Polynomial equations
(1) An ancient Babylonian farmer knows that he grows 100 bushels of wheat per hectare on his field (a hectare is an area equal to 10000 = 1002 square meters). The tax man comes along and, after a cursory glance to take in the rough size of the farmer's land, demands 1100 bushels of wheat as tax, so the farmer knows that he's going to have to devote 11 hectares to wheat grown for tax purposes alone. He decides to cut a rectangular portion out of one of his fields to plant the wheat tax whose length is ten more than the width. How much fencing will he need to fence of the four sides of his rectangular plot?
(2) Use Harriot's method to compute the roots of the following two cubic equations. The first one is already depressed. Don't forget to start by depressing the second one.
(3) Factor the following quartic into two quadratics:
Complex numbers
(4) Find the square roots of the following complex numbers: i, -2i, 5+12i.
Tangent lines and derivatives
(5) Compute the derivatives f'(x) of the polynomials f(x) given by f(x)=5x4+3x3+2x2+x+6 and f(x)=x^6+1. Find the equation of the tangent line to the curve y=3x^3+2x+7 at the point (1,12).
(6) What type of curve is given by the equation y=x2+3x? What is the derivative f'(x) of the function f(x)=x2 +3x? Graph the derivative, and compute the area under the curve y=f'(x) from 0 to some point x. Compare the expression for the area with the function f(x). What theorem predicts the result?
There are no specific right answers to the following questions: just think about them and give your ideas.
(7) Problem 5 shows that if you graph the line which is the derivative f'(x) and compute the area under that line from 0 to x, you obtain the value of the function f(x). This is the meaning of the fundamental theorem of calculus. How could one explain why this happens?
(8) Suppose you looked in a parabolic mirror (like a concave dish facing you). How is the reflection you see of yourself deformed? (Try it with a spoon.) What do you think you would see if you placed your face directly at the focus?
(9) Write a paragraph on any one of the three essay questions that you did not choose for the midterm.