James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Problems
1. One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular
to the hypotenuse as a function of the length of the hypotenuse.
2. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length
of the hypotenuse as a function of the perimeter.
3. Solve the equation | 2x 2 1 | 2 | x 1 5 | − 3.
4. Solve the inequality | x 2 1 | 2 | x 2 3 | > 5.
5. Sketch the graph of the function f sxd − | x 2 2 4| x | 1 3 | .
6. Sketch the graph of the function tsxd − | x 2 2 1 | 2 | x 2 2 4 | .
7. Draw the graph of the equation x 1 | x | − y 1 | y | .
8. Sketch the region in the plane consisting of all points sx, yd such that
| x 2 y | 1 | x | 2 | y | < 2
9. The notation maxha, b, . . .j means the largest of the numbers a, b, . . . . Sketch the graph of
each function.
(a) f sxd − maxhx, 1yxj
(b) f sxd − maxhsin x, cos xj
(c) f sxd − maxhx 2 , 2 1 x, 2 2 xj
10. Sketch the region in the plane defined by each of the following equations or inequalities.
(a) maxhx, 2yj − 1
(b) 21 < maxhx, 2yj < 1
(c) maxhx, y 2 j − 1
11. Evaluate slog 2 3dslog 3 4dslog 4 5d∙ ∙ ∙slog 31 32d.
12. (a) Show that the function f sxd − ln(x 1 sx 2 1 1 ) is an odd function.
(b) Find the inverse function of f.
13. Solve the inequality lnsx 2 2 2x 2 2d < 0.
14. Use indirect reasoning to prove that log 2 5 is an irrational number.
15. A driver sets out on a journey. For the first half of the distance she drives at the leisurely
pace of 30 miyh; she drives the second half at 60 miyh. What is her average speed on
this trip?
16. Is it true that f 8 st 1 hd − f 8 t 1 f 8 h?
17. Prove that if n is a positive integer, then 7 n 2 1 is divisible by 6.
18. Prove that 1 1 3 1 5 1 ∙∙∙ 1 s2n 2 1d − n 2 .
19. If f 0sxd − x 2 and f n11sxd − f 0s f nsxdd for n − 0, 1, 2, . . . , find a formula for f nsxd.
20. (a) If f 0sxd − 1
2 2 x and fn11 − f0 fn
8 for n − 0, 1, 2, . . . , find an expression for fnsxd and
use mathematical induction to prove it.
; (b) Graph f 0, f 1, f 2, f 3 on the same screen and describe the effects of repeated composition.
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