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Unformatted text preview: 62 Section 2.4
(b) The slope of the tangent steadily decreases as a increases. 23. Let f (t) lim
h0 17. continued
4 lim
h0 f(2 h) h f(2) lim
h0 (2 4 h) 1 2 lim
h0 [4 h (2 h)] 4h 2 100 4.9t 2.
f(2) f(2 Yes. The slope is 1 . 4 h lim h0 4h 1 4 3 4 3 4 2 2 lim
h0 h) h [100 100 4.9(2 19.6 lim
h0 h) 2] h 19.6h h [100 4.9h 2 4.9(2) 2] 100 19.6 18. No. The function is discontinuous at x because lim
x(3 /4) lim ( 19.6
h0 4.9h) f(x) cos
3 4 lim
x(3 /4) sin x
2 2 sin 19.6 The object is falling at a speed of 19.6 m/sec. but f 19. (a) 3 4 .
[(a a2 2ah h h) 2 2ah h2 2] h h2 h (a 2 2 2) a2 2 lim
h0 f(a h) h f(a) lim
h0 24. Let f (t) lim
h0 3t 2.
h) h f(10) 60h h 3h 2 f(10 300 lim
h0 lim
h0 3(10 h) 2 h 300 lim
h0 h0 lim
h0 300 lim (2a 2a h) lim (60
h0 3h) 60 The rocket's speed is 60 ft/sec. 25. Let f(r) lim
h0 (b) The slope of the tangent steadily increases as a increases. 2 2 20. (a) lim
h0 f(a h) h f(a) lim
h0 a h a r 2, the area of a circle of radius r.
h) h f(3) lim
h0 lim
h0 h 2a 2(a h) ah(a h) 2 a(a h) f(3 lim
h0 (3 9 lim
h0 h0 h)2 h 6 h h (3)2 h2 9 2 a2 lim (6 6 h) (b) The slope of the tangent is always negative. The tangents are very steep near x 0 and nearly horizontal as a moves away from the origin. 21. (a) lim
h0 The area is changing at a rate of 6 in2/in., that is, 6 square inches of area per inch of radius. 26. Let f (r)
4 3 r . 3 f(2 h) f(2) lim h h0 f(a h) h f(a) lim
h0 a 1 h 1 1 a 1 h (a 1) (a h 1) 1)(a h 1) h0 h(a lim lim lim
h0 4 (2 3 h)3 h h)3 h 12h 4 (2)3 3 h0 (a 1 1) 2 1 1)(a h 1) (a (b) The slope of the tangent is always negative. The tangents are very steep near x 1 and nearly horizontal as a moves away from the origin. 22. (a) lim
h0 4 3 4 3 4 3 4 3 lim
h0 (2 8 23 6h 2 + h 3 h 8 lim
h0 lim (12
h0 6h h 2) 12 f(a h) h f(a) lim
h0 [9 9 (a a2 lim
h0 lim
h0 h0 2ah h h)2] (9 h 2ah h 2 h h2 a 2) 9 a2 16 The volume is changing at a rate of 16 in3/in., that is, 16 cubic inches of volume per inch of radius. lim ( 2a 2a h) ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN
 Slope

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