Differentiation

See attached file for full problem description.

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 t ...continues

Volume of Revolution

1. The shaded region R, is bounded by the graph of y = x^2 and the line y = 4. a) Find the area of R. b) Find the volume of the solid generated by revolving R about the x-axis. c) There exists a number k, k>4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in par ...continues

Volume of a Rotating Solid

Let f and g be the functions given by f(x)=e^x and g(x)=ln x. b) Find the volume of the solid generated when the enclosed region of f and g between x = ½ and x = 1, is revolved about the line y = 4. c) Let h be the function given by h(x)=f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval ½ &# ...continues

Definite Integral and Application Problem : Braking System of a Vehicle

1. Find the definite integral : S sec^2(4x+1)dx from 0 to pi/8, 2. A car's braking system is described by the function : a(t)= -2 + .0025t^2 and a(t)>/= 0, a(t)=0 is the point where the brakes fail and V(0)=20 m/s. At what speed will the brakes fail (or will the vehicle stop) and at what distance will the brakes fail (o ...continues

Mixing Problem : Exponenetial Functions

A 5000 gallon aquarium is maintained with a pumping system that circulates 100 gallons of water per minute through the tank. To treat a certain fish malady, a soluble antibiotic is introduced into the inflow system. Assume that the inflow concentration of medicine is 10te-t/50 oz/gal, where t is measured in minutes. The well- ...continues

Oscillating Inflow Concentration

Make a conjecture, on the basis of physical reasoning, as to whether you expect the amount of salt in the tank to reach a constant equilibrium value as time increases. In other words, will lim(t) -> infinite Q(t) exist? (see attachment for full question)

Solve a Complicated Integral

show that (7x/x^2 + 5) + (4/3x+15) - (5/6x-24) = (45x^3-15x^2-825x-35)/((6x^2+30)(x^2+x-20)) then use that information to determine S=integral S(45x^3-15x^2-825x-35)/((6x^2+30)(x^2+x-20)) dx.

Logarithmic Function : Domain, One-to-one, Extrema

For f(x) = ln(x^2-10x+16) a) use algebraic methods, not a graph, to find the domain of f. Express in interval notation. b) Prove that f is one-to-one function or prove that f is not a one-to-one function. (can't be based on graph) c)use algebra and calculus to determine where f is increasing and where f is decreasing. ...continues

Ladder method of integration

See word file for problems regarding the ladder method of integration by parts