Questions in Section Calculus I: page 4

Use logarithmic differentiation to find the derivative of the function `y=e^x x^(2ln(x))`.

To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to obtain the cross section. Find the dimensions of the cross section with the maximum area (this will result in the trough with the largest possible volume).

A farmer wants to create a pen inside a barn. The sides of the barn will form two sides of the pen, while fencing material will be used for the other two sides. The farmer has a total of 80 feet of fencing material to enclose the pen. What is the maximum area that can be enclosed?

Sketch the graph of `y=(2-x)/(x+3)`, find extrema, intervals of increase/decrease, inflection points, intervals of concavity, etc.

Let `f(x)=(x-2)/(x+1)`.

  1. Find the interval(s) where `f(x)` is concave upward.
  2. Find the interval(s) where `f(x)` is concave downward.
  3. Find the x-coordinate(s) of any point(s) of inflection.

Find the interval(s) where `f(x)=x^3+x^2-5x+2` is increasing and the interval(s) where it is decreasing.

You are asked to design a rainwater tank in the shape of a tall box whose cross-section is a right-angled triangle. The tank is to fit against the side of a house, between the house wall and a path. The tank has a rectangular base, a vertical rectangular side that is flush with the wall of the house, a sloping rectangular side that slopes from the top of the vertical side down to the ground, and two triangular ends. The height of the tank is to be four times the width of the base of the tank, where the width is measured in the direction perpendicular to the house wall. The heavy plastic material for making the tank costs $6 per square metre. You have $30 to spend on the material.

Find the limit of the following:

  1. `lim_(x->-1)(x^2+6x+5)/(x^2-3x-4)`.

  2. `lim_(x->2)(x^2 -4x + 4)/(x^2+x-6)`.

Find the equation of all lines passing through the point `(6,-1)` for which the product of their `x` and `y` intercepts is 3.

Find an expression for composition of functions `f@g` given that: `f(x) = 1/(1+x)` and `g(x)=root(3)(x)`.

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