By Edgar Asplund; Lutz Bungart

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**Extra info for A first course in integration**

**Example text**

X) = V T ^ X , g(x) = V x ^ T 16. f{x) = —, g(x) = x2 17. f{x) = y = = > g(x) = *b I»· /(*) = ^ΪΤΪ, 1 9 ^(x) = l2x, x<0^ ( x ) = l5x, gix) = Vx~=l x<0 21. 22. 23. *24. *25. 26. 27. 28. 29. Let/(x) = 2x + 4 a n d g ( x ) = \x - 2. Show that (/° g)(x) = (g°/)(x) = x. For fix) = 3x + 2, find a function g such that (/ ° g)(x) = (g ° /)(x) = x. For/(x) = x2, find two functions g such that (/° g)(x) = x 2 - lOx + 25. For/(x) = ax + fr, find a function g such that (/°g)(x) = fe°/)(x) = x. Assume that a Φ 0.

D) Here the slope is undefined since the line is parallel to the y-axis (the x-coordinates of both points have the same value, 3). The lines are sketched in Figure 5. ■ ► X (-1,6) (2,6) (3,5) m is undefined (3,1) —>~x -►x (c) (d) FIGURE 5 Eight different lines, together with their slopes, are sketched in Figure 6. Definition 2 ANGLE OF INCLINATION If the line L is not parallel to the x-axis, then its angle of inclination Θ is the angle between 0° and 180° that it makes with the positive x-axis.

I \),r = \ 25. (3, - 2 ) ; Y = 4 26. Show that the equation x2 - 6x + y2 + 4y - 12 = 0 is the equation of a circle, and find the circle's center and radius. *27. Show that the equation x2 + ax + y2 + by + c = 0 is the equation of a circle if and only if a2 + b2 - 4 c > 0. *28. Find an equation for the unique circle that contains the points (0, - 2 ) , (6, - 1 2 ) , and (-2, -4). 4 LINES Lines will b e v e r y i m p o r t a n t to u s in the s t u d y of calculus. T w o p o i n t s (xlf yx) a n d (x2/ y 2 ) d e t e r m i n e a line.

### A first course in integration by Edgar Asplund; Lutz Bungart

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