area of R.

c) Curve C is part of the curve x^2 - y^2 = 1, Show that x^2 - y^2 = 1 can be written as the polar equation r^2 = 1/(cos^2θ - sin^2θ).

d) Use the polar equation given in part c) to set up an integral expression with respect to theta that represents the area of R.

For each of the following cost-output relationships, describe the shape (U-shape, decreasing, increasing, constant) of the average total cost and marginal cost functions
(C = total cost, Q = output):
(1) C = 42,500,000 + 2550Q
(2) C = 8.48 + 0.65Q + .00220Q2

See attached file for full problem description.
1. Find dy for each of the following functions.
dx
a) y=3x^4-6x^2+2x
b) y=3/x^2
c) y=(8x^4-5x^2-2)/4x^3
d) y=square root 5x - square root x/5
2. a) Find the slope of the tangent to the curve y=4x^3-3x^2+1 at the point where x=-1.

Determine the intervals on which the function is increasingand intervals on which the function is decreasing. Check your answers by graphing the corresponding functions.
- Please view the attachment for the rest of the solution.
Question: Determine the intervals on which the function is increasingand intervals on whic

20. The graph below displays growth of a town's population y = P(t) over the next 3 years, where t is time in months.
a. Estimate how fast the population is increasing 5 months, and 20 months from now.
b. Graph y = P'(t).
In graphs of questions 2, 4 and 6, determine which is the f(x) function and which is the derivative?

Finding tangent line for f(x) that pass through the given point ( and the point isn't on the curve)
part 1
f(X) = 4x-x^2; (2,5)
Part 2
f(x) =x^2; (1,-3)

1.) Find and equation of the tangent to the curve at the point corresponding to the given value of the parameter
x= cos t + sin 2t, y= sin t + cos 2t (t=0)
2.) Find dy/dx and d^2/dx^2 for which values of t is the curve concave upward
x= t + ln t, y = 1 - ln t

(See attached file for full problem description)
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1) Consider the following function:
a) f (x) = 9x2 - x3
b) f (x) = x + 1
x - 2
c) f (x) = x2/3 (x - 5)
for each of the above functions complete the following table. Show the work to justify your answers below the table.
f(x) is i