Chapter 1

Express the area of an equilateral triangle as a function of the length of a side.

State whether the function is odd, even, or neither. $$f(x)=\sin 3 x$$.

Evaluate. \(-\frac{7 !}{0 ! 7 !}\).

State whether the function is odd, even, or neither. $$g(x)=\tan x$$.

Show that the sum of two rational numbers is a rational number.

State whether the function is odd, even, or neither. $$f(x)=1+\cos 2 x$$.

Show that the sum of a rational number and an irrational number is irrational.

State whether the function is odd, even, or neither. $$g(x)=\sec x$$.

Show that the product of two rational numbers is a rational number.

State whether the function is odd, even, or neither. $$f(x)=x^{3}+\sin x$$.

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.

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 \(\sqrt{2}\) 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.

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

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

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.

Find the point where the lines intersect and determine the angle between the lines. $$l_{1}: 4 x-y+2=0, \quad l_{2}: 19 x+y=0$$.

Show that if a circle and a square have the same perimeter, then the circle has the larger area. Given that a circle and a rectangle have the same perimeter, which has the larger area?

Find the point where the lines intersect and determine the angle between the lines. $$l_{1}: 5 x-6 y+1=0 , \quad I_{2}: 8 x+5 y+2=0$$.

Theorem (a phony \(0: x\) ): \(1 \quad :2\). PROOF (a phony one): Let \(a\) and \(b\) be real numbers, both different from 0. Suppose now that \(a=b .\) Then \(a b=b^{2}\) \(a b-a^{2}=b^{2}-a^{2}\) \(a(b-a)=(b+a)(b-a)\) \(a=b+a\). since \(a: b,\) we have \(a=2 a\). Division by \(a,\) which by assumption is not \(0,\) gives \(1=2 . \quad \square\) What is wrong with this argument?

The setting for this Exercises is a triangle with sides \(a, b, c\) and opposite angles \(A, B, C\). Show that the area of the triangle is given by the formula \(A=\frac{1}{2} a b \sin C\).

The setting for this Exercises is a triangle with sides \(a, b, c\) and opposite angles \(A, B, C\). Confirm the law of sines: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$. HINT: Drop a perpendicular from one vertex to the opposite side and use the two right triangles formed.

Confirm the law of cosines: $$a^{2}=b^{2}+c^{2}-2 b c \cos A$$. HINT: Drop a perpendicular from angle \(B\) to side \(b\) and use the two right triangles formed.

Verify the following identities: $$\sin \left(\frac{1}{2} \pi-\theta\right)=\cos \theta, \quad \cos \left(\frac{1}{2} \pi-\theta\right)=\sin \theta$$.

Verify that $$\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$$. HINT: \(\sin (\alpha+\beta)=\cos \left[\left(\frac{1}{2} \pi-\alpha\right)-\beta\right]\).

(a) Use a graphing utility to graph the polynomials $$\begin{aligned}&f(x)=x^{4}+2 x^{3}-5 x^{2}-3 x+1,\\\&g(x)=-x^{4}+x^{3}+4 x^{2}-3 x+2.\end{aligned}$$ (b) Based on your graphs in part (a), make a conjecture about the general shape of the graphs of polynomials of degree 4. (c) Test your conjecture by graphing $$f(x)=x^{4}-4 x^{2}+4 x+2 \text { and } g(x)=-x^{4}$$. Conjecture a property shared by the graphs of all polynomials of the form $$P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$$. Make an analogous conjecture for polynomials of the form. $$Q(x)=-x^{4}+a x^{3}+b x^{2}+c x+d$$.

(a) Use a graphing utility to graph the polynomials. $$\begin{aligned}&f(x)=x^{5}-7 x^{3}+6 x+2,\\\&g(x)=-x^{5}+5 x^{3}-3 x-3. \end{aligned}$$ (b) Based on your graphs in part (a), make a conjecture about the general shape of the graph of a polynomial of degree 5. (c) Now graph $$P(x)=x^{3}+a x^{4}+b x^{3}+c x^{2}+d x+c$$ for several choices of \(a, b, c, d, e .\) (For example, try \(a=b=c=d=e=0 .)\) How do these graphs compare with your graph of \(f\) from part (a)?

(a) Use a graphing utility to graph \(f_{A}(x)=A \cos x\) for several values of \(A ;\) use both positive and negative values. Compare your graphs with the graph of \(f(x)=\cos x\). (b) Now graph \(f_{B}(x)=\cos B x\) for several values of \(B\). since the cosine function is even, it is sufficient to use only positive values for \(B\). Use some values between 0 and 1 and some values greater than \(1 .\) Again, compare your graphs with the graph of \(f(x)=\cos x\). (c) Describe the effects that the coefficients \(A\) and \(B\) have on the graph of the cosine function.

$$\text { Lel } f_{n}(x)=x^{n}, n=1.2 .3 \ldots$$ (a) Using a graphing utility, draw the graphs of \(f_{n}\) for \(n=2,4,6\) in one figure, and in another figure draw the graphs of \(f_{n}\) for \(n=1,3,5\). (b) Based on your results in part (a), make a general sketch of the graph of \(f_{n}\) for even \(n\) and for odd \(n\). (c) Given a positive integer \(k,\) conpare the graphs of \(f_{k}\) and \(f_{k+1}\) on [0,1] and on \((1, \infty)\).