91Ó°ÊÓ

Is the product of a rational number and an irrational number necessarily rational? necessarily irrational?

State whether the function is odd, even, or neither. $$h(x)=\frac{\cos x}{x^{2}+1}$$.

Show by example that the sum of two irrational numbers (a) can be rational; (b) can be irrational. Do the same for the product of two irrational numbers.

A string 28 inches long is to be cut into two pieces. one piece to form a square and the other to form a circle. Express the total area enclosed by the square and circle as a function of the perimeter of the square.

Prove that \(\sqrt{2}\) is irrational. HINT: Assume that \(\sqrt{2}=p / q\) with the fraction written in lowest terms. Square both sides of this equation and argue that both \(p\) and \(q\) must be divisible by 2.

Suppose that \(l_{1}\) and \(l_{2}\) are two nonvertical lines. If \(m_{1} m_{3}=\) \(-1,\) then \(l_{1}\) and \(l_{2}\) intersect at right angles. Show that if \(l_{1}\) and \(l_{2}\) do not interscet al right angles, then the angle \(\alpha\) between \(l_{1}\) and \(l_{2}\) (see Scction 1.4 ) is given by the formula $$\tan \alpha=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$$. HINT: Derive the identity $$\tan \left(\theta_{1}-\theta_{2}\right)=\frac{\tan \theta_{1}-\tan \theta_{2}}{1+\tan \theta_{1} \tan \theta_{2}}$$ by expressing the right side in terms of sines and cosines.

Prove that \(\sqrt{3}\) is irrational.

Find the point where the lines intersect and determine the angle between the lines. $$l_{1}: 4 x-y-3=0, \quad l_{2}: 3 x-4 y+1=0$$.

Find the point where the lines intersect and determine the angle between the lines. $$l_{1}: 3 x+y-5=0 , \quad l_{2}: 7 x-10 y+27=0$$.

Let \(S\) be the set of all rectangles with perimeter \(P .\) Show that the square is the element of \(\mathcal{S}\) with largest area.