Consider the function f: R → R defined by f(x) = cos(x), where n is an even integer. It can be shown that any positive power of cos x can be expressed as a sum of multiples of terms of the form cos(kx), where the k are non-negative integers. Note that the term where k = 0 is the constant term. It can also be shown that any odd positive power of sin a can be expressed as a sum of multiples of terms of the form sin(kx), where the k are non-negative integers. (a) What is the constant term in the expansion of f(x) described above? & A Syntax advice: Remember to use correct Maple syntax. For example, • the factorial n! may be written as n! or factorial (n) the binomial coefficient (R) • the term a* is written a^x may be written as binomial (n,k) (b) Now, consider the function g: R → R defined by g(x) = sin 975 (x)f(x). Briefly explain in words why g(x) may be written as a sum of multiples of terms of the form sin(kx), where k are non-negative integers.

Prove the existence of a number x satisfying the expression cos(x) = = x.

2) Find the EXACT value of cos if tan(0) = 3 and sin(0) < 0

Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min