## Practice Problems

Work through the odd-numbered problems 1-31. Once you have completed the problem set, check your answers.

### Problems

1. If T is the Celsius temperature of the air and v is the speed of the wind in kilometers per hour, then

$WCI = \left\{ \begin{array} {lll}T & \text { if } 0 ≤ v ≤ 6.5 \\33 - \dfrac{10.45+5.29\sqrt v - 0.279v}{22} (33-T) & \text { if } 6.5 ≤ v ≤ 72 \\ 1.6T - 19.8 & \text { if } 72 < v \end{array} \right.$

Determine the Wind Chill Index

(a) for a temperature of 0o C and a wind speed of 49 km/hr and
(b) for a temperature of 11o C and a wind speed of 80 km/hr.
(c) Write a multiline function definition for the WCI if the temperature is 11o C.

3. Use the graph of $y = g(x)$ in Fig. 19 to evaluate $g(0)$, $g(1)$, $g(2)$, $g(3)$, $g(4)$ and $g(5)$. Write a multiline function definition for $g$.

5. Use the graphs in Fig. 20 and the equation $h(x) = x - 2$ to
determine the values of

(a) $f( f( 1 ) ), f( g( 2 ) ), f( g( 0 ) ), f( g( 1 ) )$
(b) $g( f( 2 ) ), g( f( 3 ) ), g( g( 0 ) ), g( f( 0 ) )$
(c) $f( h( 3 ) ), f( h( 4 ) ), h( g( 0 ) ), h( g( 1 ) )$

7.

$f(x)=\left\{ \begin{array} {lll}x+1 & \text { if } x < 1 \\1 & \text { if } 1 ≤ x < 3 \\ 2 - x & \text { if } 3 ≤ x \end{array} \right.$

$g(x)=\left\{ \begin{array} {ll}x^2 - 3 & \text { if } x < 0 \\ INT (x) & \text { if } 0 ≤ x \end{array} \right.$

$h(x) = x - 2$

(a) Evaluate $f(x), g(x)$, and $h(x)$ for $x = -1, 0, 1, 2, 3$, and $4$ .
(b) Evaluate $f( g( 1 ) ), f( h( 1 ) ), h( f( 1 ) ), f( f( 2 ) ), g( g( 3.5 ) )$.
(c) Graph $f(x), g(x)$ and $h(x)$ for $-5 ≤ x ≤ 5$.

9. You are planning to take a one week vacation in Europe, and the tour brochure says that Monday and Tuesday will be spent in England, Wednesday in France, Thursday and Friday in Germany, and Saturday and Sunday in Italy. Let $L(d)$ be the location of the tour group on day d and write a multiline function definition for $L(d)$.

11. Write a multiline function definition for the curve $y = f(x)$ in Fig. 22.

13. Define $B(x)$ to be the area of the rectangle whose lower left corner is at the origin and whose upper right corner is at the point $(x, 1/x )$.
a) Evaluate $B(1), B(2)$ and $B(3)$.
b) Show that $B(x) = 1$ for all $x > 0$.

15. Fig. 24 is the graph of $g(x)$.

Graph (a) $g(x) - 1$,

(b) $g( x-1 )$,

(c) $| g(x) |$, and

(d) $[ g(x) ]$.

17. (a) Let $f(x) = 3x + 2$ and $g(x) = 2x + A$. Find a value for $A$ so that $f( g(x) ) = g( f(x) )$.
(b) Let $f(x) = 3x + 2$ and $g(x) = Bx - 1$. Find a value for $B$ so that $f( g(x) ) = g( f(x) )$.

19. Graph $y = f(x) = x - INT(x)$ for $-1 ≤ x ≤ 3$. This function is called the "fractional part of $x$" and is an example of a "sawtooth" graph.

21. Modify the function in example 6 to produce a "square wave" graph with a "long on, short off, long on, short off" pattern.

23. Define $g(x)$ to be the slope of the line tangent to the graph of $y = f(x)$ in Fig. 26 at $(x,y)$.

(a) Estimate $g(1), g(2), g(3)$ and $g(4)$.
(b) Graph $y = g(x)$ for $0 ≤ x ≤ 4$.

25. Pressing the COS (cosine) button on your calculator several times will produce iterates of $f(x) = cos(x)$. What number will the iterates approach if you start with $x = 1$ and press the COS button 20 or 30 times? What happens if you start with $x = 2$ or $x = 10$? (Be sure your calculator is in radian mode).

C27. Starting with $x = 1$, do the iterates of $f(x) = \dfrac{x^2 + 1}{2x}$ approach a number? What happens if you start with $x = .5$ or $x = 4$?

29. Let $f(x) = \dfrac{x}{3} + 4$(a) What are the iterates of f if you start with $x = 2? 4? 6?$

(b) Find a number $c$ so that $f(c) = c$.

(c) Find a fixed point of $g(x) = \dfrac{x}{3} + A$.

Some iterative procedures are geometric rather than numerical.

31. (Optional) Sketch the graph of $p(x)=\left\{\begin{array}{cl}3-x & \text { if } x \text { is a rational number } \\1 & \text { if } x \text { is an irrational number }\end{array}\right.$