# 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.

#### Solution Summary

A variety of functional property problems are solved with explanations.