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.

... 4. An efficiently functioning cartel achieves a monopoly price ... Thus an increase in the prices of rivals will ... where long run average cost is tangent to the ...

... and fecundity and an efficient functioning of all ... The curvature of the function describing the relationship ... than contribute to vegetative functions which are ...

... the short run with the given demand function the firm ... Thus an increase in the prices of rivals will shift ... reached where long-run average cost is tangent to the ...

... the number of cancer cells at time t increases at a ... 4. For the function f ( x) = x 2 − 4 x : a ... at x = 2 . 2x 5. Determine the equation of the tangent line at ...

... of the derivative (slope of the tangent line) of ... of radioactive material remaining, requires a logarithmic function. ... of the most useful functions in mathematics ...

... from different expertise helps in smooth functioning of the ... fact that it considers several functions in computing ... but not always, an increasing function of t ...

... around the 240, which is because sin and cos are functions. ... length of the segment AB, which is tangent to the ... respect to the vertical axis is a function of the ...

... the two very significant additional functions, beyond reduction ... scribed by the production function f(¢). Recall ... those for which the \isoquants are tangent". ...

...functions given to find the indicated trigonometric functions. ... properties of sine, cosine, and tangent.) Show all ... in the city neither increases nor decreases...