Textbook Chapter 1
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34 <strong>Chapter</strong> 1 Prerequisites for Calculus<br />
This line goes through the point (2, 1) when t 0. We determine a and b so that the<br />
line goes through (3, 5) when t 1.<br />
3 2 a ⇒ a 5 x 3 when t 1.<br />
5 1 b ⇒ b 4 y 5 when t 1.<br />
Therefore,<br />
x 2 5t, y 1 4t, 0 t 1<br />
is a parametrization of the line segment with initial point (2, 1) and terminal point (3, 5).<br />
Now try Exercise 23.<br />
Quick Review 1.4<br />
(For help, go to Section 1.1 and Appendix A1.)<br />
In Exercises 1–3, write an equation for the line.<br />
1. the line through the points (1, 8) and (4, 3) y 5 3 x 2 9<br />
<br />
3<br />
2. the horizontal line through the point (3, 4) y 4<br />
3. the vertical line through the point (2, 3) x 2<br />
In Exercises 4–6, find the x- and y-intercepts of the graph of the<br />
x-intercepts: x 3 and x 3<br />
relation.<br />
4. x y-intercepts: y 4 and y <br />
2<br />
y2<br />
x2<br />
1 5. <br />
9 1 6<br />
1 6<br />
y 4<br />
2<br />
9 1<br />
x-intercepts: x 4 and x 4<br />
y-intercepts: None<br />
6. 2y 2 x-intercept: x 1<br />
x 1<br />
1 1<br />
y-intercepts: y and y <br />
2 2<br />
In Exercises 7 and 8, determine whether the given points lie on the<br />
graph of the relation.<br />
7. 2x 2 y y 2 3<br />
(a) (1, 1) Yes (b) (1, 1) No (c) (12, 2) Yes<br />
8. 9x 2 18x 4y 2 27<br />
(a) (1, 3) Yes (b) (1, 3) Yes (c) (1, 3) No<br />
9. Solve for t. t 2x 5<br />
<br />
3<br />
(a) 2x 3t 5 (b) 3y 2t 1 t 3y 1<br />
<br />
2<br />
10. For what values of a is each equation true?<br />
(a) a 2 a (b) a 2 a (c) 4a 2 2a<br />
a 0 All reals All reals<br />
Section 1.4 Exercises<br />
1. Graph (c). Window: [4, 4] by [3, 3], 0 t 2p 9. (b) x 2 y 2 1; upper half (or y 1 ; x 2 all) 11. (b) x 2 y 2 1; upper half (or y 1 ; x 2 all)<br />
In Exercises 1–4, match the parametric equations with their graph. In Exercises 5–22, a parametrization is given for a curve.<br />
State the approximate dimensions of the viewing window. Give a parameter<br />
interval that traces the curve exactly once.<br />
any? Indicate the direction in which the curve is traced.<br />
(a) Graph the curve. What are the initial and terminal points, if<br />
1. x 3 sin(2t), y 1.5 cos t<br />
(b) Find a Cartesian equation for a curve that contains the parametrized<br />
curve. What portion of the graph of the Cartesian<br />
2. x sin 3 t, y cos 3 Graph (a).<br />
t<br />
Window: [2, 2] by [2, 2], 0 t 2p<br />
3. x 7 sin t sin(7t), y 7 cos t cos(7t)<br />
equation is traced by the parametrized curve?<br />
4. x 12 sin t 3 sin(6t), y 12 cos t 3 cos(6t)<br />
5. x 3t, y 9t 2 , t (b) y x 2 ; all<br />
6. x t, y t, t 0 (b) y x 2 ; left half (or x y; all)<br />
(a)<br />
(c)<br />
3. Graph (d). Window: [10, 10] by [10, 10], 0 t 2p<br />
4. Graph (b). Window: [15, 15] by [15, 15], 0 t 2p<br />
(b)<br />
(d)<br />
7. x t, y t, t 0 (b) y x; all (or x y 2 ; upper half)<br />
8. x (sec 2 t) 1, y tan t, p2 t p2 (b) x y 2 ; all<br />
9. x cos t, y sin t, 0 t p<br />
10. x sin (2pt), y cos (2pt), 0 t 1 (b) x 2 y 2 1; all<br />
11. x cos (p t), y sin (p t), 0 t p<br />
x<br />
12. x 4 cos t, y 2 sin t, 0 t 2p (b) 4 <br />
2<br />
<br />
y<br />
2 <br />
2<br />
1; all<br />
13. x 4 sin t, y 2 cos t, 0 t p<br />
14. x 4 sin t, y 5 cos t, 0 t 2p<br />
15. x 2t 5, y 4t 7, t (b) y 2x 3; all<br />
x<br />
13. (b) 4 <br />
2<br />
<br />
y<br />
2 <br />
2<br />
1;<br />
right half (or x 24 ; y 2 all)<br />
x<br />
14. (b) 4 <br />
2<br />
<br />
y<br />
5 <br />
2<br />
1; all